Updated: April 2024

**Math/Stats Thesis and Colloquium Topics 2024- 2025**

The degree with honors in Mathematics or Statistics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core material and skills, breadth and, particularly, depth of knowledge beyond the core material, ability to pursue independent study of mathematics or statistics, originality in methods of investigation, and, where appropriate, creativity in research.

An honors program normally consists of two semesters (MATH/STAT 493 and 494) and a winter study (WSP 031) of independent research, culminating in a thesis and a presentation. Under certain circumstances, the honors work can consist of coordinated study involving a one semester (MATH/STAT 493 or 494) and a winter study (WSP 030) of independent research, culminating in a “minithesis” and a presentation. At least one semester should be in addition to the major requirements, and thesis courses do not count as 400-level senior seminars.

An honors program in actuarial studies requires significant achievement on four appropriate examinations of the Society of Actuaries.

Highest honors will be reserved for the rare student who has displayed exceptional ability, achievement or originality. Such a student usually will have written a thesis, or pursued actuarial honors and written a mini-thesis. An outstanding student who writes a mini-thesis, or pursues actuarial honors and writes a paper, might also be considered. In all cases, the award of honors and highest honors is the decision of the Department.

Here is a list of possible colloquium topics that different faculty are willing and eager to advise. You can talk to several faculty about any colloquium topic, the sooner the better, at least a month or two before your talk. For various reasons faculty may or may not be willing or able to advise your colloquium, which is another reason to start early.

RESEARCH INTERESTS OF MATHEMATICS AND STATISTICS FACULTY

Here is a list of faculty interests and possible thesis topics. You may use this list to select a thesis topic or you can use the list below to get a general idea of the mathematical interests of our faculty.

**Colin Adams (On Leave 2024 – 2025)**

__Research interests:__ Topology and tiling theory. I work in low-dimensional topology. Specifically, I work in the two fields of knot theory and hyperbolic 3-manifold theory and develop the connections between the two. Knot theory is the study of knotted circles in 3-space, and it has applications to chemistry, biology and physics. I am also interested in tiling theory and have been working with students in this area as well.

Hyperbolic 3-manifold theory utilizes hyperbolic geometry to understand 3-manifolds, which can be thought of as possible models of the spatial universe.

__Possible thesis topics:__

- Investigate various aspects of virtual knots, a generalization of knots.
- Consider hyperbolicity of virtual knots, building on previous SMALL work. For which virtual knots can you prove hyperbolicity?
- Investigate why certain virtual knots have the same hyperbolic volume.
- Consider the minimal Turaev volume of virtual knots, building on previous SMALL work.
- Investigate which knots have totally geodesic Seifert surfaces. In particular, figure out how to interpret this question for virtual knots.
- Investigate n-crossing number of knots. An n-crossing is a crossing with n strands of the knot passing through it. Every knot can be drawn in a picture with only n-crossings in it. The least number of n-crossings is called the n-crossing number. Determine the n-crossing number for various n and various families of knots.
- An übercrossing projection of a knot is a projection with just one n-crossing. The übercrossing number of a knot is the least n for which there is such an übercrossing projection. Determine the übercrossing number for various knots, and see how it relates to other traditional knot invariants.
- A petal projection of a knot is a projection with just one n-crossing such that none of the loops coming out of the crossing are nested. In other words, the projection looks like a daisy. The petal number of a knot is the least n for such a projection. Determine petal number for various knots, and see how it relates to other traditional knot invariants.
- In a recent paper, we extended petal number to virtual knots. Show that the virtual petal number of a classical knot is equal to the classical petal number of the knot (This is a GOOD question!)
- Similarly, show that the virtual n-crossing number of a classical knot is equal to the classical n-crossing number. (This is known for n = 2.)
- Find tilings of the branched sphere by regular polygons. This would extend work of previous research students. There are lots of interesting open problems about something as simple as tilings of the sphere.
- Other related topics.

__Possible colloquium topics__: Particularly interested in topology, knot theory, graph theory, tiling theory and geometry but will consider other topics.

**Christina Athanasouli**

Research Interests: Differential equations, dynamical systems (both smooth and non-smooth), mathematical modeling with applications in biological and mechanical systems

My research focuses on analyzing mathematical models that describe various phenomena in Mathematical Neuroscience and Engineering. In particular, I work on understanding 1) the underlying mechanisms of human sleep (e.g. how sleep patterns change with development or due to perturbations), and 2) potential design or physical factors that may influence the dynamics in vibro-impact mechanical systems for the purpose of harvesting energy. Mathematically, I use various techniques from dynamical systems and incorporate both numerical and analytical tools in my work.

Possible colloquium topics: Topics in applied mathematics, such as:

- Mathematical modeling of sleep-wake regulation
- Mathematical modeling vibro-impact systems
- Bifurcations/dynamics of mathematical models in Mathematical Neuroscience and Engineering
- Bifurcations in piecewise-smooth dynamical systems

**Julie Blackwood
**

__Research Interests:__ Mathematical modeling, theoretical ecology, population biology, differential equations, dynamical systems.

My research uses mathematical models to uncover the complex mechanisms generating ecological dynamics, and when applicable emphasis is placed on evaluating intervention programs. My research is in various ecological areas including (*I*) invasive species management by using mathematical and economic models to evaluate the costs and benefits of control strategies, and (*II*) disease ecology by evaluating competing mathematical models of the transmission dynamics for both human and wildlife diseases.

__Possible thesis topics:__

- Mathematical modeling of invasive species
- Mathematical modeling of vector-borne or directly transmitted diseases
- Developing mathematical models to manage vector-borne diseases through vector control
- Other relevant topics of interest in mathematical biology

Each topic (1-3) can focus on a case study of a particular invasive species or disease, and/or can investigate the effects of ecological properties (spatial structure, resource availability, contact structure, etc.) of the system.

__Possible colloquium topics:__ Any topics in applied mathematics, such as:

- Mathematical modeling of invasive species
- Mathematical modeling of vector-borne or directly transmitted diseases
- Developing mathematical models to manage vector-borne diseases through vector control

**Xizhen Cai**

__Research Interest__: Statistical methodology and applications. One of my research topics is variable selection for high-dimensional data. I am interested in traditional and modern approaches for selecting variables from a large candidate set in different settings and studying the corresponding theoretical properties. The settings include linear model, partial linear model, survival analysis, dynamic networks, etc. Another part of my research studies the mediation model, which examines the underlying mechanism of how variables relate to each other. My research also involves applying existing methods and developing new procedures to model the correlated observations and capture the time-varying effect. I am also interested in applications of data mining and statistical learning methods, e.g., their applications in analyzing the rhetorical styles in English text data.

__Possible thesis topics:__

- Variable selection uses modern techniques such as penalization and screening methods for several different parametric and semi-parametric models.
- Extension of the classic mediation models to settings with correlated, longitudinal, or high-dimensional mediators. We could also explore ways to reduce the dimensionality and simplify the structure of mediators to have a stable model that is also easier to interpret.
- We shall analyze the English text dataset processed by the Docuscope environment with tools for corpus-based rhetorical analysis. The data have a hierarchical structure and contain rich information about the rhetorical styles used. We could apply statistical models and statistical learning algorithms to reduce dimensions and gain a more insightful understanding of the text.

__Possible colloquium topics: __I am open to any problems in statistical methodology and applications, not limited to my research interests and the possible thesis topics above.

**Duncan Clark**

**Duncan Clark**

My research focuses on problems where complex data generating processes produce data where observations are inextricably linked to other observations. I am interested in network generation processes, Bayesian social network models, causal inference for social networks and spatial point process models. In general I am interested in problems where applying an “off the shelf” statistical method does not yield sensible results. I have primarily worked with applications in the social sciences and public health policy, though I am very open to expanding to other areas, especially if the problem relates to relational data.

Possible Thesis Topics

- Comparing methods for social network modeling. There are many different approaches for considering network data, this thesis would understand some of the most common, apply them to network data, and describe the strengths and weaknesses of different approaches, and perhaps propose some extensions.
- An investigation of near degeneracy in Exponential-family random graph models (ERGM) and related models. ERGMs are widely used to model social network data, yet have a fundamental problem known as near-degeneracy. We would learn what it is, why its problematic, and investigate possible options augmenting ERGMs to try to prevent this degeneracy.
- Causal Inference with observational and or/complex data. Researchers often try to tease out causal conclusions, e.g. A caused B rather than A is associated with B, in situations with either a complex treatment assignment mechanism, or simply observational data. As causal claims are very powerful, especially for making important decisions, there is much interest in being able to make these conclusions. This work would investigate modern methods for causal inference in difficult settings, understanding where things can go wrong and where Statistics can unlock powerful causal insights.
- Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology to answer an interesting scientific question.

Possible Colloquium Topics:

- Topics in causal inference
- Methods for social network analysis
- Bayesian methods
- Topics with spatial point processes
- Any applied statistical research

**Richard De Veaux **

__Research interests:__ Statistics.

My research interests are in both statistical methodology and in statistical applications. For the first, I look at different methods and try to understand why some methods work well in particular settings, or more creatively, to try to come up with new methods. For the second, I work in collaboration with an investigator (e.g. scientist, doctor, marketing analyst) on a particular statistical application. I have been especially interested in problems dealing with large data sets and the associated modeling tools that work for these problems.

__Possible thesis topics:__

- Human Performance and Aging.I have been working on models for assessing the effect of age on performance in running and swimming events. There is still much work to do. So far I’ve looked at masters’ freestyle swimming and running data and a handicapped race in California, but there are world records for each age group and other events in running and swimming that I’ve not incorporated. There are also many other types of events.
- Variable Selection. How do we choose variables when we have dozens, hundreds or even thousands of potential predictors? Various model selection strategies exist, but there is still a lot of work to be done to find out which ones work under what assumptions and conditions.
- Problems at the interface.In this era of Big Data, not all methods of classical statistics can be applied in practice. What methods scale up well, and what advances in computer science give insights into the statistical methods that are best suited to large data sets?
- Applying statistical methods to problems in science or social science.In collaboration with a scientist or social scientist, find a problem for which statistical analysis plays a key role.

__Possible colloquium topics:__

- Almost any topic in statistics that extends things you’ve learned in courses — specifically topics in Experimental design, regression techniques or machine learning
- Model selection problems

**Thomas Garrity (On Leave 2024 – 2025)**

__Research interest:__ Number Theory and Dynamics.

My area of research is officially called “multi-dimensional continued fraction algorithms,” an area that touches many different branches of mathematics (which is one reason it is both interesting and rich). In recent years, students writing theses with me have used serious tools from geometry, dynamics, ergodic theory, functional analysis, linear algebra, differentiability conditions, and combinatorics. (No single person has used all of these tools.) It is an area to see how mathematics is truly interrelated, forming one coherent whole.

While my original interest in this area stemmed from trying to find interesting methods for expressing real numbers as sequences of integers (the Hermite problem), over the years this has led to me interacting with many different mathematicians, and to me learning a whole lot of math. My theses students have had much the same experiences, including the emotional rush of discovery and the occasional despair of frustration. The whole experience of writing a thesis should be intense, and ultimately rewarding. Also, since this area of math has so many facets and has so many entrance points, I have had thesis students from wildly different mathematical backgrounds do wonderful work; hence all welcome.

__Possible thesis topics: __

- Generalizations of continued fractions.
- Using algebraic geometry to study real submanifolds of complex spaces.

__Possible colloquium topics:__ Any interesting topic in mathematics.

**Leo Goldmakher**

__Research interests:__ Number theory and arithmetic combinatorics.

I’m interested in quantifying structure and randomness within naturally occurring sets or sequences, such as the prime numbers, or the sequence of coefficients of a continued fraction, or a subset of a vector space. Doing so typically involves using ideas from analysis, probability, algebra, and combinatorics.

__Possible thesis topics:____ __

Anything in number theory or arithmetic combinatorics.

__Possible colloquium topics:__ I’m happy to advise a colloquium in any area of math.

**Susan Loepp
**

__Research interests:__ Commutative Algebra. I study algebraic structures called commutative rings. Specifically, I have been investigating the relationship between local rings and their completion. One defines the completion of a ring by first defining a metric on the ring and then completing the ring with respect to that metric. I am interested in what kinds of algebraic properties a ring and its completion share. This relationship has proven to be intricate and quite surprising. I am also interested in the theory of tight closure, and Homological Algebra.

__Possible thesis topics: __

Topics in Commutative Algebra including:

- Using completions to construct Noetherian rings with unusual prime ideal structures.
- What prime ideals of C[[
*x*_{1},…,*x*]] can be maximal in the generic formal fiber of a ring? More generally, characterize what sets of prime ideals of a complete local ring can occur in the generic formal fiber._{n} - Characterize what sets of prime ideals of a complete local ring can occur in formal fibers of ideals with height
*n*where*n*≥1. - Characterize which complete local rings are the completion of an excellent unique factorization domain.
- Explore the relationship between the formal fibers of
*R*and*S*where*S*is a flat extension of*R*. - Determine which complete local rings are the completion of a catenary integral domain.
- Determine which complete local rings are the completion of a catenary unique factorization domain.

__Possible colloquium topics:__ Any topics in mathematics and especially commutative algebra/ring theory.

**Steven Miller**

For more information and references, see http://www.williams.edu/Mathematics/sjmiller/public_html/index.htm

__Research interests__: Analytic number theory, random matrix theory, probability and statistics, graph theory.

My main research interest is in the distribution of zeros of L-functions. The most studied of these is the Riemann zeta function, Sum_{n=1 to oo} 1/n^s. The importance of this function becomes apparent when we notice that it can also be written as Prod_{p prime} 1 / (1 – 1/p^s); this function relates properties of the primes to those of the integers (and we know where the integers are!). It turns out that the properties of zeros of L-functions are extremely useful in attacking questions in number theory. Interestingly, a terrific model for these zeros is given by random matrix theory: choose a large matrix at random and study its eigenvalues. This model also does a terrific job describing behavior ranging from heavy nuclei like Uranium to bus routes in Mexico! I’m studying several problems in random matrix theory, which also have applications to graph theory (building efficient networks). I am also working on several problems in probability and statistics, especially (but not limited to) sabermetrics (applying mathematical statistics to baseball) and Benford’s law of digit bias (which is often connected to fascinating questions about equidistribution). Many data sets have a preponderance of first digits equal to 1 (look at the first million Fibonacci numbers, and you’ll see a leading digit of 1 about 30% of the time). In addition to being of theoretical interest, applications range from the IRS (which uses it to detect tax fraud) to computer science (building more efficient computers). I’m exploring the subject with several colleagues in fields ranging from accounting to engineering to the social sciences.

__Possible thesis topics: __

- Theoretical models for zeros of elliptic curve L-functions (in the number field and function field cases).
- Studying lower order term behavior in zeros of L-functions.
- Studying the distribution of eigenvalues of sets of random matrices.
- Exploring Benford’s law of digit bias (both its theory and applications, such as image, voter and tax fraud).
- Propagation of viruses in networks (a graph theory / dynamical systems problem). Sabermetrics.
- Additive number theory (questions on sum and difference sets).

__Possible colloquium topics: __

- Theoretical models for zeros of elliptic curve L-functions (in the number field and function field cases).
- Studying lower order term behavior in zeros of L-functions.
- Studying the distribution of eigenvalues of sets of random matrices.
- Exploring Benford’s law of digit bias (both its theory and applications, such as image, voter and tax fraud).
- Propagation of viruses in networks (a graph theory / dynamical systems problem). Sabermetrics.
- Additive number theory (questions on sum and difference sets).

Plus anything you find interesting. I’m also interested in applications, and have worked on subjects ranging from accounting to computer science to geology to marketing….

**Ralph Morrison
**

__Research interests:__ I work in algebraic geometry, tropical geometry, graph theory (especially chip-firing games on graphs), and discrete geometry, as well as computer implementations that study these topics. Algebraic geometry is the study of solution sets to polynomial equations. Such a solution set is called a variety. Tropical geometry is a “skeletonized” version of algebraic geometry. We can take a classical variety and “tropicalize” it, giving us a tropical variety, which is a piecewise-linear subset of Euclidean space. Tropical geometry combines combinatorics, discrete geometry, and graph theory with classical algebraic geometry, and allows for developing theory and computations that tell us about the classical varieties. One flavor of this area of math is to study chip-firing games on graphs, which are motivated by (and applied to) questions about algebraic curves.

__Possible thesis topics__: Anything related to tropical geometry, algebraic geometry, chip-firing games (or other graph theory topics), and discrete geometry. Here are a few specific topics/questions:

- Study the geometry of tropical plane curves, perhaps motivated by results from algebraic geometry. For instance: given 5 (algebraic) conics, there are 3264 conics that are tangent to all 5 of them. What if we look at tropical conics–is there still a fixed number of tropical conics tangent to all of them? If so, what is that number? How does this tropical count relate to the algebraic count?
- What can tropical plane curves “look like”? There are a few ways to make this question precise. One common way is to look at the “skeleton” of a tropical curve, a graph that lives inside of the curve and contains most of the interesting data. Which graphs can appear, and what can the lengths of its edges be? I’ve done lots of work with students on these sorts of questions, but there are many open questions!
- What can tropical surfaces in three-dimensional space look like? What is the version of a skeleton here? (For instance, a tropical surface of degree 4 contains a distinguished polyhedron with at most 63 facets. Which polyhedra are possible?)
- Study the geometry of tropical curves obtained by intersecting two tropical surfaces. For instance, if we intersect a tropical plane with a tropical surface of degree 4, we obtain a tropical curve whose skeleton has three loops. How can those loops be arranged? Or we could intersect degree 2 and degree 3 tropical surfaces, to get a tropical curve with 4 loops; which skeletons are possible there?
- One way to study tropical geometry is to replace the usual rules of arithmetic (plus and times) with new rules (min and plus). How do topics like linear algebra work in these fields? (It turns out they’re related to optimization, scheduling, and job assignment problems.)
- Chip-firing games on graphs model questions from algebraic geometry. One of the most important comes in the “gonality” of a graph, which is the smallest number of chips on a graph that could eliminate (via a series of “chip-firing moves”) an added debt of -1 anywhere on the graph. There are lots of open questions for studying the gonality of graphs; this include general questions, like “What are good lower bounds on gonality?” and specific ones, like “What’s the gonality of the n-dimensional hypercube graph?”
- We can also study versions of gonality where we place -r chips instead of just -1; this gives us the r^th gonality of a graph. Together, the first, second, third, etc. gonalities form the “gonality sequence” of a graph. What sequences of integers can be the gonality sequence of some graph? Is there a graph whose gonality sequence starts 3, 5, 8?
- There are many computational and algorithmic questions to ask about chip-firing games. It’s known that computing the gonality of a general graph is NP-hard; what if we restrict to planar graphs? Or graphs that are 3-regular? And can we implement relatively efficient ways of computing these numbers, at least for small graphs?
- What if we changed our rules for chip-firing games, for instance by working with chips modulo N? How can we “win” a chip-firing game in that context, since there’s no more notion of debt?
- Study a “graph throttling” version of gonality. For instance, instead of minimizing the number of chips we place on the graph, maybe we can also try to decrease the number of chip-firing moves we need to eliminate debt.
- Chip-firing games lead to interesting questions on other topics in graph theory. For instance, there’s a conjectured upper bound of (|E|-|V|+4)/2 on the gonality of a graph; and any graph is known to have gonality at least its tree-width. Can we prove the (weaker) result that (|E|-|V|+4)/2 is an upper bound on tree-width? (Such a result would be of interest to graph theorists, even the idea behind it comes from algebraic geometry!)
- Topics coming from discrete geometry. For example: suppose you want to make “string art”, where you have one shape inside of another with string weaving between the inside and the outside shapes. For which pairs of shapes is this possible?

__Possible Colloquium topics:__ I’m happy to advise a talk in any area of math, but would be especially excited about talks related to algebra, geometry, graph theory, or discrete mathematics.

**Shaoyang Ning (On Leave 2024 – 2025)
**

__Research Interest__: Statistical methodologies and applications. My research focuses on the study and design of statistical methods for integrative data analysis, in particular, to address the challenges of increasing complexity and connectivity arising from “Big Data”. I’m interested in innovating statistical methods that efficiently integrate multi-source, multi-resolution information to solve real-life problems. Instances include tracking localized influenza with Google search data and predicting cancer-targeting drugs with high-throughput genetic profiling data. Other interests include Bayesian methods, copula modeling, and nonparametric methods.

__Possible thesis topics:__

- Digital (disease) tracking: Using Internet search data to track and predict influenza activities at different resolutions (nation, region, state, city); Integrating other sources of digital data (e.g. Twitter, Facebook) and/or extending to track other epidemics and social/economic events, such as dengue, presidential approval rates, employment rates, and etc.
- Predicting cancer drugs with multi-source profiling data: Developing new methods to aggregate genetic profiling data of different sources (e.g., mutations, expression levels, CRISPR knockouts, drug experiments) in cancer cell lines to identify potential cancer-targeting drugs, their modes of actions and genetic targets.
- Social media text mining: Developing new methods to analyze and extract information from social media data (e.g. Reddit, Twitter). What are the challenges in analyzing the large-volume but short-length social media data? Can classic methods still apply? How should we innovate to address these difficulties?
- Copula modeling: How do we model and estimate associations between different variables when they are beyond multivariate Normal? What if the data are heavily dependent in the tails of their distributions (commonly observed in stock prices)? What if dependence between data are non-symmetric and complex? When the size of data is limited but the dimension is large, can we still recover their correlation structures? Copula model enables to “link” the marginals of a multivariate random variable to its joint distribution with great flexibility and can just be the key to the questions above.
- Other cross-disciplinary, data-driven projects: Applying/developing statistical methodology to answer an interesting scientific question in collaboration with a scientist or social scientist.

__Possible colloquium topics:__ Any topics in statistical methodology and application, including but not limited to: topics in applied statistics, Bayesian methods, computational biology, statistical learning, “Big Data” mining, and other cross-disciplinary projects.

**Anna Neufeld**

__Research interests:__ My research is motivated by the gap between classical statistical tools and practical data analysis. Classic statistical tools are designed for testing a single hypothesis about a single, pre-specified model. However, modern data analysis is an adaptive process that involves exploring the data, fitting several models, evaluating these models, and then testing a potentially large number of hypotheses about one or more selected models. With this in mind, I am interested in topics such as (1) methods for model validation and selection, (2) methods for testing data-driven hypotheses (post-selection inference), and (3) methods for testing a large number of hypotheses. I am also interested in any applied project where I can help a scientist rigorously answer an important question using data.

__Possible thesis topics:__

- Cross-validation for unsupervised learning. Cross-validation is one of the most widely-used tools for model validation, but, in its typical form, it cannot be used for unsupervised learning problems. Numerous ad-hoc proposals exist for validating unsupervised learning models, but there is a need to compare and contrast these proposals and work towards a unified approach.

- Identifying the number of cell types in single-cell genomics datasets. This is an application of the topic above, since the cell types are typically estimated via unsupervised learning.
- There is growing interest in “post-prediction inference”, which is the task of doing valid statistical inference when some inputs to your statistical model are the outputs of other statistical models (i.e. predictions). Frameworks have recently been proposed for post-prediction inference in the setting where you have access to a gold-standard dataset where the true inputs, rather than the predicted inputs, have been observed. A thesis could explore the possibility of post-prediction inference in the absence of this gold-standard dataset.
- Any other topic of student interest related to selective inference, multiple testing, or post-prediction inference.

- Any collaborative project in which we work with a scientist to identify an interesting question in need of non-standard statistics.

__Possible colloquium topics: __

- I am open to advising colloquia in almost any area of statistical methodology or applications, including but not limited to: multiple testing, post-selection inference, post-prediction inference, model selection, model validation, statistical machine learning, unsupervised learning, or genomics.

**Allison Pacelli
**

__Research interests:__ Math Education, Math & Politics, and Algebraic Number Theory.

*Math Education. *Math education is the study of the practice of teaching and learning mathematics, at all levels. For example, do high school calculus students learn best from lecture or inquiry-based learning? What mathematical content knowledge is critical for elementary school math teachers? Is a flipped classroom more effective than a traditional learning format? Many fascinating questions remain, at all levels of education. We can talk further to narrow down project ideas.

*Math & Politics. *The mathematics of voting and the mathematics of fair division are two fascinating topics in the field of mathematics and politics. Research questions look at types of voting systems, and the properties that we would want a voting system to satisfy, as well as the idea of fairness when splitting up a single object, like cake, or a collection of objects, such as after a divorce or a death.

*Algebraic Number Theory. *The Fundamental Theorem of Arithmetic states that the ring of integers is a unique factorization domain, that is, every integer can be uniquely factored into a product of primes. In other rings, there are analogues of prime numbers, but factorization into primes is not necessarily unique!

In order to determine whether factorization into primes is unique in the ring of integers of a number field or function field, it is useful to study the associated class group – the group of equivalence classes of ideals. The class group is trivial if and only if the ring is a unique factorization domain. Although the study of class groups dates back to Gauss and played a key role in the history of Fermat’s Last Theorem, many basic questions remain open.

** **__Possible thesis topics:__

- Topics in math education, including projects at the elementary school level all the way through college level.
- Topics in voting and fair division.
- Investigating the divisibility of class numbers or the structure of the class group of quadratic fields and higher degree extensions.
- Exploring polynomial analogues of theorems from number theory concerning sums of powers, primes, divisibility, and arithmetic functions.

__Possible colloquium topics:__ Anything in number theory, algebra, or math & politics.

**Anna Plantinga**

__Research interests:__ I am interested in both applied and methodological statistics. My research primarily involves problems related to statistical analysis within genetics, genomics, and in particular the human microbiome (the set of bacteria that live in and on a person). Current areas of interest include longitudinal data, distance-based analysis methods such as kernel machine regression, high-dimensional data, and structured data.

__Possible thesis topics:__

- Impacts of microbiome volatility. Sometimes the variability of a microbial community is more indicative of an unhealthy community than the actual bacteria present. We have developed an approach to quantifying microbiome variability (“volatility”). This project will use extensive simulations to explore the impact of between-group differences in volatility on a variety of standard tests for association between the microbiome and a health outcome.
- Accounting for excess zeros (sparse feature matrices). Often in a data matrix with many zeros, some of the zeros are “true” or “structural” zeros, whereas others are simply there because we have fewer observations for some subjects. How we account for these zeros affects analysis results. Which methods to account for excess zeros perform best for different analyses?
- Longitudinal methods for compositional data. When we have longitudinal data, we assume the same variables are measured at every time point. For high-dimensional compositions, this may not be the case. We would generally assume that the missing component was absent at any time points for which it was not measured. This project will explore alternatives to making that assumption.
- Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology (or variations on existing methods) to answer an interesting scientific question.

__Possible colloquium topics:__

Any topics in statistical application, education, or methodology, including but not restricted to:

- Topics in applied statistics.
- Methods for microbiome data analysis.
- Statistical genetics.
- Electronic health records.
- Variable selection and statistical learning.
- Longitudinal methods.

**Cesar Silva
**

__Research interests__: Ergodic theory and measurable dynamics; in particular mixing properties and rank one examples, and infinite measure-preserving and nonsingular transformations and group actions. Measurable dynamics of transformations defined on the p-adic field. Measurable sensitivity. Fractals. Fractal Geometry.

__Possible thesis topics:__ *Ergodic Theory.* Ergodic theory studies the probabilistic behavior of abstract dynamical systems. Dynamical systems are systems that change with time, such as the motion of the planets or of a pendulum. Abstract dynamical systems represent the state of a dynamical system by a point in a mathematical space (phase space). In many cases this space is assumed to be the unit interval [0,1) with Lebesgue measure. One usually assumes that time is measured at discrete intervals and so the law of motion of the system is represented by a single map (or transformation) of the phase space [0,1). In this case one studies various dynamical behaviors of these maps, such as ergodicity, weak mixing, and mixing. I am also interested in studying the measurable dynamics of systems defined on the p-adics numbers. The prerequisite is a first course in real analysis. *Topological Dynamics.* Dynamics on compact or locally compact spaces.

__Possible colloquium topics:__

Topics in mathematics and in particular:

- Any topic in measure theory. See for example any of the first few chapters in “Measure and Category” by J. Oxtoby. Possible topics include the Banach-Tarski paradox, the Banach-Mazur game, Liouville numbers and s-Hausdorff measure zero.
- Topics in applied linear algebra and functional analysis.
- Fractal sets, fractal generation, image compression, and fractal dimension.
- Dynamics on the p-adic numbers.
- Banach-Tarski paradox, space filling curves.

**Mihai Stoiciu **

__Research interests:__ Mathematical Physics and Functional Analysis. I am interested in the study of the spectral properties of various operators arising from mathematical physics – especially the Schrodinger operator. In particular, I am investigating the distribution of the eigenvalues for special classes of self-adjoint and unitary random matrices.

__Possible thesis topics:__

Topics in mathematical physics, functional analysis and probability including:

- Investigate the spectrum of the Schrodinger operator. Possible research topics: Find good estimates for the number of bound states; Analyze the asymptotic growth of the number of bound states of the discrete Schrodinger operator at large coupling constants.
- Study particular classes of orthogonal polynomials on the unit circle.
- Investigate numerically the statistical distribution of the eigenvalues for various classes of random CMV matrices.
- Study the general theory of point processes and its applications to problems in mathematical physics.

__Possible colloquium topics: __

Any topics in mathematics, mathematical physics, functional analysis, or probability, such as:

- The Schrodinger operator.
- Orthogonal polynomials on the unit circle.
- Statistical distribution of the eigenvalues of random matrices.
- The general theory of point processes and its applications to problems in mathematical physics.

**Elizabeth Upton **

__Research Interests:__ My research interests center around network science, with a focus on regression methods for network-indexed data. Networks are used to capture the relationships between elements within a system. Examples include social networks, transportation networks, and biological networks. I also enjoy tackling problems with pragmatic applications and am therefore interested in applied interdisciplinary research.

__Possible thesis topics: __

- Regression models for network data: how can we incorporate network structure (and dependence) in our regression framework when modeling a vertex-indexed response?
- Identify effects shaping network structure. For example, in social networks, the phrase “birds of a feather flock together” is often used to describe homophily. That is, those who have similar interests are more likely to become friends. How can we capture or test this effect, and others, in a regression framework when modeling edge-indexed responses?
- Extending models for multilayer networks. Current methodologies combine edges from multiple networks in some sort of weighted averaging scheme. Could a penalized multivariate approach yield a more informative model?
- Developing algorithms to make inference on large networks more efficient.
- Any topic in linear or generalized linear modeling (including mixed-effects regression models, zero-inflated regressions, etc.).
- Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology to answer an interesting scientific question.

__Possible colloquium topics: __

- Any applied statistics research project/paper
- Topics in linear or generalized linear modeling
- Network visualizations and statistics