Math/Stats Thesis and Colloquium Topics

April 2017

Math/Stats 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/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 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.

 

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.  There is also 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.

 

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.  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 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

 

Investigate superinvariants, which are related to the standard invariants given by bridge number, unknotting number, crossing number and braid number.

 

Generalize concepts known for knots to virtual knots.

 

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

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:

 

1) Mathematical modeling of invasive species

 

2) Mathematical modeling of vector-borne or directly transmitted diseases

 

3) Developing mathematical models to manage vector-borne diseases through vector control

 

4) 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, especially those related to biology

 

 

Andrew Bydlon

Research interests:  Commutative algebra and algebraic geometry.

 

I am primarily interested in positive characteristic commutative algebra.  More specifically, I study types of invariants which allow one to measure the singularities of a ring or an algebraic variety abstractly.  These techniques include the multiplier and test ideal, tight closure, F-signature, and the Hilbert-Kunz Multiplicity.  In addition, I often use these tools to relate the singularities of a variety to a general hypersurface and vice-versa (for example, studying a surface by studying ‘most’ of the curves contained within it).

 

Possible thesis topics:

 

  • How do singularities restrict to a hypersurface? g. F—injective or F-rational singularities

 

  • Behavior of singularities in flat families.

 

  • Open questions pertaining to the test ideal and non-F-pure ideal.

 

  • Measuring the smallest degree hypersurfaces through a given collection of points.

 

  • Questions over finite fields.

 

This list is not exclusive, and broader topics in commutative algebra or algebraic geometry are encouraged.

 

Possible colloquium topics:

 

Any topics in algebra and especially commutative algebra or algebraic geometry.

 

 

Cory Colbert

Possible colloquium topics:

Any topic in abstract algebra or number theory.

 

Richard De Veaux (On Leave 2017-2018)

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

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 (On Leave Fall 2017)

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 and possible thesis topics: Representation theory, enumerative and algebraic combinatorics, graph theory, and poset theory.

 

Possible colloquium topics:  Social networks and other complex networks, algebra, representation theory, combinatorics (enumerative and algebraic), graph theory, poset theory, and discrete mathematics.

 

 

Brianna Heggeseth

Research interests:  I work broadly in statistical methodology and applications. My current work involves the study and development of statistical models that can uncover structure and patterns in data specifically in longitudinal and spatial data that have inherent dependence.  Additionally, I work in collaboration with scientists in a variety of fields such as epidemiology and environmental science to apply statistical methods to answer scientific research questions.

 

Possible thesis topics:

 

  • Cluster analysis for longitudinal data and functional data

 

  • Regression trees for longitudinal data

 

  • Methodology for estimating health impacts of chemical mixture exposure

 

  • Applying statistical methods to real data sets in innovative ways, usually in collaboration with a scientist or social scientist.

 

Possible colloquium topics:

 

Any topics in statistics, such as

 

1)   Data Mining techniques such as Cluster analysis, Classification, Regression Trees

 

2)   Longitudinal or spatial methodology

 

3)    Applying statistical methods to real data sets in innovative ways

 

 

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.

 

  • Optimal control theory.

 

  • Evolutionary dynamics.

 

  • Game Theory.

 

  • Dynamical systems.

 

Possible colloquium topics:

 

Any topics in mathematics, dynamical systems, mathematical models in life sciences, engineering, and other fields, applied mathematics in general.

 

 

Bernhard Klingenberg

Possible thesis topics:

 

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

Research Interests: Commutative and Homological Algebra.  Homology and cohomology groups are used to measure the size and shape of algebraic objects like commutative rings or topological spaces.  Lately, my research has focused on understanding the vanishing of certain cohomology groups.  As a result I also have a growing interest in algebraic topology and the representation theory of finite dimensional algebras.

 

Possible Colloquium topics:   I’d be happy to advise any topics in Algebra.

 

 

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:

  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 what sets of prime ideals of a complete local ring can occur in formal fibers of ideals with height n where n ≥1.

3. Characterize which complete local rings are the completion of an excellent unique factorization domain.

4. Compute the tight closures of specific ideals in rings.

5. Explore the relationship between the formal fibers of R and S where S is a flat extension of R. For which ideals in excellent rings does tight closure and completion commute?

 

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 and discrete geometry 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 is 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.

 

  • 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?

 

  • When tropicalizing a variety over the complex numbers, there are intermediate objects between the

variety and the tropical variety called amoebas.  What can we say about their geometry?

 

  • 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.

 

  • 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?

 

  • Develop algorithms for computing properties of algebraic and/or tropical 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 number 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.

 

Allison Pacelli (On Leave Spring 2018)

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:

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.

4)    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 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:

  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.

4.   P-adic dynamics. P-adic numbers, dynamics on the p-adics.

5. Banach-Tarski paradox, space filling curves.

6. Random walks

 

 

Chad Topaz

Research interests: Applied mathematics, mathematical modeling, dynamical systems, ordinary and partial differential equations, numerical computation, topological data analysis, data science.

 

My research uses diverse mathematical tools to examine complex systems in the natural and social sciences. I am most interested in systems that form patterns or display other sorts of collective behavior. Examples include: biological aggregations such as insect swarms and fish schools; chemical reaction-diffusion systems; fluid surface waves; spatial vegetation patterns in ecology; gender representation in STEM fields; and much more.

 

Possible thesis topics:

 

1) Topological data analysis of agent-based models. Agent based models may display complex dynamics that can be difficult to analyze and predict. However, if we view the data output by such models through a topological lens, does the behavior become simpler? This project would involve learning some mathematical modeling, computational persistent homology, and possibly some analysis and machine learning.

 

2) Modeling vegetation patterns. Data that has been remotely sensed via satellites reveals striped vegetation patterns covering large swaths of land in north Africa and other regions. This project would involve learning about and possibly creating mathematical models for these patterns, tying the models closely to large data sets, and using the models to assess the factors most responsible for the formation of the patterns.

 

3) Locust swarms. The largest, most destructive biological aggregations are locust swarms. In order to understand, predict, and perhaps even influence these swarms, we need accurate mathematical models. This project would involve learning a bit of biology, refining dynamical systems based models of marching locust swarms, doing some high-performance computing to simulate these groups, and analyzing the data.

 

4) Data science, gender, and STEM. Women are underrepresented at nearly every level of the profession in STEM fields. In this project, we will use data science tools to assess patterns of gender representation in various subfields and at various levels of professional development.

 

Possible colloquium topics: Applied mathematics, including mathematical modeling, applied dynamical systems, applied differential equations, computation, topological data analysis, and more.

 

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.

 

 

Laura 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 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?
  • Spatial/spatio-temporal analysis 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 optimization 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 generalize 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