# Faculty Interests

Here is a list of faculty interests and possible thesis topics.  You may use this list to select a thesis topic, to choose a colloquium topic or you can use the list below to get a general idea of the mathematical interests of our faculty.  Some of you may also wish to prepare ahead and be ready to give your colloquium talk by the beginning of the fall semester.

Research interest:
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:

1. 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 axis.
2. Investigate superinvariants, which are related to the standard invariants given by bridge number, unknotting number, crossing number and braid number.
3. 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.
4. Determine how many knots must exist in a given graph, no matter how that graph is placed in space.
5. Investigate toroidally alternating knots, knots which can be placed on a torus such that their crossings alternate between over and under.  When is a toroidally alternating knot trivial?
6. Investigate which knots have totally geodesic Seifert surfaces.
7. 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.
8. Investigate geodesics on hyperbolic surfaces.  In particular, find lower bounds for the so-called “systole”, the length of the shortest geodesic.
9. Explore how cusp diagrams determine a hyperbolic 3-manifold.

Possible Colloquium topics:
Topics in mathematics, including but not restricted to subjects related to the thesis topics above.

Research Interests:
I am interested in the interactions of algebra with topology and geometry, most recently with configuration spaces, graphs, polyhedra, and tilings.  I am also interested in aspects of origami, biology, cartography and visual display of information.

Possible thesis topics:

1. Spaces of particle collisions.
2. Combinatorics of polyhedra and polytopes in higher dimensions.
3. Computational geometry.
4. Mathematical Biology

Possible Colloquium topics:
Anything involving pictures.

## Richard DeVeaux

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:

1. 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.
2. Ensemble Methods. One of the exciting areas to come out of machine learning applied to statistical modeling is model combination. How can we take the output of many models and combine them for better accuracy and prediction?
3. 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, including but not restricted to subjects related to the topics above.

## 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:

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

Possible Colloquium topics:
Any interesting topic in mathematics.

## Stewart Johnson

Research interest:
Dynamical systems, ordinary differential equations, mathematical modelling, control theory, mathematical biology, evolutionary dynamics.

Possible thesis topics:

1. Mathematical modelling using dynamical systems and differential equations.
2.  Continuous or hybrid dynamical systems.
3. Optimal control theory.
4. Evolutionary dynamics.
5. Mathematical biology.

Possible Colloquium topics:
Topics in mathematics, including but not restricted to subjects related to the thesis topics above, and in particular: Mathematical models in life sciences, engineering, and other fields.  Applied math in general.

## Bernhard Klingenberg

Research Interests:
Statistics and Biostatistics
Topics in Categorical Data Analysis and Simultaneous Inference
For the most up to date information on my research, visit http://www.williams.edu/~bklingen>www.williams.edu/~bklingen.

Possible thesis topics:

1. Construction of simultaneous confidence intervals for binomial proportions.
2. Models for longitudinal observations (Generalized Linear Mixed Models).
3. Conditional inference for generalized linear models with asymmetric link functions.
4. Any applied data analytic method that involves more advanced statistics.
5. Statistical consulting with potential collaboration of another scientist, economist or social scientist.

Possible Colloquium topics:
Any topic in statistics; Bootstrap and Permutation tests; Exact methods for 2×2 contingency tables (Fisher’s exact test); An applied statistics project.

## 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.

Possible thesis topics:
Topics in Commutative Algebra including:

1. 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.
2. Characterize which complete local rings are the completion of an excellent unique factorization domain.
3. Compute the tight closures of specific ideals in rings.
4. For which ideals in excellent rings does tight closure and completion commute?

Possible Colloquium topics:

Topics in mathematics, including but not restricted to subjects related to the thesis topics above.

## Steven Miller

For more information and references, see http://www.williams.edu/go/math/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:

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

Possible Colloquium topics:
same as the above, 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….

## Frank Morgan

Research interests:  Geometry, minimal surfaces such as soap bubbles, manifolds with density.

Possible thesis topics:

1. Isoperimetric problems and soap bubble clusters of various sorts.  The 1990 “SMALL” Geometry Group proved the so-called Double Bubble Conjecture in R2.  In 1999/2000 my collaborators and I proved the Double Bubble Conjecture in R3, and the Geometry Group extended the proof to bubbles in R4.  Many similar open problems involve optimal structures.
2. Manifolds with density (like the density in physics that you use to compute mass, used in recent proof of Poincaré Conjecture).  Basic properties and isoperimetric problems.
3. Tilings of the plane. We’d like to prove the least-perimeter way to tile the plane with pentagons.

No special background needed, just a desire and willingness to work.

Colloquium topics:
All kinds of geometry and analysis, the Big Questions, teaching,  Monthly articles, or anything else you are interested in.  There’s always something good in the math news or popular media.

## Allison Pacelli

Research Interests:
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.

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:

1.  Investigating the divisibility of class numbers of quadratic fields and higher degree extensions.
2. Investigating the structure of the class group.
3. Exploring polynomial analogues of theorems from number theory concerning sums of powers, primes, divisibility, and arithmetic functions.

Possible Colloquium topics:
Topics in mathematics, including but not restricted to, subjects related to the thesis topics above; in particular: anything in algebra or number theory and many things in mathematical politics 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.

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.

Possible Colloquium topics:
Topics in mathematics, including but not restricted to subjects related to the thesis topics above, and in particular:

1. 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.
2. Topics in applied linear algebra and functional analysis.
3. Fractal sets, fractal generation, image compression, and fractal dimension.

## 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:

1. 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.
2. Study particular classes of orthogonal polynomials on the unit circle.
3. Investigate numerically the statistical distribution of the eigenvalues for various classes of random CMV matrices.
4. Study the general theory of point processes and its applications to problems in mathematical physics.

Possible Colloquium topics:
Topics in mathematics, including but not restricted to subjects related to the thesis topics above.

## Qing Wang

Research Interests:
The primary focus of my thesis research has been a set of topics concerning U-statistics (a class of unbiased estimators) and their practical implementation. This area is quite old, with many important results first appeared in Hoeffding (1948).  More generally, I am interested in modern nonparametric methodology in statistics.

My research interests include applications of U-statistic methods in risk estimation and cross-validation. Cross-validation is a technique widely used to validate a statistical methodology on an independent sample in order to guard against overfitting problems. It has many important applications in areas such as nonparametric statistics and machine learning. Although it is unconventional, I treat cross-validation as a U-statistic methodology and have discovered several important results. I am continuing to study the applications of the cross-validation criterion in kernel density estimation and model selection.

In addition, I am interested in resampling schemes which help to answer the question how one can gain more information of the true distribution by drawing subsamples from the data set repeatedly. I also have interest in interpolation and extrapolation techniques with applications in variance estimation, bandwidth selection, quartile estimation, and many other topics.

Possible thesis topics:

1. Kernel density estimation: the past, the present, and the future.
2. Studying the bootstrapping method (i.e. resampling with replacement). Investigate its limitations and the effect of resampling size on its performance.
3. Studying the subsampling method (i.e. resampling without replacement) along with comparison to bootstrapping method.
4. Application of U-statistic cross-validation methodology in the nonparametric kernel regression estimator.
5. Investigate the limitations of established model selection criteria for mixture models. Compare the performance of the U-statistic model selection criterion with others in such a situation based on a real data set.
6. Apply statistical modeling techniques on real data sets in agricultural science, social science, actuarial science, or other areas.

Possible Colloquium topics:

Topics in statistics, including but not restricted to subjects related to the thesis topics above.