Updated: May 2020

**Math/Stats Thesis and Colloquium Topics **

THE DEGREE WITH HONORS IN MATHEMATICS OR STATISTICS

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 “mini-thesis” 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 and giving a second colloquium talk. Written work is a possible component.

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 2020 – 2021)**

__Research interests__: Topology. 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. 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
*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. - Investigate the stick number of knots, which is the least number of sticks glued end-to-end to make a given knot. Still unknown for two twisted strands. Can also consider lattice stick knots, where all sticks are parallel to the
*x,y,z*axes. - Investigate superinvariants, which are related to the standard invariants given by bridge number, unknotting number, crossing number and braid number.
- Investigate geometric degree of knots, which is the greatest number of times a plane intersects a knot minimized over all ways to put the knot in space.
- Determine how many knots must exist in a given graph, no matter how that graph is placed in space.
- Investigate knots in a thickened surface. When are they hyperbolic?
- Investigate which knots have totally geodesic Seifert surfaces.
- Investigate the width and cusp thickness of quasi-Fuchsian surfaces in hyperbolic 3-manifolds. Quasi-Fuchsian surfaces generalize totally geodesic surfaces. Show that many surfaces in knot complements are quasi-Fuchsian.
- Investigate geodesics on hyperbolic surfaces. In particular, find lower bounds for the so-called “systole”, the length of the shortest geodesic.
- Explore how cusp diagrams determine a hyperbolic 3-manifold.
- Other related topics.

__ __

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

**Julie Blackwood (On Leave Spring 2021)**

__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. My research focuses on resolving issues with large and/or high dimensional datasets. One of my research topics is variable selection for high dimensional data. I am interested in traditional and modern approaches to select a smaller number of variables from a large candidate set. I am especially interested in selecting variables for survival models, which study how risk factors influence the time until the occurrence of a certain event of interest. Another part of my research studies repeated measurements of subjects over time, or the so called “intensive longitudinal data.” My research involves applying existing methods and developing new procedures to model the correlated observations and capture the time-varying effect. Additionally, I am also interested in analyzing the rhetorical styles in English text data using statistical methods.

__Possible thesis topics: __

- Building survival models for some time-to-event of interest. The traditional survival models usually study the relationship between several risk factors and time to the occurrence (or recurrence) of a certain disease. We may evaluate whether the treatment is effective, identify important risk factors and compare the survival features among different demographic groups.

On the other hand, survival models can be extended to be used in other settings, e.g. modeling large dynamic networks. We could answer questions like how current network structures predict future development of the network.

- Variable selection using modern techniques by penalization. For example, for survival models in the previous topic, we could include all potentially relevant risk factors (or network features) initially, then select the most important ones for a simpler model with easy interpretations. Other examples are not restricted to models previously mentioned.
- Applying statistical models/methods on time-varying repeated measurements. We may examine the time-varying profile of the measurements over time, identify a smooth function to capture the major pattern and even compare between individuals/groups. We could also study the time-varying effects of predictor variables on the response variable.
- Analyzing English text data. We shall analyze English text dataset processed by the environment called “Docuscope” with tools for corpus-based rhetorical analysis. The data have hierarchical structure and contain very rich information about the rhetorical styles being used. We could apply statistical models to reduce dimensions and have a more insightful understanding of the text.

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

**Josh Carlson**

__Research interests:__ Graph theory and combinatorics. In particular, I study topics such as zero forcing, cops and robbers, and throttling.

I am interested in parameters that model spread or infection across a network. An infection process on a graph uses initial resources (often vertices that are infected in some way) to progressively spread the infection in each time step. Zero forcing is an example that iteratively changes the color of vertices to blue. Throttling is the idea of balancing the cost of the initial resources with the cost of the time steps in order to be as cost-effective as possible. Throttling numbers measure this extreme cost-effectiveness. The game of cops and robbers is connected to searching algorithms and can also be throttled.

__Possible thesis topics:__

- Explore throttling for a variant of zero forcing called “loop zero forcing”.
- Can we count the number of initial zero forcing sets that realize throttling numbers in various graph families?
- Investigate throttling for variants of cops and robbers. Alternatively, what happens to standard cop-throttling on directed or oriented graphs?
- Is there a structural characterization of cop-throttling numbers? What graph operations monotonically decrease the cop-throttling number?

__Possible colloquium topics:__ I am happy to advise a colloquium talk in any topic related to graph theory and combinatorics. I will also advise topics in the intersection of linear algebra and graph theory including combinatorial matrix theory and spectral graph theory.

**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’ swim data and a handicapped race in California, but there are world records for each age group and every events in running and swimming that I’ve not incorporated. Masters’ events in running would be another source for data.
- Social Networks and unstructured data. Classical statistics deals with quantitative and categorical variables, but what happens when the variables have even less structure? Using text mining and other recent work from statistics and machine learning can we figure out how people feel about a topic by analyzing what they say as well as their actions?
- 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.
- Text Mining. Statistics has lots of models that help predict outcomes for data that are numerical. But what if the data are text? What can we say about documents based only on the words they contain? Can we use comments in surveys to help answer questions traditionally modeled only by quantitative variables?
- 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:__

Topics in statistics, such as:

- Human Performance and Aging.
- Social Networks and unstructured data
- Variable Selection.
- Text Mining.
- Problems at the interface
- Applying statistical methods to problems in science or social science.

**Thomas Garrity (On Leave 2020 – 2021)**

__Research interest:__ Geometry and Number Theory.

I work in algebraic and differential geometry and in number theory. I am interested in the geometry of functions (polynomials for algebraic geometry and differentiable functions for differential geometry) and in the Hermite problem (which asks for ways to represent real numbers so that interesting algebraic properties can be easily identified).

__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.

**Eva Goedhart (On Leave Fall 2020 but taking thesis students)**

__Research interests__: Elementary and algebraic number theory.

My main research focus is in solving families of Diophantine equations, these are polynomial equations in 2 or more variables that are solving in integers. Methods for solving such equations vary widely; some needing complicated structures that required 350 years of mathematics to solve, while others can be solved by simply looking at whether the numbers have to even or odd. Most families of equations require something in the middle and often require a mix of various methods.

__Possible thesis topics__:

- Solving a family of equations requiring Lehmer pairs and Lehmer numbers. This is an elementary number theory approach and would require some modular arithmetic.
- Solving a special case of a family of equations that require continued fractions and inequalities. This will involve writing a few small computer programs, analyzing the data, and algebraically solving some equations, and manipulation of inequalities.
- Solving a family of equations using logarithms and continued fractions. This will require manipulation of algebraic formulas and a little bit of computer programming.
- Other: I’m open to finding topics of interest in elementary number theory, such as happy numbers and the like.

__Possible colloquium topics__: I’m happy to advise a wide variety of topics in number theory and in algebra.

**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.

**Pamela Harris**

__Research interest:__ My research is in the area of algebraic combinatorics. I like to use combinatorial arguments and techniques to enumerate, examine, and investigate the existence of discrete mathematical structures with certain properties. Areas of interest for these applications are in algebra, discrete geometry, number theory, and graph theory, but there is no limit to the applications: if you can count it, I am interested!

__Possible thesis topics:__ Here are some sample ideas for things we could investigate:

- Finding new closed formulas for certain classes of vector partition functions.
- Finding combinatorial bijections between partition functions and other combinatorial objects. For example, some recent Williams student thesis work connected partition functions to juggling sequences!
- Determine the (discrete) volume of families of flow polytopes.
- Create and enumerate new integer sequences and their properties. These sequences would arise from generalizing known combinatorial families of objects. For example, generalized parking functions and generalized happy numbers.
- Compute bounds and whenever possible exact (t,r) broadcast domination numbers for specific families of graphs.
- Study permutations via their peaks, valleys, pinnacles, vales, and other such qualities.
- Investigate graph labeling/coloring problems.
- q-counting problems. This refers to the ability to introduce a parameter ”q” when counting objects. Then the answers to q-counting problems are polynomials, whose evaluation at q=1 recovers the total number of objects you were interested in counting.
- An integer lattice (r,s) point is visible from the origin if it is the only lattice point on the straight line segment connecting the origin to that point. Investigate generalized lattice point visibility problems where you view the points through other interesting curves and not just straight lines.

__Possible colloquium topics:__ Any topic with a combinatorics flavor

**Stewart Johnson**

__Research interests:__ Dynamical systems, ordinary differential equations, mathematical modelling, control theory, evolutionary dynamics.

__Possible thesis topics: __

- Mathematical modelling using dynamical systems and differential equations.
- Continuous or hybrid dynamical systems.
- Optimal control theory.
- Evolutionary dynamics.
- Game Theory.

__Possible colloquium topics:__

Any topics in mathematics, mathematical models in life sciences, engineering, and other fields, applied mathematics in general, such as:

- Mathematical modelling using dynamical systems and differential equations.
- Continuous or hybrid dynamical systems.
- Optimal control theory.
- Evolutionary dynamics.
- Game Theory.

**Bernhard Klingenberg**

** **

__Research interests:__ I’m interested in the analysis and statistical modeling of categorical response data, ranging from simple binary or longitudinal ordinal response to multivariate binary observations. Such data are common in the social and political sciences, and in medical and public health research. One particular application is in drug safety, where several adverse events (and their severity) are observed and compared between a treated group and an untreated group. Such settings naturally lead to considerations of multiplicity, the compounding of errors when conducting multiple inferences, and so a further interest of mine is in multiple comparison procedures for categorical responses. Since I believe it is better to estimate effect rather than just to establish significance, I’m interested in methodology for constructing simultaneous confidence intervals.

__Possible thesis topics:__

Any and all methods for categorical data analysis

__Possible colloquium topics:__

Any topic in statistics, such as:

- Bootstrap and permutation tests
- Exact methods for categorical data
- Simultaneous Inference and Multiple Comparisons

**Haydee Lindo (On Leave 2020 – 2021)**

__Research interests:__ Commutative Algebra, Homological Algebra, Representation Theory

I work in commutative algebra, using tools from homological algebra and representation theory. I’m interested in generalizations of trace maps of matrices to modules over commutative rings. I apply the theory of these trace maps to study the vanishing of cohomology groups and to find new characterizations of commutative Noetherian rings.

__Possible thesis topics:__ Any topics in commutative algebra.

__Possible colloquium topics:__ I’m happy to advise any colloquium on any topic in mathematics.

**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:

- 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, which is the study of solution sets to polynomial equations. Such a solution set is called a variety. In particular, I focus on tropical geometry, which is a “skeletonized” version of algebraic geometry. We 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. I also study computational algebraic geometry, which uses various algorithmic tools to better study the geometry of varieties. I’m especially interested in this area when working over non-Archimedean fields, like the p-adics.

__Possible thesis topics:__ Anything related to the fields of tropical and algebraic geometry, or non-Archimedean fields. Here are a few example questions/topics.

- Given a subset of Euclidean space that “looks” like a tropicalization, when does it actually arise as the tropicalization of an algebraic variety?
- Given the equations for an algebraic variety, find a tropicalization that preserves the most information about the curve. (Such a tropicalization is called “faithful”.)
- A tropical curve can be thought of as a finite graph with lengths on each edge. Which such graphs arise as tropical curves, under various restrictions? For instance, with the tropical curve being embedded in the Euclidean plane.
- Tropical plane curves are dual to subdivisions of polygons with vertices at points with integer coordinates. What can we say about these polygons, and what do they tell us about the geometry of our tropical curves?
- We can model important problems from algebraic geometry using chip-firing games from graphs. There are tons of open problems about these games: given a graph, how can we most efficiently win a game? How can we compute numbers that measure the difficulty of these games? When can we win a game with chips spread out on the graph, versus concentrated on certain vertices? How can we relate these chip-firing games to other graph theoretic subjects, like treewidth?
- Study the combinatorics of higher-dimensional tropical varieties. For instance, a tropical surface of degree 4 contains a distinguished polyhedron with at most 63 facets. Which polyhedra are possible?
- Many results in classical algebraic geometry have analogs in tropical geometry that do not trivially follow from the original result. Prove such results in the tropical world, and study how the classical and tropical results interact.
- 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.)
- An algebraic variety is defined by an ideal in a polynomial ring. How can we find nice sets of generators for this ideal? What do these generators tell us about the associated tropical variety?
- Many objects that arise in linear algebra (such as pairs of commuting matrices) can be described using polynomial equations. How can we find nice collections of generating polynomials? What do they let us compute about the varieties?

__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, or graph theory. For instance, there are many results in classical algebraic geometry (about the geometry of plane curves, for instance) that would be perfect for a colloquium.

**Shaoyang Ning**

__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.

**Allison Pacelli**

__Research interests:__ Algebraic Number Theory and Math Education

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.

I am also very interested in the beautiful analogies between the integers and polynomials over a finite field and between number fields and function fields.

** **__Possible thesis topics:__

- Investigating the divisibility of class numbers of quadratic fields and higher degree extensions.
- Investigating the structure of the class group.
- Exploring polynomial analogues of theorems from number theory concerning sums of powers, primes, divisibility, and arithmetic functions.
- Topics in math education.

__Possible colloquium topics:__ I’m interested in advising any topics in algebra, number theory, or mathematics and politics, including voting and fair division.

**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 high-dimensional data, distance-based analysis methods such as kernel machine regression, feature selection, statistical learning, and structured data.

__Possible thesis topics:__

- 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?
- Microbiome volatility analysis. Sometimes the variability of a microbial community is more indicative of an unhealthy community than the actual bacteria present. This project will develop a method to characterize microbiome variability (“volatility”) and test for association with health outcomes.
- 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.
- Variable selection and statistical learning.
- Longitudinal methods.

**Cesar Silva (On leave 2020-2021)**

__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.

*Probability.* Markov shifts, information theory.

__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.
- P-adic dynamics. P-adic numbers, dynamics on the p-adics.
- Banach-Tarski paradox, space filling curves.
- Random walks.

**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.

**Chad Topaz **

__Research interests:__ Applied mathematics (dynamical systems, differential equations, mathematical modeling, topological data analysis), data science, social justice.

My research uses diverse mathematical tools to examine complex systems in the natural sciences and in the realm of social justice. Recent problems I have worked on include: biological aggregations such as insect swarms and fish schools; chemical reaction-diffusion systems; spatial vegetation patterns in ecology; diversity of artists in major U.S. art museums; power structures in the STEM community; sentencing disparities in the criminal justice system; and much more.

__Possible thesis topics:__

Projects in topological data analysis applied to natural and social systems. Data science and applied mathematics used for social justice. So that I can best serve you as an advisor, any thesis topic with me should make use of tools I am familiar with (see above) and the topic itself shouldn’t stray too far from areas I work on. Talk to me if you are interested.

__Possible colloquium topics:__

Applied mathematics, including mathematical modeling, applied dynamical systems, applied differential equations, computation, topological data analysis, quantitative social justice, and more.

**Laurie Tupper **

__Research interests: __Statistics, both application and methodology. I work with scientists and engineers in a range of areas on data sets of interest to them, adapting and developing statistical methods to answer research questions. Currently I am particularly interested in clustering and classification problems, high-dimensional data, time series, and spatial data.

__Possible thesis topics:__

- Clustering and distance. When classifying or clustering a dataset, both the clustering algorithm and the measurement of dissimilarity between observations are important. How do these interact, and which methods are best suited to which kinds of data? For data that can be treated in multiple ways (for example, as high-dimensional or spatio-temporal), what are the effects of using corresponding distance measures? How do we describe those effects?
- Spatial/spatio-temporal analysis, including with non-Euclidean distance. In a transportation network or a city, distances aren’t “as the crow flies.” How can we analyze dependence between points in this context?
- Concise expressions of complicated data. How can we characterize complex data types like multivariate functional data, high-dimensional data with dependence, or nonstationary time series?
- New combinations of statistical methods and real data sets, in collaboration with a scientist or social scientist.

__Possible colloquium topics: __Statistical methodology, application, and education, including but not restricted to subjects related to the topics above.

**Daniel Turek**

__Research Interests: __Bayesian statistics. My research involves studying and designing efficient algorithms for Markov chain Monte Carlo (MCMC) sampling of hierarchical (graphical) models. How can we optimally assign samplers based upon model structure? Under what circumstances are different sampling strategies advantageous? The larger goal is to design an automated procedure for producing a highly efficient, problem-specific, MCMC algorithm.

__Possible thesis topics:__

- Implementation of existing MCMC sampling algorithms into new, flexible MCMC software, to make them easily usable and testable.
- Testing a variety of discrete-valued sampling algorithms in different hierarchical modeling scenarios, to determine decision criteria for when different algorithms are the most efficient.
- Designing a decision-rule system for MCMC algorithm creation. This would involve designing a framework to define rules based upon model and algorithm properties (e.g. continuous or discrete? conjugate or non-conjugate? univariate or multivariate?, etc.), and incorporating this into flexible MCMC software to allow a higher-level of customization for MCMC algorithm creation.
- Parameterization of a (finite and well-defined) set of valid MCMC algorithms. A formal mathematical definition of such a set will allow for application of discrete optimisation algorithms to automate algorithm selection.
- Application of simulated annealing to such a parameterization (described above) to produce a valid exploratory algorithm. A first such algorithm has been created, but we will generalise this further, and broaden the space of MCMC algorithms considered.

__Possible colloquium topics:__

- Any applied statistics research project
- Bayesian analyses
- Testing various Bayesian sampling algorithms
- Model uncertainty and model averaging
- Bayesian model averaging

**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

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