# Past Mathematics Conferences

The Department of Mathematics previously hosted conferences and archived conference information is available on this page. Some past conference talks are on the college’s Youtube page.

## 2024 International Fibonacci Conference

### General Information

Harvey Mudd College hosted the 21st International Fibonacci Conference from July 8-12, 2024. The Fibonacci Association holds conferences typically every two years. The Association and Conference focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.

The organizers were Professors Art Benjamin and Steven J. Miller (Williams College). For more information, visit here.

## 2022 Southern California Applied Mathematics Symposium

### General Information

Harvey Mudd College hosted SOCAMS 2022 on Saturday, May 21, 2022. The conference aimed to bring together researchers from universities throughout Southern California, working in all areas of applied mathematics, for a one-day exchange of ideas in an informal and collaborative atmosphere.

The organizers were Professors Jamie Haddock and Heather Zinn-Brooks. For more information, visit socams.org.

## 2011 HMC Mathematics Conference on Broadening Participation in the Mathematical Sciences

### General Information

The Department of Mathematics at Harvey Mudd College hosted the twelfth annual Harvey Mudd College Mathematics Conference on Friday & Saturday, February 4–5, 2011. The theme of the conference was Broadening Participation in the Mathematical Sciences.

The panel was comprised of individuals from industry and from all academic levels (students, postdoctoral scholars, junior faculty, senior faculty) discussing the question:How can one work toward broadening participation in mathematics at all levels of one’s career?

The conference was sponsored by the Department of Mathematics at Harvey Mudd College and the National Science Foundation(NSF).

The organizers were Professors Dagan Karp, Talithia Williams, and Ursula Whitcher.

### Abstracts

#### Judit Camacho: “What Does Mathematics Have to Do withYou,Me, andUS?”

As the United States (US) becomes more diverse, there is an increasing need to have all of our populations included in the scientific enterprise, especially those from historically underrepresented communities. Mathematics, which forms the language of science, is a golden key to accessing all that the scientific world has to offer. You can play a vital role in providing this key to the US, helping everyone in our nation fully participate in society, and rebuilding this nation’s economic infrastructure and science leadership.

During this presentation we will explore what organizations like SACNAS are doing to help individuals and communities from underrepresented backgrounds gain access to science and mathematics education, research, teaching careers, and positions of leadership. We will also share ideas of how you can participate in local, regional, and national efforts to improve participation of all communities in mathematics.

#### Illya Hicks: “Are You Ready For Some Football?!!”

With the occurrence of this conference aligned with Super Bowl weekend, it is befitting to focus on a unique perspective on broadening participation in the mathematical sciences: football. Maybe football can help. Believe it or not, football and mathematics have some similarities. We will explore these similarities and discuss ways that mathematicians can learn from football how to engage more citizens from groups that are currently underrepresented in the discipline. Hence, the focus will be on two things I am passionate about: football and diversity within mathematics.

#### Phillip Kutzko: “Just Walk Away, René—Cultural Issues in Broadening Participation in Mathematics”

Science, as we know it today, developed in a particular time and place for reasons that have never fully been explained. The concept of a function—a concept that underlies all of modern science—first appears in Descartes’La Géometriein 1637; within a generation, Newton and Leibniz had developed the calculus and Newton had laid the foundation for modern physics. Similar transformative advances occurred shortly thereafter in chemistry, biology and medicine. This is the context in which we do science today; a West European, Cartesian context in an increasingly non-European nation.

The Western approach to science embodies certain cultural values, among them skepticism, objectivity, secularism and a belief in progress as an unmixed virtue. These values are by no means universally accepted, either internationally or within our own country. Further, they have sometimes been used to justify aggression and sometimes worse by Europeans and their descendants in the Americas against other ethnic groups and even against groups within European society. This, it would seem, is reason enough for underrepresented minority groups and other Americans who have not historically been invited to the table to steer clear of European science.

Any approach toward broadening participation in science that fails to take into account this cultural context can only go so far. Examples are afforded by standardized testing and affirmative action each of which is ultimately motivated by the same goal: to remove impediments to access caused by overt ethnic and class discrimination (standardized tests) and by the consequences of such discrimination (affirmative action). Both have been valuable in extending access to ethnic and national groups who have found Western science culturally appealing as well as to individuals with similar proclivities from underrepresented groups; indeed, the use of standardized testing transformed the populations doing science during the Sputnik era while affirmative action has been responsible for similar transformations in more recent times. However, these and other strategies that have focused largely on removing barriers to inclusion may be nearing the limit of their utility.

One of the distinctive features of our math department’s initiative to broaden participation in our graduate program is the awareness we have developed of the cultural context in which our effort takes place. I will discuss this cultural context in my talk and argue that an understanding of this context can lead to new strategies, strategies which, in our case, have transformed a traditional mathematics department in an ethnically homogeneous state into what some have called a model for what an American math department should look like in the twenty-first century.

### Speakers & Schedule

#### Friday, February 4

#### 7:00 PM

Robert Bell ’72 (AT&T)

Lessons from the Netflix Prize

#### Saturday, February 5

#### 9:30 AM

Illya Hicks(Rice University)

Are You Ready For Some Football?!!

#### 11:00 AM

Judit Camacho(SACNAS)

What Does Mathematics Have to Do withYou,Me, andUS?

#### 12:15 PM

Lunch

#### 1:30 PM

Poster session

#### 2:30 PM

Phillip Kutzko(University of Iowa)

Just Walk Away, René—Cultural Issues in Broadening Participation in Mathematics

#### 4:00 PM

#### Panel Discussion: “How can one work toward broadening participation in mathematics at all levels of one’s career?”

Candice Price: Graduate Student, University of Iowa

David Uminsky ’03: Postdoc, UCLA

Robin Wilson: Cal Poly Pomona

Ami Radunskaya: Pomona College

Herbert Medina: Loyola Marymount University

Robert Bell ’72: AT&T

## 2010 Melvin Henriksen Memorial Conference

### General Information

TheDepartment of Mathematics at Harvey Mudd College hosted the Melvin Henriksen Memorial Conference in remembrance of the life and work of Melvin Henriksen, Professor Emeritus at Harvey Mudd College.

The conference was sponsored by theDepartment of MathematicsatHarvey Mudd Collegeand theClaremont Center for the Mathematical Sciences (CCMS).

The organizers were Professors Asuman Aksoy, Sandy Grabiner, and Andrew Bernoff.

### About Melvin Henriksen

Melvin Henriksen, professor of mathematics emeritus at Harvey Mudd College, passed away on October 14 in Albuquerque, New Mexico, at the age of 82. A significant portion of his life was spent at Harvey Mudd College where he served as a professor of mathematics from 1969 to 1997. After retiring, he remained an active member of the mathematics community in Claremont and beyond.

Henriksen was well-known in the mathematics community for his work on the study of rings of continuous functions, which involves the interplay of algebra and topology. As a major innovator in a part of topology developed mostly in the second half of the twentieth century, his work on “rings of continuous functions” helped create a new field of mathematics that combines topology with modern algebra.

Henriksen graduated from the City College of New York with a bachelor’s degree in mathematics in 1948. He received his master’s and Ph.D. at the University of Wisconsin in 1948 and 1951. While serving as a faculty member at HMC, Henriksen also served briefly as a visiting professor in the department of mathematics at Wesleyan University. Before coming to Mudd, he taught math at Case Western Reserve University, Purdue University, Wayne State University, University of Alabama and the University of Wisconsin.

Henriksen belonged to the American Mathematical Society, the Mathematical Association of America and the National Association of Mathematics. He also directed 11 Ph.D. dissertations, seven at Purdue University, one at Case Western Reserve University and three at Claremont Graduate University.

### Speakers & Schedule

#### 8:15 AM

Coffee

#### 9:00 AM

Official Welcome by Andrew Bernoff, Harvey Mudd College

#### 9:10 AM

W. Wistar Comfort, Wesleyan University

Remembering Mel Henriksen and (Some of) His Theorems

#### 10:20 AM

Mel’s Life as a Mathematician

#### 11:00 AM

Coffee

#### 11:15 AM

Suzanne Larson, Loyola Marymount University-Los Angeles

SV Spaces: A Case Study in Mel’s Enthusiasm for Mathematics

#### 11:40 AM

Frank Smith, Kent State University

Mel and His Regular Ideals

#### 12:05 PM

Ted C.K. Chinburg, University of Pennsylvania

Mel, Blues Piano, and Automorphisms of Power-Series Rings

#### 12:30 PM

Lunch Break, Aviation Room

#### 1:40 PM

Garth Dales, University of Leeds

Large Fields from Algebras of Continuous Functions—Ramifications of a Seminal Paper of Erdős, Gillman, and Henriksen

#### 2:45 PM

Mathematical Reminiscences

Hank Krieger, Harvey Mudd College

Mel at HMC

Sandy Grabiner, Pomona College

Mel and the Claremont Mathematics Community

Don Johnson, New Mexico State University

Mel in the Mathematical World

#### 3:45 PM

Coffee Break

#### 4:00 PM

R. Grant Woods, University of Manitoba

Mel Henriksen in Winnipeg: 25 Years of Anecdotes and Theorems

#### 4:25 PM

Ralph Kopperman, City College of New York

How Mel Drove Me to Asymmetry

#### 4:50 PM

Joanne Walters-Wayland, OCCTAL,

Chapman University

Framing ‘Mel’

## 2010 HMC Mathematics Conference on the Mathematics of Environmental Sustainability and Green Technology

### General Information

TheDepartment of Mathematics at Harvey Mudd College hosted the eleventh annual Harvey Mudd College Mathematics Conference(previously known as the Mt. Baldy Mathematics Conference). The theme of this year’s conference was The Mathematics of Environmental Sustainability and Green Technology.

On Friday night, there was a panel of researchers from the Claremont Colleges and representatives from environmental groups in Claremont. On Saturday, there were four speakers from a variety of disciplines who posed problems and challenges of interest to the mathematics community. This collection of talks from the perspective of mathematics, engineering, physics, and atmospheric science provided an overview of the cutting edge of research in energy and sustainability.

The conference was sponsored by theDepartment of Mathematics at Harvey Mudd College, the Claremont Center for the Mathematical Sciences(CCMS), and the National Science Foundation(NSF).

The organizer was Rachel Levy.

### Abstracts

#### Harry Atwater: Light–Matter Interactions for Solar Energy Conversion

Solar energy is currently enjoying substantial growth and investment, owing to worldwide sensitivity to energy security and the importance of renewable energy as a means to mitigate carbon emissions.

In this talk I will describe approaches to control of light-matter interactions leading to enhanced absorption in solar photovoltaic structures. Conventionally, it is thought that semiconductor photovoltaic absorbers should have a physical thickness comparable to the “optical thickness” to enable nearly complete light absorption and photocarrier current collection.

Solar cell design and material synthesis considerations are strongly dictated by this simple optical thickness requirement. Dramatically reducing the absorber layer thickness or volume confers several fundamental and practical benefits, including increased open circuit voltage and conversion efficiency, and also expansion of the scope and quality of absorber materials that are suitable for photovoltaic devices by, for example, enabling efficient photocarrier collection across short distances in low dimensional structures. Semiconductor wire array solar cells have a geometry that both facilitates photogenerated carrier collection and enhanced light absorption; results for enhanced optical absorption and carrier collection in Si wire array solar cells will be given, and limits to enhanced absorption will be explored.

To date, little systematic thought has been given to the question of how plasmonic and metamaterial structures might be exploited to advantage in photovoltaics. I will describe design approaches using metallic nanostructures to excite localized and propagating surface plasmons which can dramatically increase the optical path length in thin active photovoltaic layers to enhance overall photoabsorption. Examples of plasmon-enhanced absorption in thin Si, GaAs and InGaN solar cells will be described. Future metamaterial optical design directions for dramatic reduction of solar cell active volume will be outlined.

#### Kenneth M. Golden: “Climate Change and the Mathematics of Transport in Sea Ice”

Sea ice is both an indicator and agent of climate change. It also hosts extensive algal and bacterial communities, which sustain life in the polar oceans. The dramatic decline of the summer Arctic ice pack is perhaps the most visible, large-scale change on Earth’s surface in recent years. Most global climate models, however, have underestimated this decline, while the Antarctic sea-ice pack has increased. We will discuss some key sea ice processes which must be better represented in climate models, such as snow-ice formation and the evolution of melt ponds and ice pack reflectance. Recent mathematical advances in characterizing the porous microstructure of sea ice, and fluid flow through it, shed new light on these processes. Our work will help in predicting and monitoring the impact of global warming on sea ice and the response of polar ecosystems. Video from a 2007 Antarctic expedition where we measured fluid and electrical transport in sea ice will be shown.

#### Julie K. Lundquist: “Harnessing the Power of the Wind”

Wind energy offers the promise of a robust and inexhaustible domestic energy source. Not only do wind turbines provide power with minimal greenhouse gas emissions, but virtually no fresh water is required for power production. Wind energy capacity in the U.S. now produces enough electricity to power the equivalent of approximately 7 million households (American Wind Energy Association, 2009). Despite impressive recent growth, wind energy still constitutes less than 2% of US electricity sources. Several attainable technical challenges must be surmounted if electricity generated by wind is to provide a significant source of domestic power.

This presentation will highlight the mathematical aspects of some of these challenges. The integration of large fractions of fluctuating quantities of renewable energy into power grids required accurate prediction of power availability. These predictions in turn depend on accurate forecasts of wind and atmospheric conditions, nuanced understanding of the impacts of complex terrain on flow and turbulence in the lower atmosphere, and even delineation of the impacts of turbines on each other through turbulent wake effects. As society considers large-scale implementation of wind energy, an assessment of the local environmental impacts of turbines will also be required.

#### Ron Lloyd: “Modeling Problems in the Green Economy”

Any problem that requires new ways of thinking in order to be solved, also requires breakouts in the analytical models used to evaluate potential solutions. A green economy challenges engineers with converting one form of energy—sunlight, wind, waves, heat—into useful forms with reliable and economically viable transformation technologies. This talk will focus on three examples from the green economy that require new analytical models to properly evaluate their technical and economic viability.

Electric vehicles—Conventional internal combustion engines operate at 20–25% efficiencies, with a tremendous amount of waste heat to shed. Battery electric vehicles, while more efficient, have very limited on-board energy storage and most ancillary systems are a load on the vehicle’s battery system. Vehicle energy use models need a major overhaul to support electric vehicle development.

Energy data load and storage—The confluence of Smart Grid, telematics, distributed energy, wireless technologies, and increasing capacities for network traffic and data storage create an explosion of time-sequenced data to be stored and mined in the current and future economies. Predicting storage requirements and system load from data transfer and user mining is an increasingly difficult challenge.

Sustainability—Macroeconomics focuses on large-scale societal systems driving the economy. On a global scale, this presumes that economic health is predicated on growth. Constant growth, if predicated on resource consumption, is inconsistent with most views of “sustainability”—a steady state energy system, with finite resources in the biosphere. Reconciling these two fundamental viewpoints is one of the most important macro problems of 21st century society.

### Speakers & Schedule

#### Friday, January 29

#### 7:00 PM

Opening Panel: Forum on Research and Organizational Efforts in Sustainability and Green Technology

#### Panelists

Dan Davids: HMC Alumnus and President of Plug In America

Jeffrey Byron: Commissioner of the California Energy Commission

Devon Hartman: Co-founder, HartmanBaldwin Design/Build

Freeman Allen: Professor Emeritus of Chemistry, Pomona College, and Representative from Sustainable Claremont

Richard Elderkin: Professor of Mathematics, Pomona College

Richard Haskell: Professor of Physics and Director of the Center for Environmental Studies, Harvey Mudd College

Examples of questions that will be addressed:

- Given a limited amount of resources to invest in sustainability (such as time and money), what are the most important ways to spend them? For example, if you wanted to donate $1000 or $1,000,000? Or if you wanted to volunteer 3 hours per month?
- If you are a student, what are the most important things you can do to prepare yourself for a career in sustainability and green technology?
- What are some of the most exciting cutting edge developments happening now?
- What are the most important behaviors people should engage in to help sustainability efforts (recycling, water conservation, less driving, buy local…)?

#### Saturday, January 30

#### 8:45 AM

Registration

#### 9:15 AM

Welcome and introductory remarks (Galileo Hall)

#### 9:30 AM

Harry Atwater(California Institute of Technology)

Light–Matter Interactions for Solar Energy Conversion.

#### 10:30 AM

Poster session and refreshments

#### 11:00 AM

Ken Golden(University of Utah)

Climate Change and the Mathematics of Transport in Sea Ice.

#### 12:00 PM

Lunch

#### 1:30 PM

Julie Lundquist(University of Colorado at Boulder)

Harnessing the Power of the Wind.

#### 2:30 PM

Poster session and refreshments

#### 3:00 PM

Ron Lloyd(Fat Spaniel Technologies)

Modeling Problems in the Green Economy.

#### 4:00 PM

Closing remarks.

## 2008 HMC Mathematics Conference on Nonlinear Functional Analysis

### General Information

The mathematics department at Harvey Mudd Collegehosted the tenth annual Harvey Mudd College Mathematics Conference(previously known as the Mt. Baldy Mathematics Conference) on Saturday, October 25, 2008.

This year’s topic wasnonlinear functional analysis. Keynote speakers discussed applications of nonlinear functional analysis and future directions for such research.

The conference was sponsored by the mathematics department at Harvey Mudd College, the Claremont Center for the Mathematical Sciences (CCMS), and the National Science Foundation.

The organizer was Alfonso Castro.

### Abstracts

#### Peter Bates: “Mathematical Excursions Inspired by Materials Science”

Equations for a material that can exist stably in one of two homogeneous states are derived from a microscopic or lattice viewpoint with the assumption that the evolution follows a gradient flow of the free energy with respect to some metric. Alternatively, Newtonian dynamics can be considered.The resulting lattice dynamical systems are analyzed, as are equations on the continuum where the lattice interaction energy is viewed as an approximation to a Riemann integral. Several techniques of nonlinear functional analysis are used to examine the well-posedness of the equations, while others give asymptotics or quantitative behavior of special solutions, such as traveling waves or pulses.

#### Robert Borrelli & Courtney Coleman (Harvey Mudd College):

Forty Years of Differential Equations in the Claremont Colleges

Harvey Mudd College graduated its first class in 1961, and in the intervening 47 years the mathematics curriculum at HMC has undergone many changes. In the beginning there was no differential equations course in the mathematics core, little or no use of computers in any mathematics course and no graduate program in mathematics. HMC and the Claremont Colleges have come a long way since then. In this talk we will trace the evolution of differential equations in the HMC mathematics curriculum and cite some milestones along the way.

#### Mónica Clapp: “Classical and Recent Results on Elliptic Problems with Critical Nonlinerarity”

Elliptic equations with critical nonlinearity arise in fundamental questions in Differential Geometry like Yamabe’s problem or the prescribed scalar curvature problem. In this talk we shall consider the model problem

\[ \begin{aligned} -\Delta u=\left\vert u\right\vert ^{2^{\ast}-2}u\text{ in }\Omega,\qquad u=0 \text{ on } \partial\Omega, \end{aligned} \]where $ \Omega $ is a bounded smooth domain in $\mathbb{R}^{N}$ $N\geq3$, and $ 2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent.

Despite its simple form, this problem has been an amazing source of open problems and new ideas. It has a rich geometric structure: It is invariant under the group of Möbius transformations. This fact turns it into an interesting and challenging problem because, though it can be expressed as a variational problem, the usual variational methods cannot be applied to it in a straightforward manner due to the lack of compactness caused by the Möbius invariance.

We shall review some well known results about existence and nonexistence of solutions to this problem, and present some new multiplicity results for positive and sign changing solutions, recently obtained in collaboration with M. Musso, F. Pacella, A. Pistoia, and T. Weth.

#### Yanyan Li: A Geometric Problem and the Hopf Lemma

A classical result of A.D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in $R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1} = \mathrm{constant}$ in case $M$ satisfies: for any two points

$(X’, X_{n+1}), (X’, \widehat X_{n+1})$ on $M$, with $X_{n+1} > \widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, we establish it under some additional conditions. Some variations of the Hopf Lemma will also be presented. Several open problems will be described. These are joint works with Louis Nirenberg.

#### Ratnasingham Shivaji: Population Dynamics with Diffusion

We study the steady state distribution of reaction diffusion equations arising in population dynamics. In particular, we consider strong Allee effect type growth rates and constant yield harvesting in bounded habitats. Assuming the exterior of the habitat is completely hostile, we establish existence and multiplicity results for positive solutions. We obtain our results via the method of sub super solutions.

#### Zhi-Quiang Wang: A Twisting Condition, Resonances, and Periodic Solutions of Hamiltonian Systems

In this talk, we first discuss a bit of background from the Poincaré-Birkhoff theorem to its applications of finding periodic solutions of Hamiltonian systems. The Poincaré-Birkhoff theorem assures the existence of two fixed points for any area preserving homeomorphisms $F$ on a two-dimensional annulus that satisfy a boundary twist condition. This boundary twist condition states that $F$ advances points on the outer edge of the annulus positively and points on the inner edge negatively. Then we discuss analogues of the twisting condition of the Poincaré-Birkhoff theorem in the setting of higher dimensional Hamiltonian systems, which are related to resonances of the Hamiltonian functions near zero and infinity. Finally we present some recent work on the existence of periodic solutions of Hamiltonian systems under a twisting condition which resembles more in spirit of the classical one in the Poincaré-Birkhoff theorem. This talk is intended for undergraduate students.

### Speakers & Schedule

Talks were given on the evening of Friday, October 24, and from 9:00 AM to 5:00 PM on Saturday, October 25, 2008, in theGalileo HallsatHarvey Mudd College.

#### Friday, October 24

#### 7:00 PM

Robert Borrelli & Courtney Coleman (Harvey Mudd College)

Forty Years of Differential Equations in the Claremont Colleges

#### Saturday, October 25

#### 8:30 AM

Registration

#### 9:00 AM

Peter Bates(Michigan State University)

Mathematical Excursions Inspired by Materials Science

#### 10:00 AM

Mónica Clapp (Universidad Nacional Autónoma de México)

Classical and Recent Results on Elliptic Problems with Critical Nonlinearity

#### 11:00 AM

Break and Poster Session

#### 11:30 AM

Yanyan Li(Rutgers University)

A Geometric Problem and the Hopf Lemma

#### 12:30 PM

Lunch and Poster Session

#### 2:00 PM

Ratnasingham Shivaji(Mississippi State University)

Population Dynamics with Diffusion

#### 3:00 PM

Break and Poster Session

#### 3:30 PM

Zhi-Qiang Wang(Utah State University)

A Twisting Condition, Resonances, and Periodic Solutions of Hamiltonian Systems

## 2007 HMC Mathematics Conference on Public Sector Operations Research

### General Information

The mathematics department at Harvey Mudd College hosted the ninth annual HMC Mathematics Conference(previously known as the Mt. Baldy Mathematics Conference) on Friday–Saturday, September 28–29, 2007. The topic this year was public-sector operations research. Keynote speakers discussed applications of operations research models in the public sector and future directions for such research.

The conference was sponsored by the mathematics department at Harvey Mudd College and the National Science Foundation.

The organizer wasSusan Martonosi.

### Abstracts

#### Arnold Barnett(Massachusetts Institute of Technology): Is It Really Safe to Fly?

We discuss several data analyses about the mortality risk of passenger air travel. These analyses offer estimates of actual risk levels, as well as comparative findings about different periods, regions, and airlines. A question of practical importance that we confront is:

Given that two airlines fly a route nonstop, is there ever any reason related to safety to prefer one over the other?

We also consider some questions about air safety that cannot be answered with the data available now. Even here, however, we suggest that operations research reasoning provides some insight about what the answers might be.

**Bio**

Arnold Barnett’s research specialty is applied mathematical modeling with a focus on problems of health and safety. His early work on homicide was presented to President Gerald Ford at the White House, and his analysis of U.S. casualties in Vietnam was the subject of a column by William F. Buckley. He has received the President’s Award and the Expository Writing Award from INFORMS in 1996 and in 2001, and is a Fellow of INFORMS. Barnett has been called the “nation’s leading expert” on aviation safety, and in 2002 received the President’s Citation from the Flight Safety Foundation for “truly outstanding contributions on behalf of safety.” He has written op-ed pieces forThe New York Times, The Wall Street Journal, The Boston Globe,andUSA Today. He has been honored by students at MIT Sloan for outstanding teaching on ten occasions.

#### Margaret L. Brandeau(Stanford University): OR and Public Health: A Little Help Can Go a Long Way

How should the Centers for Disease Control and Prevention revise national immunization recommendations so that gaps in immunization coverage will be filled in a cost-effective manner? What is the most cost effective way to use limited HIV prevention and treatment resources? To what extent should local communities stockpile antibiotics for response to a potential bioterror attack? This talk will describe examples from past and ongoing OR-based analyses of public health policy questions. We provide perspectives on key elements of a successful policy analysis, ways in which such analysis can influence policy, and modeling and policy challenges for the future.

**Bio**

Dr. Brandeau is Professor of Management Science and Engineering. She holds a B.S. in Mathematics and an M.S. in Operations Research from the Massachusetts Institute of Technology, and a Ph.D. in Engineering-Economic Systems from Stanford. She is an operations researcher and policy analyst with extensive background in the development of applied mathematical and economic models, and a distinguished investigator in HIV. Among other awards, Professor Brandeau has received a Presidential Young Investigator Award from the National Science Foundation, the Pierskalla Prize from the Institute for Operations Research and Management Science (INFORMS) for research excellence in health care management science, and the departmental Outstanding Teaching Award. She also holds a patent on a method for operation assignment in printed circuit board assembly.

#### Richard C. Larson(Massachusetts Institute of Technology): Designing and Managing Critical Infrastructures at the Intersection of Engineering/OR, Management and the Social Sciences

Infrastructures comprise society’s lifelines. They provide connected networks facilitating transportation, communication, and daily life needs such as electricity, natural gas, water, and sewage disposal. Infrastructural breakdown implies societal breakdown in some form. And “infrastructure” is not limited to bricks, mortar, steel and fiber-optic cables. A functioning society operates with complex inter-connected social networks that also comprise critical infrastructure. A breakdown in social networks can be just as damaging as a collapsed bridge, a massive blackout or a broken fiber optic cable.

The design and operation of infrastructural systems requires deep knowledge on many fronts. Being society’s lifelines, with many infrastructural systems incorporating all the complex and oftentimes beautiful complexities of human behavior, narrow purely technocratic approaches to the analysis of infrastructural systems are usually naively inadequate. Solutions almost always need to be created from multiple disciplines. At MIT we have drawn together three required disciplines under a new umbrella called “Engineering Systems.” This new field operates at the Venn diagram intersection of the fields of traditional engineering/OR, management and social sciences. Simultaneous attention to all three circles of the Venn diagram can pay off handsomely in infrastructural system design and operation.

In this presentation, we offer examples of MIT ESD infrastructure system research within the context of the Venn diagram overlap. We attempt to demonstrate that improved systems will emerge from an engineering and OR design approach that considers management and social science issues at the outset. These issues may be as important as Newton’s laws of physics or Faraday’s laws of electricity or even Little’s Law for queues! They are not “side constraints”.

Our first two examples deal with demand management for critical infrastructures. Here we are speaking of congested infrastructural systems in which we want to shave peak time demands and fill in the valleys of demand. Examples are drawn from electric power distribution—using demand-dependent pricing, and from urban transportation—in which we discuss road-use and parking-use pricing schemes. In each case, there are rich complex narrowly technocratic engineering and OR issues to work out. But equally if not more important are human issues relating to how users of these systems will respond to alternative proposed system implementations. And the management of such systems is no small matter.

Our third example is drawn from disaster preparedness and response. Natural disasters, human caused accidental disasters, and terrorist acts can create havoc, damaging physical as well as social infrastructures. While we have studied a variety of disasters, our focus here is on hurricanes (also called typhoons and cyclones). The idea is that one obtains from aircraft and satellites updated information at given intervals on the state of the hurricane—the location of its center (the “eye”), its maximum wind speed, its movement vector, etc. As it approaches landfall, disaster management decision makers have a sequence of action timing decisions to make, related to mobilization of resources, positioning of supplies and equipment, and evacuation of coastal residents. The technical aspect of this decision problem under uncertainty can be captured as a complex stochastic dynamic programming problem. But there is so much more, not the least of which is the propensity of the population to follow evacuation orders. Here, previous highly publicized false alarms may give rise to a population-wide condition known as the “Boy-Who-Cries-Wolf syndrome.” This is a real first-order “social science” effect that needs to be incorporated into the mathematical planning analysis.

A fourth example is drawn from another type of disaster, what we call a Low Probability, High Consequence (LPHC) event. We select Pandemic Influenza, a condition that affects the world about three times per century, with one of those times being as terrible as the 1918–1919 great influenza, often misnamed the Spanish Influenza. For the most serious of these pandemics, the low probability is about 1 chance in 100 on an annual basis. The high consequence is that hundreds of millions of people worldwide could die from the infection. The death toll could be greater than a full nuclear exchange between nuclear-armed states. The engineering/OR challenge is to design a “’flu preparedness and response system” that minimizes illnesses and deaths. But at the heart of this system is human behavior: Will citizens of a country adhere to strongly recommended protocols related to “social distancing” and hygienic procedures? A major flu pandemic can have devastating effects on traditional physical infrastructure systems, as needs for maintenance and repair may go unanswered due to workers being sick or staying at home to care for sick loved ones and/or to isolate themselves from others who are contagious.

A fifth and final example deals with technology-enabled, networked distance learning in developing countries. We use as a case study an MIT-created volunteer effort, LINC, Learning International Networks Consortium,http://linc.mit.edu. The underlying principle of LINC is this:With today’s computer and telecommunications networked technologies, every young person can have a quality education regardless of his or her place of birth.The design and implementation of an e-learning system in a developing country has many of the quantitative optimization problems usually associated with OR, but it also has myriad issues related to management and social science.

Following the cases, we attempt to sum up lessons learned and to characterize how to undertake engineering systems activity in the Venn diagram intersection. We also discuss if we should collectively allow OR to be stereotyped as the “quant” part of the problem or if, like Philip M. Morse and other OR co-founders in the 1940’s and ’50’s, we wish to go back to the roots of OR and greatly broaden its definition and role to include all in what many are now labeling “engineering systems.”

**Bio**

Dr. Larson is the Mitsui Professor of Engineering Systems in the department of Civil and Environmental Engineering and the founding director of the Center for Engineering Systems Fundamentals at the Massachusetts Institute of Technology. He received his Ph.D. from MIT.

The majority of his career has focused on operations research as applied to services industries. He is author, co-author or editor of six books and author of over 75 scientific articles, primarily in the fields of technology-enabled education, urban service systems (esp. emergency response systems), queueing, logistics and workforce planning. His first book,Urban Police Patrol Analysis(MIT Press, 1972) was awarded theLanchester Prizeof the Operations Research Society of America (ORSA). He is co-author, with Amedeo Odoni, ofUrban Operations Research, Prentice Hall, 1981.

He served as President of ORSA, (1993-4), and is Past-President ofINFORMS, INstitute for Operations Research and the Management Sciences. He has served as consultant to the World Bank, the United Nations, Johnson Controls, EDS, United Artists Cinemas, Union Carbide Corp., Rand Corp., the Kuwait Foundation for the Advancement of Science, Predictive Networks, WebCT, Hibernia College in Ireland, Hong Kong University and the U.S. Department of Justice. With outside companies on which he serves as board member, most recentlyStructured Decisions Corporation, Dr. Larson has undertaken major projects with Citibank, American Airlines, Actmedia/Turner Broadcasting, the U.S. Postal Service, the City of New York, Jenny Craig, Conagra, Diebold, BOC and other firms and organizations. Dr. Larson’s research on queues has not only resulted in new computational techniques (e.g., the Queue Inference Engine and the Hypercube Queueing Model), but has also been covered extensively in national media (e.g., ABC TV’s20/20). Dr. Larson has served as Co-Director of theMIT Operations Research Center(over 15 years in that post). Dr. Larson was first listed in Who’s Who in America in 1982. He is a member of the National Academy of Engineering and is an INFORMS Founding Fellow. He has been honored with the INFORMS President’s Award and the Kimball Medal.

From 1995 to mid 2003, Dr. Larson served as Director of MIT’s CAES, Center for Advanced Educational Services. Dr. Larson’s position at CAES focused on bringing technology-enabled learning to students living on the traditional campus and to those living and working far from the university, perhaps on different continents. During the years 1995–1999 he built the center from two to seven business units, encompassing MIT’s production and R&D capabilities in educational technologies and its two major lifelong learning academic programs. His center produced the world’s most ambitious point-to-point distance learning program, theSingapore MIT Alliance. He has been invited to give lectures on the future of technology-enabled education in testimony before the House Committee on Science (Washington, D.C.) and in North and South America, Asia, Africa and Europe. He has served as Principal Investigator of several of MIT’s most ambitious technology-enabled learning programs, includingPIVoT—the web-basedthe Physics Interactive Video Tutor,Masters’ Voices(sponsored by the Ford Motor Company),MIT World, “Inventing the Global Classroom,” “Good Clinical Practices” and “Fungal Infections” (the last two sponsored by the Pfizer Corporation). He is Founding Director ofLINC, Learning International Networks Consortium, an MIT-based international project that has just held its third international symposium. Dr. Larson also served as founding co-director of theForum the Internet and the University, a not-for-profit organization affiliated with the Forum for the Future of Higher Education.

#### Eva K. Lee(Georgia Institute of Technology): Modeling and Optimizing the Public Health Infrastructure for Emergency Response to Bioterror Attacks and Infectious Disease Outbreaks

City readiness and emergency response for biodefense and infectious disease outbreaks involves multilevel strategic and operational planning of the public health infrastructure. We provide an overview of infrastructure networks and highlight some challenges in emergency preparedness and response. We then describe our experience with the Centers for Disease Control and Prevention, and with state and county public health and emergency response administrators on large-scale dispensing of prophylactic medication in response to biological and infectious disease outbreaks. Strategic stockpile, medical supply distribution, locations of dispensing facilities, optimal facility staffing and resource allocation, routing of the population, and various logistics, transportation and dispensing modalities will be discussed and analyzed.

**Bio**

Dr. Lee is an Associate Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology, and Director of the Center for Operations Research in Medicine and HealthCare. She is also a Senior Research Professor at the Atlanta VA Medical Center. Dr. Lee earned a Ph.D. at Rice University in the Department of Computational and Applied Mathematics, and received her undergraduate degree in Mathematics from Hong Kong Baptist University, where she graduated with Highest Distinction. Dr. Lee was awarded a NSF/NATO postdoctoral fellowship and a postdoctoral fellowship from Konrad-Zuse-Zentrum Informationstechnik Berlin in 1995 for Scientific Computing. In 1996, she received the NSF Presidential Young Investigator Award for research on integer programming and parallel algorithms and their applications to medical diagnosis and cancer treatment. She was the first IE/OR recipient for the prestigious Whitaker Foundation Biomedical Grant for Young Investigators, awarded for her work on a novel approach for combining biological imaging and optimal treatment design for prostate cancer. In 2004, she was selected as one of the Extraordinary Women Engineers. In 2005, she received the INFORMS Pierskalla Best Paper Award for research excellence in Health Care and Management Science for her work on emergency response and planning, large-scale prophylaxis dispensing, and resource allocation for bioterrorism and infectious disease outbreaks. In 2006, she was chosen by the American Mathematical Society as the mathematician ambassador to speak and discuss individually to congressional leaders about her research advances in medical and healthcare domain, and the importance of mathematics in scientific advances. In 2007, she was inducted as the Franz Edelman Award Laureate, and together with Dr. Marco Zaider at Memorial Sloan Kettering Cancer Center, their work on “Operations Research Advances Cancer Therapeutics” was selected as the winner of the 2007 Franz Edelman award. She has received seven patents on innovative medical decision systems and devices.

Lee works in the area of mathematical programming and large-scale computational algorithms with a primary emphasis on medical/healthcare decision analysis and logistics operations management. In medicine and healthcare, she has developed real-time computer decision-support systems to help analyze large-scale biological, DNA/genomic and clinical data to assist in health prediction, early disease prediction and diagnosis, optimal treatment design and drug delivery, treatment outcome analysis and prediction, and efficient and cost-effective healthcare delivery and operations logistics and processes.

In logistics, Lee’s research focuses on optimization and algorithmic advances for optimal operations planning and resource allocation. She has developed decision support systems for inventory control, large-scale truck dispatching, scheduling, and transportation logistics, telecommunications, portfolio investment, and emergency treatment responses and facility layout and planning.

#### Stephanie Snyder(USDA Forest Service): Applying Operations Research Tools to Important Public Sector Problems in Forestry and Habitat Conservation

Public land managers face a host of complex resource allocation, project selection, scheduling and routing problems. Decisions about the timing and location of forest management activities, fire crew and equipment allocation and deployment, and land acquisition to create protected habitat reserves are just a few of the problems which can and are increasingly being addressed through the use of optimization decision models and operations research techniques. I’ll provide an overview of some of the major forestry and habitat conservation problems which have been addressed with operations research techniques as well as discuss emerging and unsolved areas of research in these fields where OR can make a difference.

**Bio**

Dr. Snyder is an Operations Research Analyst with the US Forest Service, Northern Research Station in St. Paul, MN. She held previous positions with the Minnesota Department of Transportation as an analyst and with the US Army Corps of Engineers’ Institute for Water Resources as a policy analyst. She has a BA in physical geography from Boston University and an MA and PhD in applied operations research from the Department of Geography and Environmental Engineering at Johns Hopkins University. Her research focuses primarily on the development of discrete optimization decision models applied to practical, public-sector problems in timber harvest scheduling and habitat reserve design. She is an active member of INFORMS and has served on the editorial board ofForest Science.

### Speakers & Schedule

#### Friday, September 28

#### 7:00 PM

Arnold Barnett(Massachusetts Institute of Technology)

Is It Really Safe to Fly?

#### Saturday, October 23

#### 8:30 AM

Registration

#### 9:00 AM

Welcome and introductory remarks (Galileo Hall)

#### 9:15 AM

Margaret L. Brandeau(Stanford University)

OR and Public Health: A Little Help Can Go a Long Way

#### 10:15 AM

Poster session and refreshments

#### 10:45 AM

Richard C. Larson(Massachusetts Institute of Technology)

Designing and Managing Critical Infrastructures at the Intersection of Engineering/OR, Management and the Social Sciences

#### 11:45 AM

Lunch

#### 1:30 PM

Eva K. Lee(Georgia Institute of Technology)

Modeling and Optimizing the Public Health Infrastructure for Emergency Response to Bioterror Attacks and Infectious Disease Outbreaks

#### 2:30 PM

Poster session and refreshments

#### 3:00 PM

Stephanie Snyder(USDA Forest Service)

Applying Operations Research Tools to Important Public Sector Problems in Forestry and Habitat Conservation

#### 4:00 PM

Open Discussion: Future Directions in Public Sector O.R.

### Advertising Photos

## 2006 HMC Mathematics Conference on Enumerative Combinatorics

### General Information

The Department of Mathematics at Harvey Mudd College hosted the eighth annual Harvey Mudd College Mathematics Conference (previously known as the Mt. Baldy Mathematics Conference) on Saturday, October 7, 2006. The topic this year was enumerative combinatorics.

Keynote speakers discussed new developments and applications of enumerative combinatorics. Speakers made the first portion of their talks accessible to a general mathematical audience so we encouraged anyone who wished to learn more about research in this area to attend.

The conference was sponsored by the Department of Mathematics at Harvey Mudd College with support from the National Science Foundation. The organizer wasArthur Benjamin.

### Abstracts

#### George E. Andrews(Pennsylvania State University): Partition Analysis and the Search for Modular Forms

This talk will introduce P.A. MacMahon’s method of partition analysis. We will the discuss some of its applications including our most recent discovery of certain partitions related to directed graphs that have modular forms as generating functions and consequently have a variety of interesting related arithmetical problems and prospects. This work is joint with Peter Paule and Axel Riese.

#### Ken Ono(University of Wisconsin): Freeman Dyson’s Challenge for the Future: The Story of Ramanujan’s Mock Theta Functions

The legend of Ramanujan is one of the most romantic stories in the modern history of mathematics. It is the story of an untrained mathematician, from south India, who brilliantly discovers tantalizing examples of phenomena well before their time. Indeed, the legacy of Ramanujan’s work (as a whole) is well documented and includes direct connections to some of the deepest results in modern number theory such as the proof of the Weil Conjectures, and the proof of Fermat’s Last Theorem. However, one final problem remained. In his last letter to Hardy (written on his death bed), Ramanujan gave examples of 17 functions he referred to as “mock theta functions”. Without a definition and without good clues, number theorists were unable to make any real sense out of these peculiar functions. Nevertheless, these examples make important appearances in many disparate areas of mathematics, a fact which inspired Freeman Dyson to proclaim:

“Mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure… This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions.”

—Freeman Dyson, 1987

In this lecture I will outline the history of Ramanujan’s final enigma, and I will announce the solution.

#### Igor Pak(MIT): The Nature of Partition Bijections

The study of partition identities has a long history going back to Euler, with applications ranging from analysis to number theory, from enumerative combinatorics to probability. Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find “hidden symmetries”, etc.

In the talk I will present a number of partition bijections and discuss their applications. The talk assumes no background whatsoever, and hopefully will be somewhat entertaining.

#### Igor Pak(MIT): MacMahon’s Master Theorem

MacMahon’s Master Theorem is a classical combinatorial result celebrated by its applications to binomial identities. In this talk I will give an introduction to the subject, present an algebraic and a direct bijective proof of the theorem. If time permits, I will also discuss various extensions of the Master Theorem.

#### Carla D. Savage(North Carolina State University): The Binomial Essence of Lecture Hall Partitions

Alecture hall partitionis a sequencex1,x2, … ,xnof nonnegative integers satisfyingx1/n ≥x2/(n-1) ≥ … ≥xn/1. Lecture hall partitions were introduced in 1997 by Bousquet-Mélou and Eriksson who showed that they are in one-to-one correspondence with partitions into odd parts less than 2n. Since then, several generalizations and refinements of this result have been discovered. In this talk we view lecture hall partitions from three different perspectives (integer-analogs,q-series identities, and Sylvester’s bijection) to uncover some new connections.

#### Doron Zeilberger(Rutgers University): Beautiful and Insightful Computer-Generated Bijective Proofs

Computer-Generated proofs do not have to be artificial, and many human-made proofs are not that natural. It is possible to take “artificial” proofs and naturalize them, and we can even cheat and pretend that we did it all by ourselves. (Joint work with Philip Matchett Wood.)

### Speakers & Schedule

#### Friday, October 6

#### 1:15 PM

Igor Pak(MIT)

The Nature of Partition Bijections

#### 4:15 PM

Ken Ono(University of Wisconsin)

Freeman Dyson’s Challenge for the Future: The Story of Ramanujan’s Mock Theta Functions

#### Saturday, October 7

#### 8:30 AM

Registration

#### 9:00 AM

Welcome and introductory remarks (Galileo Hall)

#### 9:15 AM

George E. Andrews(Pennsylvania State University)

Partition Analysis and the Search for Modular Forms

#### 10:15 AM

Poster session and refreshments

#### 11:00 AM

Igor Pak(MIT)

MacMahon’s Master Theorem

#### 12:00 PM

Lunch

#### 2:00 PM

Carla D. Savage(North Carolina State University)

The Binomial Essence of Lecture Hall Partitions

#### 3:00 PM

Poster session and refreshments

#### 3:45 PM

Doron Zeilberger(Rutgers University)

Beautiful and Insightful Computer-Generated Bijective Proofs

## 2005 HMC Mathematics Conference on Scientific Computing

### General Information

The Department of Mathematics at Harvey Mudd College hosted the seventh annual Harvey Mudd College Mathematics Conference (previously known as the Mt. Baldy Mathematics Conference) on Saturday, November 5, 2005. The topic was scientific computing. Keynote speakers discussed new developments in numerical analysis and techniques and how they have led to novel applications.

The conference was sponsored by the Department of Mathematics at Harvey Mudd College with support from theNational Science Foundation and in cooperation with the Society for Industrial and Applied Mathematics (SIAM). The organizer was Darryl Yong.

### Abstracts

#### Randy LeVeque(University of Washington, Seattle): Finite-Volume Methods and Software for Hyperbolic PDEs and Conservation Laws

Hyperbolic systems of partial differential equations frequently arise when modeling phenomena involving wave propagation or advective flow. Finite-volume methods are a natural approach for conservation laws since they are based directly on integral formulations and are applicable to problems involving shock waves and other discontinuities. High-resolution shock-capturing methods developed originally for compressible gas dynamics can also be applied to other hyperbolic systems, even if not in conservation form. I will describe a robust class of wave-propagation methods that have been implemented in the CLAWPACK (Conservation LAW PACKage) software for solving hyperbolic problems in one, two, and three space dimensions. Adaptive mesh refinement capabilities are also included. This software has been applied to a variety of problems in diverse fields, including gas dynamics, multiphase flows, linear and nonlinear elasticity, combustion, biological flows, and numerical relativity. Some examples will be shown from recent work on geophysical flows modeling volcanoes and tsunamis.

#### Adrian Lew(Stanford University): Modeling and Simulation of Highly Deformable Materials

How is the design of a car tire optimized? How can we learn about the relation between the structure of muscle and its function, in healthy and diseased tissue? Will a barrier or container withstand an explosive impact? The assortment and intricacy of behaviors observed in these examples is originated in strong nonlinearities present in each problem. Complex material response capable of undergoing important shape changes is underlying all these questions, and numerical methods to capture them are the subjects of this talk.

I will begin by presenting a class of time-integration algorithms termed Asynchronous Variational Integrators (AVI), designed to accelerate the simulation of multi-physics problems with multiple time scales, as well as their very recent parallel versions (PAVI). The algorithms have a beautiful geometric origin which endows them with a number of desirable properties, to be discussed. I will later comment very briefly about some general aspects of nonlinear elasticity, and show recent developments in my group in the creation of discontinuous Galerkin methods for its simulation. I will demonstrate the performance of the new methods through selected examples.

#### Linda Petzold(University of California, Santa Barbara): Bridging the Scales in Biochemical Simulation

In microscopic systems formed by living cells, the small numbers of reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. An analysis tool that respects these dynamical characteristics is the stochastic simulation algorithm (SSA), a numerical simulation procedure that is essentially exact for chemical systems that are spatially homogeneous or well stirred. Despite recent improvements, as a procedure that simulates every reaction event, the SSA is necessarily inefficient for most realistic problems. There are two main reasons for this, both arising from the multiscale nature of the underlying problem: (1) stiffness, i.e. the presence of multiple timescales, the fastest of which are stable; and (2) the need to include in the simulation both species that are present in relatively small quantities and should be modeled by a discrete stochastic process, and species that are present in larger quantities and are more efficiently modeled by a deterministic differential equation (or at some scale in between). We will describe several recently developed techniques for multiscale simulation of biochemical systems, and outline some of the technical challenges that still need to be addressed.

#### Jane Wang(Cornell University): Computational Modeling of Insect Flight

Most living species, include protozoa, bacteria, insects, birds, and fish, locomote via the interactions between fluids and moving surfaces. Whether using cilia, flagella, wings, or fins, the locomotion is governed by the Navier-Stokes equation coupled to moving boundaries. The computational methods I will describe are two examples of solving this class of problems. The first method is designed to solve a single flapping wing using a conformal mesh to resolve the wing tip, and the second method is an immersed interface method to simulate multiple wing interactions. I will discuss some of our recent results on the efficiency of flapping flight, the trajectories of a piece of falling paper, and the fore-hind wing interactions in dragonfly flight.

### Speakers & Schedule

#### 8:30 AM

Registration

#### 9:00 AM

Welcome and Introductory Remarks (Galileo Hall)

#### 9:15 AM

Randy LeVeque(University of Washington, Seattle)

Finite-Volume Methods and Software for Hyperbolic PDEs and Conservation Laws (slides)

#### 10:15 AM

Poster Session and Refreshments

#### 11:00 AM

Adrian Lew(Stanford University)

Modeling and Simulation of Highly Deformable Materials

#### 12:00 PM

Lunch at Hoch-Shanahan Dining Hall (Upper West Dining Area)

#### 2:00 PM

Jane Wang(Cornell University)

Computational Modeling of Insect Flight

#### 3:00 PM

Poster session and refreshments

#### 3:45 PM

Linda Petzold(University of California, Santa Barbara)

Bridging the Scales in Biochemical Simulation

## 2004 Mt. Baldy Conference on Geometry, Algebra, and Phylogenetic Trees

### General Information

The Department of Mathematics at Harvey Mudd College hosted the sixth annual Mt. Baldy Mathematics Conference on Saturday, October 23, 2004. The topic was “Geometry, Algebra, and Phylogenetic Trees”. Keynote speakers discussed mathematical methods (e.g., from geometry and algebra) for the problem of reconstructing phylogenetic trees (e.g., a “tree of life” representing relationships between species) from data (e.g., genomic data).

The conference was sponsored by the Mathematics Department at Harvey Mudd College. The organizer was Francis Edward Su.

### Abstracts

#### Bernd Sturmfels(UC-Berkeley): Phylogenetic Algebraic Geometry

Many widely used statistical models of evolution are algebraic varieties, that is, solutions sets of polynomial equations. We discuss this algebraic representation and its implications for the construction of maximum likelihood trees. The ensuing interaction between algebraic geometry and phylogenetics is a two-way street: computational biologists may be benefit from new algebraic tools, while algebraic geometers will find a rich source of open problems concerning objects reminiscent of classical projective varieties.

#### Michael Hendy(Massey University, New Zealand): Hadamard Conjugation and the Molecular Clock

Molecular phylogenetics is the art of inferring evolutionary trees (phylogenies) from comparative biological sequence data. As a mathematician, the challenge is to make a science of this art. Instances of that art include the reconstruction algorithms of Maximum Parsimony (MP) and Maximum Likelihood (ML).

A phylogeny has a time dimension, the molecular clock hypothesis asserts that there is a common rate of substitutional change along branches of the tree. It is common to ignore this hypothesis with MP and ML.

I will show how Hadamard conjugation was developed as a tool to analyse the accuracy of MP under a molecular clock, and how it has been useful for other applications, including the theory of invariants and in analysing ML.

#### David Bryant(McGill University, Canada): Probabilistic Models for Splits Graphs

Splits graphs are generalisations of trees that permit the representation of conflict and ambiguity. One of the biggest challenges in splits graphs is their interpretation. If we want to go beyond splits graphs as a simple means of representation (and I believe we should), we need some way of modelling evolution in terms of a splits graph. I will discuss recent progress in this area, and outline connections with graphical models and the Hadamard transform.

#### Susan Holmes(Stanford): Using Phylogenetic Trees to Analyse Microarrays

Microarrays provide data on what genes are transcribed at a given time and we follow cell development and differentiation by looking at the gene expression patterns. We will present cases when hierarchical clustering and phylogenetic trees can be used to anlalyse gene expression data and show how using the bootstrap and distances between trees we can capture information on which genes or groups of genes are responsible for parts of the tree.

This will be illustrated on data from a study on T-cells done in collaboration with Peter Lee from Stanford’s School of Medicine.

### Poster Session Abstracts

#### Nicholas Eriksson (UC-Berkeley): Reconstruction of Phylogenetic Trees with Algebraic Invariants

We present a new method of deducing phylogenetic trees using algebraic invariants and numerical linear algebra. Unlike distance based methods, our method utilizes all information in an alignment. Simulations studies show that our algorithm is promising. It can currently handle at least 20 taxa trees and work is under way to improve it.

The algebraic invariants of a Markov model on a treeTare the algebraic relations between the joint probabilities of the observations at the leaves ofT. For any Markov model, some relations are given by rank conditions on matrices built by flattening the probability distribution along a split in the tree. Although there are exponentially many coordinates to the distribution and exponentially many splits, we provide an algorithm which is polynomial time in the number of taxa for fixed sequence length.

In order to reconstruct phylogenies using algebraic invariants, we start with a multiple alignment of the $n$ sequences. From this, we build the observed distribution. Then the matrices corresponding to splits can be tested to see if they satisfy the invariants. The singular value decomposition (SVD) is used to measure how far a matrix is from having the desired rank. Although these matrices are of exponential size, their SVD can be computed effectively since they are typically very sparse.

Our algorithm looks at alln-choose-2 possible cherries of the tree and picks the one that gives the best score using the SVD. The two taxa with the best score are collapsed and the algorithm continues recursively until a tree is built.

#### Daniel Ford (Stanford): Probability Measures on Phylogenetic Trees

The Yule, Uniform and Comb are common probability distributions on phylogenetic trees/cladograms. They all have the property that choosing a random tree withkleaves and deleting a random leaf gives a sample from the distribution on trees withk-1 leaves. In practice evolutionary trees are longer than Yule tree but flatter than Uniform trees.

Fortunately there is a parametrised family of probabilities which smoothly interpolates between these three families. This is easy to construct and perform statistical tests with.

#### Akemi Kashiwada (Harvey Mudd College): The Shapley Value of Phylogenetic Trees

(joint work with Claus-Jochen Haake and Francis E. Su)

Phylogenetic trees represent theoretical evolutionary relationships among various species. Mathematically they can be described as weighted binary trees and the leaves represent the taxa being compared and the weight of edge is some notion of distance between its endpoints. If we think of phylogenetic trees as cooperative games where the worth of a coalition is the weight of the subtree spanned by its members, then we can use the Shapley value, an important game theoretic concept, to solve this game. We prove that the Shapley value of tree games is characterized by five axioms, and discuss what these mean in a biological context. We also determine the linear transformation M that gives the Shapley value in terms of the edge weights of the tree, and show how the Shapley value of an-leaf tree game can be reconstructed from the Shapley value of its (n–1 )-leaf subtrees. We determine the null spaces ofM, which depend on the topology of the tree.

#### Dan Levy (UC-Berkeley): Reconstructing Phylogenetic Trees from Dissimilarity Maps

(joint work with Francis E. Su, Ruriko Yoshida)

Algorithms for reconstructing a phylogenetic tree from sequence data typically rely on first finding a distance matrix: a set of values indicating the dissimilarity for each pair of elements. However, one may choose to consider instead the dissimilarity of groups of elements. We may define an m-dissimilarity map as a functionDthat takes subsets ofm-elements to a non-negative real number. This may be thought of as a measure of how dissimilar a set ofmelements are or, in terms of the tree, as the sum of the edge weights of the subtree spanned by those leaves.

Pachter and Speyer (2004) gave theoretical conditions under which a tree may be constructed from its dissimilarity map; in particular a tree may be reconstructed from its m-dissimilarity map ifn> 2m-2 . Their method proceeds by first finding the splits, recovering the topology from the splits using Buneman indices, and then using the distances and topology to determine the edge weights. In contrast, our method constructs the topology of the tree and determines the edge weights simultaneously.

We achieve this result through an analogue of the four point condition for m-dissimilarity maps. We say that a pair of leaves (i,j) is a sub-cherry in the subtreeTiffiandjare inTand there is only one vertex of degree 3 on the unique path fromitojinT. We call this intermediate vertex the sub-cherry node. Our four point condition form-dissimilarity maps locates sub-cherries in subtrees withm+ 2 leaves and determines the distance from the sub-cherry node to the rest of the subtree. By locating sub-cherries and determining sub-cherry node distances, we may reconstruct both the tree and the edge weights provided thatn> 2m-2.

In the case thatm= 3, we have exploited certain symmetries to provide a fast algorithm for reconstructing phylogenetic trees from 3-dissimilarity maps. This algorithm has a time complexity of O(n^2). We have also written and tested a C++ implementation of this algorithm.

Phylogenetic tree reconstruction fromm-dissimilarity maps may provide a more accurate approach to determining the maximum likelihood tree by utilizing the more accuratem-subtree weights as opposed to pairwise distances. As demonstrated by the casem= 3, these algorithms are computationally competitive with distance based methods and may serve as a viable alternative to neighbor joining or quartet reconstructions.

#### NickD. Pattengale (New Mexico): Algorithmic Decomposition of Most Parsimonious Sets

(joint work with Bernard M.E. Moret)

Parsimony search typically finds several most parsimonious trees. Our goal is take a set of most parsimonious trees as input and algorithmically identify strongly supported evolutionary hypotheses across the set. We do not require that algorithms produce a single tree, thereby overcoming some of the limitations of consensus methods. Additionally, in contrast to most consensus methods, our method is sensitive as to how edge weights differ across the input set. Our approach has thus far yielded an interesting tree decomposition that in turn gives rise to a functional mapping between trees. We expect that the functional mapping will prove useful as a bounding technique during parsimon search itself. We also consider generalizations of the technique in which parsimony scores across the input set are not necessarily homogeneous, but are within a constant bound from most parsimonious.

#### Raazesh Sainudiin (Cornell): Enclosing the Maximum Likelihood of the Simplest DNA Model Evolving on Fixed Topologies: Towards a Rigorous Framework for Phylogenetic Inference

An interval extension of Felsenstein’s recursive formulation for the likelihood function (Felsenstein, 1981) of the simplest Markov model of DNA evolution (Jukes and Cantor, 1969) on unrooted phylogenetic trees with a fixed topology is used to obtain rigorous enclosure(s) of the global maximum likelihood. Validated global maximizer(s) inside any compact set of the parameter space which is the set of all branch lengths of the tree are thus obtained. The algorithm is an adaptation of a widely applied global optimization method using interval analysis (Hansen, 1980) for the phylogenetic context. The method is applied to enclose the maximizer(s) and the global maximum for the simplest DNA model evolving on trees with 2, 3, and 4 leaves.

#### Sagi Snir (UC-Berkeley): Convex Recolorings of Strings and Trees: Definitions, Hardness Results, and Algorithms

(joint work with Shlomo Moran)

A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex colorings of trees arise in areas such as phylogenetics, linguistics, etc. e.g., a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree.

When a coloring of a tree is not convex, it is desirable to know “how far” it is from a convex one, and what are the convex colorings which are closest to it. In this paper we study a natural definition of this distance–the recoloring distance, which is the minimal number of color changes at the vertices needed to make the coloring convex. We show that finding this distance is NP-hard even for a colored string (a path), and for some other interesting variants of the problem. In the positive side, we present algorithms for computing the recoloring distance under some natural generalizations of this concept: The first generalization is the uniform weighted model, where each vertex has a weight which is the cost of changing its color. The other is the non uniform model, in which the cost of coloring a vertexvby a colordis an arbitrary nonnative number cost(v,d). Our first algorithms find optimal convex recolorings of strings and bounded degree trees under the non uniform model in time which, for any fixed number of colors, is linear in the input size. Next we improve these algorithms for the uniform model to run in time which is linear in the input size for a fixed number of bad colors, which are colors which violate convexity in some natural sense. Finally, we generalize the above result to hold for trees of unbounded degree.

#### Mariel Vazquez (UC-Berkeley): Genome Comparison Allowing Complex Rearrangements

(joint work with Dan Levy, Rainer Sachs)

Radiation cytogenetics deals with rearrangements of the genome caused by ionizing-radiation. We have developed a mathematical framework, related to the theory of cubic multigraphs, for characterizing such rearrangements. We here apply our work to a study of comparative genomics in evolution where one considers each genome as a short sequence of synteny blocks and analyzes the rearrangements taking one sequence of blocks into the other. In traditional studies of evolution of genomes the rearrangements (apart from fusions and fissions) are usually reduced to a sequence of reactions involving just two DNA breaks. We here explore the possibility that large-scale insertions and transpositions and other rearrangements involving 3 or more breaks occur.

#### Ruriko Yoshida (Duke): Neighbor Joining with Subtree Weights

(joint work with Dan Levy and Lior Pachter)

The Neighbor-Joining algorithm is a recursive procedure for reconstructing phylogenetic trees that is based on a transformation of pairwise distances between leaves for identifying cherries in the tree (two nodes are a cherry if there is exactly one intermediate vertex on the path between them). We show that estimates of the weights ofm-leaf subtrees are more accurate than pairwise distances, and derive a generalization of the cherry picking criterion which uses such weights. This leads to an improved neighbor-joining algorithm whose total running time is still polynomial in the number of taxa. In my presentation, I will remind an outline of The Neighbor-Joining algorithm with pairwise distances, introduced by Saitou-Nei and Studier-Keppler in 1987 and then, I will describe the Neighbor-Joining algorithm with the weights ofm-leaf subtrees and a generalization of the cherry picking criterion.

### Speakers & Schedule

#### 9:10 AM

Welcome

#### 9:15 AM

Michael Hendy(Massey University, New Zealand)

Hadamard Conjugation and the Molecular Clock

#### 10:15 AM

Poster Session and Refreshments

#### 11:00 AM

Susan Holmes(Stanford)

Using Phylogenetic Trees to Analyse Microarrays

#### 12:00 PM

Lunch

#### 2:00 PM

David Bryant(McGill University, Canada)

Probabilistic Models for Splits Graphs

#### 3:00 PM

Poster Session and Refreshments

#### 4:00 PM

Bernd Sturmfels(UC-Berkeley)

Phylogenetic Algebraic Geometry

## 2003 Harvey Mudd College Mathematical Biology Conference

### General Information

The Department of Mathematics at Harvey Mudd College hosted the fifth annual Mt. Baldy Mathematics Conference on Saturday, November 8, 2003. The topic this year was mathematical biology and the program included five keynote speakers and a panel discussion on future trends in mathematical biology.

This year’s organizers were Lisette de Pillis and Jon Jacobsen.

### Speakers & Schedule

#### 8:30 AM

Registration

#### 9:00 AM

Carlos Castillo-Chavez(Cornell University)

Problems at the Interface of Epidemics and Homeland Security

#### 10:00 AM

Sally Blower(UCLA)

Predicting the Public Health Impact of Imperfect HIV Vaccines: The Red Queen Principle

#### 11:00 AM

Jim Keener(Utah)

The Topology of Defibrillation

#### 12:00 PM

Lunch

#### 2:00 PM

Zvia Agur(Institute for Medical BioMathematics, Israel)

Frequency of Drug Administration Determines Both Efficacy and Toxicity of Cytotoxic and Supportive Cancer Drug Therapy—Prospective Validation of a Mathematical Theory

#### 3:00 PM

Bard Ermentrout(Pittsburgh)

Lures of the Ring: Synchrony and Dynamics on the Circle

#### 4:00 PM

Coffee Break

#### 4:30 PM

Panel Discussion

Moderated byJohn Milton(Chicago)

### Photographs

## 2002 Mt. Baldy Conference on Applied Algebra and Combinatorics

### General Information

The Department of Mathematics at Harvey Mudd College hosted the fourth annual Mt. Baldy Mathematics Conference on Saturday, November 2, 2002. The topic this year was Applied Algebra and Combinatorics. There were four keynote speakers.

The organizer was Michael Orrison.

### Abstracts

#### Ron Graham(University of California, San Diego): Guessing Secrets

We will describe a variant of the familiar “20 Questions” problem in which one tries to discover the identity of some unknown “secret” by asking binary questions. In this variation, there is now a set of two (or more) secrets to use in supplying the answers, which in any case must always be truthful. We will discuss a number of algorithms for dealing with this problem, although we are still far from a complete understanding of the situation. Problems of this type have recently arisen in connection with certain Internet traffic routing applications, although it turns out that such problems have in fact occurred in the literature nearly 40 years ago.

#### Cheryl Praeger(University of Western Australia): Finding and Using the Automorphism Group of a Finite Vertex-Transitive Graph

Finding the full automorphism group of a graph is notoriously difficult, even if the graph is known to be vertex-transitive. However the level of information needed depends on how we wish to apply it and what problems we hope to solve. I shall discuss several families of finite vertex-transitive graphs. The techniques employed to study them range from elementary combinatorial arguments to sophisticated group theory involving finite simple groups.

#### Fan Chung Graham(University of California, San Diego): Random Graphs and Internet Graphs

Many very large graphs that arise in Internet and telecommunications applications share various properties with random graphs (while some differences remain). We will discuss some recent developments and mention a number of problems and results in random graphs and algorithmic design suggested by the study of these “massive” graphs.

#### Persi Diaconis(Stanford University): Computational Aspects of Polya Theory

Polya theory is about counting “things” up to symmetry (e.g., unlabeled trees of various flavors). I will review and show how the computer has changed things. Fortunately, there are still things for mathematicians to do.

### Speakers & Schedule

#### 9:00 AM

Registration

#### 10:00 AM

Ron Graham(University of California, San Diego)

Guessing Secrets

#### 11:00 AM

Break

#### 11:15 AM

Cheryl Praeger(University of Western Australia)

Finding and Using the Automorphism Group of a Finite Vertex-Transitive Graph

#### 12:15 PM

Lunch

#### 1:45 PM

Fan Chung Graham(University of California, San Diego)

Random graphs and Internet graphs

#### 2:45 PM

Break

#### 3:00 PM

Persi Diaconis(Stanford University)

Computational aspects of Polya theory

## 2001 Mt. Baldy Conference on Differential Geometry

### General Information

The Department of Mathematics at Harvey Mudd College hosted the third annual Mt. Baldy Mathematics Conference on Saturday, November 17, 2001. The topic this year was Differential Geometry. There were four keynote speakers in the areas of calibrated geometry, geometric analysis, Riemannian geometry, classical geometry and topology with applications to plasma physics.

The organizer was Weiqing Gu.

### Abstracts

#### Rick Schoen (Stanford): Special Lagrangian Geometry

Calibrated submanifolds include complex submanifolds of Kaehler manifolds as well as special Lagrangian submanifolds. This latter class of submanifolds is of substantial interest in string theory. This talk will give a general introduction to the subject with particular emphasis on existence questions special Lagrangian and minimal Lagrangian submanifolds. We will describe deformation results, gluing methods, heat equation methods, and variational methods for constructing them.

#### Herman Gluck (U. Pennsylvania): Geometry, Topology and Plasma Physics

In this talk, I’ll report on the work of a number of people at Penn in recent years involving mathematical methods in plasma physics. From the physics point of view, our goal is to determine and study the persistent plasma states observed in astrophysical, solar and laboratory settings. From the mathematics point of view, our goal is to develop the tools to carry this out, and to work on a number of mathematical problems suggested by this enterprise. Key roles in the story are played by the notion of “helicity” of a vector field, which measures the extent to which the field lines wrap and coil around one another, and spectral problems for the curl operator. Because helicity of vector fields is the analogue of “writhing number” of knots, the methods we use also provide an upper bound for the writhing number of a given strand of DNA.

#### Peter Li (UC Irvine): Complete Manifolds with Positive Spectrum”

In this talk, we will discuss how the Laplace operator is used to determine certain topological information of a complete non-compact manifold. Hodge theory asserts that when a manifold is compact, then its homology group can be computed using harmonic forms. Unfortunately, a similar theory does not exist for non-compact manifolds. However, for certain situations, one can still control some topological aspect using the analytic property of the Laplace operator. An example of this is a generalization of a theorem of Witten-Yau.

#### Peter Petersen (UCLA): Pinching and Compactness

The talk will be a survey of what types of results one can expect when curvatures of a manifold are pinched. Pinching of curvatures traditionally means that curvatures lie in some small interval on the real line. More recently it has also been generalized to include situations where certain L^p norms of curvatures are small. We will discuss which types of pinching conditions imply compactness in certain topologies and which don’t and also how this affects what we can say about the underlying structure of the manifold.

### Speakers & Schedule

#### 9:00am

Registration, Olin Science Center

#### 9:45am

Dean Sheldon Wettack

Welcome

#### 10:00am

Rick Schoen (Stanford)

Special Lagrangian Geometry

#### 11:00am

Break

#### 11:15am

Herman Gluck (U Penn)

Geometry, Topology and Plasma Physics

#### 12:15pm

Lunch

#### 1:45pm

Peter Li (UC Irvine)

Complete Manifolds with Positive Spectrum

#### 2:45pm

Break

#### 3:00pm

Peter Petersen (UCLA)

Pinching and Compactness

## 2000 Mt. Baldy Conference on Applied Mathematics

### General Information

The Department of Mathematics at Harvey Mudd College hosted the second annual Mt. Baldy Mathematics Conference on Saturday, October 28th, 2000. The topic this year was Applied Mathematics. There were four keynote speakers, discussing the applications of mathematics to problems arising in physics and biology.

The organizers are Andrew Bernoff and Anette Hosoi.

### Abstracts

#### Andrea Bertozzi (Duke University): Undercompressive Shocks in Thin Film Flow

Nonlinear hyperbolic conservation laws have solutions with propagating `shocks’ or discontinuities. Classically admissible shocks in scalar hyperbolic conservation laws satisfy a well known `entropy condition’, in which characteristics enter the shock from both sides. Undercompressive shocks, in which characteristics pass through the shock, arise in e.g. combined dispersive/diffusive limits of scalar laws with non-convex flux functions. We show that fourth order diffusion alone produces undercompressive fronts, yielding such unusual behavior as double shock structures from simple jump initial data. Thermal/gravity driven thin film flow is described by such equations and the signature of undercompressive fronts have been observed in recent experiments. Mathematically, the undercompressive front is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a cascade of bifurcations parameterized by the shock speed.

Michael Cross (CalTech): Theoretical Challenges of Spatiotemporal Chaos

Spatially extended nonequilibrium systems may show deterministic disordered dynamics which can be characterized as chaos with an extensive fractal dimension (i.e. dimension scaling with the system size). We call this dynamics spatiotemporal chaos. Natural examples, such as the earth’s atmosphere, are widespread. In this talk I discuss some of the interesting issues in defining, characterizing, and understanding spatiotemporal chaos, and the theoretical challenges in addressing these issues, drawing on ideas from the study of low dimensional chaos and the statistical mechanics of equilibrium systems. Experimental examples and comparisons between theory and experiment will also be discussed.

#### L. Mahadevan (MIT): Folding, Wrinkling and Crumpling

Bending a thin elastic sheet is easier than stretching it, a fact that arises more from geometry than physics. Keeping this simple idea from structural mechanics in mind, I will give various examples involving the large deformations of thin films that are relevant to phenomena over a range of scales, from micro-patterning to geology. A feature that often arises in the ensuing deformation patterns is strain localization in the vicinity of a point. Therefore, I will discuss the shape, response and stability of these singularities, and their influence on the evolution of the patterns. Finally, I will explain and exploit an analogy that allows us to understand the folding, wrinkling and crumpling of thin fluid sheets.

#### Michael Ward (University of British Columbia): The Stability and Dynamics of Spikes for a Reaction-Diffusion System

Many classes of singularly perturbed reaction-diffusion systems possess localized solutions where the gradient of the solution is very large only in the vicinity of certain points in the domain. An example of a problem where such spikes occur is the Geirer-Meinhardt (GM) activator-inhibitor system modeling biological morphogenesis. In the limit of a small activator diffusivity, this system has been used to model many situations including spot-type patterns on sea-shells and head formation in the Hydra. Most of the previous work on this system over the past twenty years has been based either on full numerical simulations or on a linearized Turing-type stability analysis around spatially homogeneous steady-state solutions. However, this type of linearized analysis is not appropriate for determining the stability of spike-type patterns. In this talk we will survey some recent results on the existence and stability of symmetric and asymmetric equilibrium spike patterns for the GM model. The inhibitor diffusivity is found to be a critical parameter. A key result that is obtained is that there exists a sequence of critical valuesDnof the inhibitor diffusivityDfor which ann-spike symmetric equilibrium solution is stable ifD < Dnand unstable ifD > Dn. An explicit formula forDnis given. The dynamics of spike patterns is also characterized in a one-dimensional domain and partial results are obtained in a multi-dimensional context. The mathematical tools used include asymptotic analysis, spectral analysis of nonlocal eigenvalue operators, dynamical systems, and numerical and matrix analysis.

Some of this work is joint with David Iron (graduate student at UBC), and Prof. Juncheng Wei (Chinese University of Hong Kong).

### Speakers & Schedule

#### 9:00am

Registration, Olin Science Center

#### 9:45am

Dean Sheldon Wettack

Opening Remarks and Welcome

#### 10:00am

Michael Cross (CalTech)

Theoretical Challenges of Spatiotemporal Chaos

#### 11:15am

Andrea Bertozzi (Duke University)

Undercompressive Shocks in Driven Films

#### 12:15pm

Lunch

#### 1:45pm

L. Mahadevan (MIT)

Folding, Wrinkling, and Crumpling

#### 3:00pm

Michael Ward (University of British Columbia)

The Stability and Dynamics of Spikes for the Gierer-Meinhart Model”

## 1999 Mt. Baldy Conference on Analysis

### General Information

The Department of Mathematics at Harvey Mudd College hosted the inaugural Mt. Baldy Mathematics Conference on Saturday, November 20, 1999. The topic was Analysis.

There were five keynote speakers, in the areas of complex analysis, harmonic analysis, geometric measure theory, and probability.

The organizer was Lesley Ward.

### Abstracts

#### Michael Christ (UC Berkeley): On One-Dimensional Schroedinger Operators with Slowly Decaying Potentials

The spectrum and (generalized) eigenfunctions of a time-independent Schroedinger operator

-d2/dx2+ V(x)

determine the nature of the associated time evolution described by Schroedinger’s equation. They depend on the potential in a highly nonlinear manner.

We study the asymptotic behavior of generalized eigenfunctions in one dimension, for slowly decaying potentials, and show that for almost everykthere exists a generalized eigenfunction behaving roughly likeexp(i k x)for largex. The eigenfunctions are expressed as sums of infinite series of multilinear operators applied to the potential, and these series are shown to converge for almost every energy, with respect to Lebesgue measure, for a wide class of potentials that is nearly optimal for this conclusion. In particular, potentials in Lebesgue spaces are treated.

The proof is related to an ancient theorem of Menshov, Paley, and Zygmund about almost everywhere convergence of Fourier integrals and more general orthogonal expansions. We present a simple and rather general maximal theorem which implies their result, and has other applications. This is joint work with Alexander Kiselev.

#### Clifford Earle (Cornell University): Variation of Moduli Under Holomorphic Motions

The theory of holomorphic motions originated as a tool for applying the method of quasiconformal variations to problems in complex dynamics, but it also can be used to study questions in geometric function theory. An important early example is Burt Rodin’s 1986 paper “Behavior of the Riemann Mapping Function Under Complex Analytic Deformations of the Domain”.

Recently, in joint projects with Sudeb Mitra and with Adam Epstein, we have used holomorphic motions and quasiconformal variations to answer some questions posed by Dieter Gaier at a 1996 conference on function theory at Oberwolfach. Rodin’s theorem can be applied to one of these questions, which concerns the conformal radius of variable slit domains. We shall discuss two of Gaier’s questions, with particular emphasis on the conformal radius problem.

#### Robert Hardt (Rice University): Limits Associated with Sequences of Smooth Maps to Spheres

Smooth maps between Riemannian manifolds are often not strongly dense in classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 Ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. With T. Riviere, we also treat the case of finite 3-energy maps from the 4 Ball to the 2 sphere. Here the notion of the Hopf degree rather than ordinary topological degree is relevant. The objects produced now may no longer have finite mass, but are nevertheless describable as graphs of atomic measure-valued functions on the reals which are essentially contained in countably 1-rectifiable sets. The crucial compactness of these functions is obtained by a fractional maximal function estimate. There are related associated variational problems for the mappings and for the currents.

#### Thomas M. Liggett (UCLA): Growth Models on Trees

I will discuss a system whose state at any time is a random connected subset of the vertices of a homogenous tree. It evolves by adding neighbors at (exponential) rate beta, and deleting leaves at rate 1. There is a phase transition in beta: the system dies out if beta is small, and explodes if beta is large. I will explain how one can compute the critical value for this phase transition exactly. One inequality involves a simple combinatorial argument. The other is harder, and uses a number of analytic and combinatorial tools. One of the former stems from an old (1952) paper of Royden on the existence of Green functions. The talk is based on the following papers:

- A. Puha. A reversible nearest particle system on the homogeneous tree,J. Th. Probab.12 (1999) 217-254.
- A. Puha. Critical exponents for a reversible nearest particle system on the binary tree, Ann. Probab., to appear.
- T. M. Liggett. Monotonicity of conditional distributions and growth models on trees, to appear.

#### Brad Osgood (Stanford University): Recent Results on the Geometry of Univalence Criteria

We will discuss the geometry of univalence criteria associated with the Schwarzian derivative, from the 1949 result of Nehari that started the subject, to recent results on John domains, quasiconformal, and homeomorphic extensions. A natural setting is a generalization of the Schwarzian that allows for conformal changes of metric, and a common theme is convexity, especially convexity in the hyperbolic metric.

### Speakers & Schedule

#### 9:00am

Registration, Olin Science Center

#### 9:30am

Michael Christ (UC Berkeley)

On One-Dimensional Schroedinger Operators with Slowly Decaying Potentials

#### 10:45am

Clifford Earle (Cornell University)

Variation of Moduli Under Holomorphic Motions

#### 12:00pm

Lunch

#### 1:30pm

Robert Hardt (Rice University)

Limits Associated with Sequences of Smooth Maps to Spheres

#### 2:45pm

Thomas M. Liggett (UCLA)

Growth Models on Trees

#### 4:00pm

Brad Osgood (Stanford University)

Recent Results on the Geometry of Univalence Criteria