# Mathematics Research

The department of mathematics encourages student research in a number of ways. Mathematics majors are required to complete a capstone project by either joining a year-long team Clinic project or working with a faculty advisor on a thesis project. The department also offers summer research opportunities hosted at HMC.

## 2020 Summer Research

### Hosted at HMC

- HMC NSF Research Experiences for Undergraduates (REU) in Data Science program
- HMC student research group on Locust Swarming with Professors Bernoff and Weinburd

### Hosted Elsewhere

- UCLA REU in Applied and Computational Mathematics (Cofounded by Professor Bernoff!)

## Faculty Research Interests

## Arthur Benjamin

Combinatorial proofs of Fibonacci identities and other interesting sequences; Mathematics of games, gambling, and puzzles.

## Andrew Bernoff

Modeling of Physical and Biological Systems, Applications of Dynamical Systems, Fluid Mechanics, Self-Similarity and Scaling

#### Topic I: Models of Discrete and Continuous Swarming

**Overview:** Swarming of organisms is a ubiquitous phenomena in the world around us; flocks of birds, schools of fish and locust swarms are examples of systems we have considered. There are several projects available that fit in as part of an ongoing effort to understand the dynamics of increasingly detailed models of these social structures. Some sample projects are listed below.

**Background Needed:** Some PDE’s, comfort with numerical methods and computing.

**Skills Learned:** Modeling of biological systems, familiarity with energy and variational methods, bifurcation theory, integral equations, numerical and asymptotic methods.

#### Sample Project: Mathematical Modeling of Heterogeneous Swarms

Most swarm models assume that all individuals are identical and interchangeable but in nature individuals are heterogeneous. For example, if one assumes that visual attraction to larger individuals is stronger in a popular model of fish schools, larger individuals drift to the swarms periphery and smaller individuals drift to the interior. This behavior is observed in nature and biologically can be interpreted as protection of juveniles. This project will be a combination of biological modeling, numerical simulation, and mathematical analysis of models of heterogeneous swarms.

#### Sample Project: Bifurcation Problems in Discrete and Continuous Swarms

Discrete, Agent-based models of swarms often exhibit steady states that while irregular suggest highly symmetric states such as spheres and rotating mills. One can study these problems by deriving continuous (PDE) versions of the agent-based models and looking for a hierarchy of bifurcations and steady-states. This project will touch on both kinetic theory of particle systems and dynamical systems methods for examining bifurcations with symmetry.

#### Sample Project: Mathematical Modeling of Nearest Neighbor Interactions in Two Dimensions

The most popular continuum models of biological aggregation have an undesirable feature that the maximum density observed in equilibrium configurations is proportional to the biomass. Recently, we have examined a strategy in one dimension for modeling nearest neighbor interactions that leads to aggregations with a maximum density. In higher dimensions, this strategy is more difficult to implement because individuals are not well-ordered and the idea of nearest neighbors is ill-defined. In this project we will use a Voronoi decomposition to identify nearest neighbors and to derive appropriate continuum models in two dimensions.

#### Topic II: Kinetic Monte Carlo Methods for Diffusive Capture

**Description:** Consider the pistil of a flower waiting to catch a grain of pollen, a lymphocyte waiting to be stimulated by an antigen to produce antibodies, or an anteater randomly foraging for an ant nest to plunder. Each of these problems can be modeled as a diffusive process with a mix of reflecting and absorbing boundary conditions. One can characterize the agent (pollen, antigen, anteater) finding its target (pistil, lymphocyte, ant nest) as a first passage time (FPT) problem for the distribution of the time when a particle executing a random walk is absorbed. Numerically simulating these problems involves analytically solving the diffusion equation in a variety of domains and then sampling the exit time distributions associated with the solutions. Our simulations and asymptotics for these problems are slowly and steadily reproducing more realistic geometries for the biological capture problems they model.

#### Topic III: Energy Driven Pattern Formation

**Description:** Many physical systems are driven by pairwise particle interactions; for example in thin liquid and solid layers, pairs of molecules interact electromagnetically. In swarming, the gathering of insects is often modeled via a pairwise social force. These systems can exhibit elaborate changes of morphologies as system parameters change. An example of this is the circle-dogbone-labyrinth transition seen in Langmuir films as the domain size increases (see Figure). In this project we will investigate the relationship between theses morphology changes and the form of the pairwise potential via energy methods and bifurcation theory. While many examples of these transitions have been observed for specific systems, characterizing these transitions for generic potentials remains a fertile area for study.

**Background Needed:** Some PDE’s, comfort with numerical methods and computing.

**Skills Learned:** Familiarity with energy and variational methods, bifurcation theory, integral equations, numerical and asymptotic methods.

If you are interested send me an e-mail (ajb@hmc.edu).

## Alfonso Castro

Solvability of ordinary and partial differential equations using both elementary integration methods and functional analytic tools. The intermediate value theorem and its generalizations to several variables, the contraction mapping principle, variational methods, and the implicit function theorem are examples of techniques used in these studies. Go to Mathscinet to see the nature of my research. Papers with C.M. Tan, B. Preskill and E. Fischer are the result of senior thesis work at HMC.

## Lisette de Pillis

Mathematical biology, including tumor modeling, immunology modeling, blood coagulation modeling, diabetes treatments, optimal control, HIV/AIDS modeling, epidemiology modeling; numerical linear algebra.

Preparation needed: Linear algebra; Differential equations; some coding (Matlab, e.g.).

### Summer 2020 Project – Please apply through REU website

*We plan to explore techniques, such as data assimilation methods, to connect data sets to dynamical systems models, with the goal of creating a testbed for emerging treatments for type 1 diabetes. Techniques such as deep learning promise to predict the future without mechanistic knowledge, learning the structure directly from the data, but they are limited in settings with sparse, irregular, inaccurate data, and they do not bring much understanding about the patient or the disease beyond their predictions. Data assimilation (DA), known also as state-space models, point processors, Kalman filters, and linear dynamical systems, is a regression method meant to solve inverse problems (estimating model parameters), often in the context of solving a forward problem (issuing a forecast) and was initially developed in contexts of space travel by Kalman and Brownian motion by Thiele. Conceptually, DA takes a model believed to represent a system being studied and synchronizes the model with data by estimating states and parameters of the system. In essence, DA is a family of regression methods that project data onto a physiologically meaningful mechanistic model the way linear regression projects data onto simpler linear models and the way deep learning projects data onto more flexible but data-hungry neural networks. DA has advanced over the decades from its original formulation as a linear, stochastic method to more recently developed nonlinear approaches. We will work through an example of DA with published data from a type 2 diabetes patient that has a two-day sample of sparse finger-prick glucose measurements and meal data. Our hope is to extend this approach to glucose measurements in mouse models and eventually human models of type 1 diabetes to forecast point of disease onset using data from literature and from our collaborators.*

### Possible Capstone Projects

- Diabetes type 1: modeling effective treatment and prevention strategies with differential equations. Strong computing skills as we as exposure to Matlab and to mathematical modeling required.
- Cancer/immunology/cancer-treatment modeling with differential equations.
- Cancer/immunology/cancer-treatment modeling with cellular automata.
- Modeling blood clotting mechanisms and treatments with differential equations.
- Epidemiological modeling with differential equations.
- Exploration of large data sets (gene expression, flow cytometry data) to identify/predict disease stage progression.

**Figure**: In this simulation, we explore the effects of “tumor gluttony” (how rapidly tumor cells consume nutrients relative to healthy cells) on tumor morphology. We see that increased tumor gluttony leads to tumors that are less compact and more papillary. Papillary tumors tend to be more dangerous.

If you wish to discuss any of these projects, feel free to contact me at depillis@g.hmc.edu. Please put `RESEARCH INTEREST` in your subject line to make sure I see your email.

## Weiqing Gu

- Differential geometry, Grassmann manifolds, characteristic classes and applications to string theory.
- Big data analysis. There are big data problems everywhere in this world. There is an urgent need to apply machine learning and advanced mathematical techniques to extract patterns and insights from large and complex collections of digital data and to achieve data-to-decision since traditional statistical methods do not suffice. In this faculty-student research or a student senior thesis project we will identify patterns and anomalies in time series data including unmanned aerial vehicle (UAV) data, stock data, and video data. We will also try to use these patterns to forecast the behavior of the time series in the immediate future. We will further develop techniques to make some optimal or semi-optimal decisions based on the knowledge learned from the data. This research project will introduce or enhance big data analytics of our students who will be faced with big data issues in their future workplace.
- Using linear algebra and geometric techniques to build new algorithms to provide mathematical fundamentals for the development of Machine Learning and AI.

## Jon Jacobsen

Student initiated topics are welcome and often some of the most enjoyable thesis projects! A few other project suggestions are outlined below.

**The tacit dimension in learning mathematics:**Michael Polanyi’s theory of personal knowledge views all knowledge — regardless of how formalized a discipline’s techniques — as personal in nature, involving processes that rely on an individual’s effort to dwell in particulars and develop an embodied subject coherence from which to attend to knowledge of things seen in its light. Neither the processes nor particulars involved can be entirely specified and this “tacit dimension” lies at the core of all knowledge. In this project we will study learning mathematics through the lens of Polanyi’s theory. Of particular interest are relations to identity, access, achievement, rigor, and one’s sense of belonging with Polanyi’s theory.**Generalized Julia Sets:**Applying Newton’s method to find the roots of a complex polynomial leads to iterating a rational function in the complex plane and the associated theory of Julia sets (fractal boundaries for the basins of attraction). This experimental in nature project would study dynamics for Newton’s method associated with finding roots of vector fields from R^n to R^n, which can be viewed as a generalization of the special R^2 to R^2 case of complex maps. Both situations can be realized as discretizations of a continuous Newton method differential equation. Since we’ve left the realm of complex dynamics, many questions remain including what are the natural candidates for Julia-like sets, their geometric properties (e.g., fractal dimension), relationships between the discrete and continuous problem, and so on. This project would involve significant computational explorations in addition to learning about the theory of complex dynamics.**Computational models in spatial ecology:**this project will require substantial computational skills (esp. matlab) to study population dynamics for integrodifference models in ecology. We have developed a model that allows one to study persistence of a population in terms of calculating the spectral radius of an associated linear operator, however this can be difficult in general and we have established certain alternate metrics that may be equally useful. The researcher would study these models, learn the associated mathematical framework, and develop numerical approaches to better understand their behavior and predictions for persistence in terms of temporal and spatial variations in the model. Of particular importance/challenge is efficient calculation of the metrics and making tools that allow an interested modeler to engage with the models.

## Dagan Karp

Algebraic Geometry and Mathematics Education. Dagan Karp’s research is focused on** (a) **disciplinary research in mathematics, specifically combinatorial algebraic geometry and geometry inspired by theoretical physics, and **(b)** research in mathematics education. Possible thesis research areas include

**Toric geometry in Gromov-Witten theory**, a combinatorial approach to an area of mathematics born from interaction with theoretical physics,**Tropical geometry**, a beautiful combinatorial approach to algebraic geometry, and**Equity centered pedagogy**, rehumanizing, decolonizing, and queering the learning, teaching, and sharing of mathematics.

Prof. Karp also welcomes students with interests in related subjects, and has supervised theses in such subjects as commutative algebra, geometric representation theory, and personalized assessment practices in mathematics education.

## Susan Martonosi

### Summer 2019 research projects:

The prevalence and propagation of “fake news” has garnered international attention following the 2016 U.S. presidential election. The mechanisms by which fake and/or biased news articles are propagated are an active area of research, particularly as social media outlets such as Facebook are increasingly being asked to play an active role in fake news detection and deterrence. This proposed research project will develop a probability model and optimization framework to determine the optimal distribution of bias and truthfulness of articles produced by a malicious agent to maximize propagation within a population having known belief distribution. This work will provide insights into the optimal characteristics of biased and/or “fake” news, which can then be used within a game theoretic framework to develop defensive strategies.**Fake News (Model Development: Apply to Prof. Martonosi directly):**Same as above.**Fake News (Data Science component: Apply through HMC Data Science REU):****The data science student researchers will assist in validating our models against publicly available social media data.**

### Thesis project possibilities:

Operations research, data analytics, applied probability, homeland security, network optimization, humanitarian logistics.

Aid organizations like the World Health Organization and the Gates Foundation negotiate prices for pediatric vaccines distributed to low income countries. However, while they try to negotiate as low a price as possible, they also want to ensure sufficient profitability for the vaccine manufacturers so that they will remain in the market, preventing a monopoly for any given vaccine. This problem then becomes a**Game theory models for pediatric vaccine pricing:**between the organizations and the vaccine manufacturers. We will use different methods of game theory to find equilibria that can inform policy.**game**The prevalence and propagation of “fake news” has garnered international attention following the 2016 U.S. presidential election. The mechanisms by which fake and/or biased news articles are propagated are an active area of research, particularly as social media outlets such as Facebook are increasingly being asked to play an active role in fake news detection and deterrence. This proposed research project will develop a probability model and optimization framework to determine the optimal distribution of bias and truthfulness of articles produced by a malicious agent to maximize propagation within a population having known belief distribution. This work will provide insights into the optimal characteristics of biased and/or “fake” news, which can then be used within a game theoretic framework to develop defensive strategies.**Fake News:**Research in this area can involve graph theory, algorithm design, network optimization, operations research, etc.**Network disruption methods:**- Operations research and data analytics techniques applied to a problem of interest to you.

## Michael Orrison

Harmonic analysis on finite groups, algebraic voting theory, and applications of the representation theory of finite groups.

## Mohamed Omar

Professor Omar will be on sabbatical and will subsequently not be available for research in summer 2019, summer 2020, nor during the 2019-2020 academic year.

## Nicholas Pippenger

My main interests are discrete mathematics, probability, and applications to communications and computation. Some more specific areas include

**Stochastic Service Systems**: customers arrive randomly at servers, queueing (or perhaps just departing) if all servers are busy. There are a variety of problems concerning performance of such systems (also the time to search for an idle server).**Circuit Switching Networks**: customers request paths over which they may communicate in a network; again there are performance and search-time problems.**Combinational and Sequential Logic Circuits**: find small and fast circuits for various computational problems.

I am also interested in discussing ideas for topics that you may have!

## Francis Su

Geometric and topological combinatorics, especially as applied to problems in mathematical economics, fair division, voting. This includes: the geometry of triangulations of convex polytopes, intersection problems for convex sets, Sperner’s lemma and and problems from combinatorial topology.

## Talithia Williams

Statistical disease modeling, Environmental statistics, Space – time data analysis.

### Description

Prevalence and incidence are two important measures of the impact of a disease. For many diseases, incidence is the most useful measure for response planning. We have developed a model to estimate incidence of progressive diseases, with an application to cataract disease in Africa. Initial results suggest different behavior of unilateral and bilateral incidence that might teach us something about the normal course of cataract disease. For example, how long before people with unilateral cataract usually develop bilateral cataract? Does this time depend on age, geographic region, gender or other factors that influence unilateral incidence? There are several natural extensions to this current body of work that would make for an exciting thesis project.

### Possible Thesis Projects

- Developing a data clustering methodology.
- Modeling age-dependent mortality due to cataract.
- Applying the current model to other progressive diseases, such as cancer.
- Developing a model that incorporates factors that influence cataracts (i.e. diabetes in the population)

## Darryl Yong

Mathematics education, applied mathematics, perturbation theory, partial differential equations