Math/Stats Thesis and Colloquium Topics

Updated: April 2021

Math Thesis and Colloquium Topics 

THE DEGREE WITH HONORS IN MATHEMATICS

The degree with honors in Mathematics 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 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 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.

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 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

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. I am also interested in tiling theory.

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 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.
  • Show that if infinitely many Dehn fillings on a manifold are hyperbolic, then the manifold is hyperbolic.
  • 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.)
  • 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.
  • Other related topics.

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

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

 

Thomas Garrity

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.

 

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:

  • What prime ideals of C[[x1,…,xn]] 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.
  • 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.
  • Using completions to construct Noetherian rings with unusual prime ideal structures.

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. 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? How about r^th gonalities–what’s the complexity of computing those? 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.

 

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.

 

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. 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.
  • 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