Faculty members at Union conduct research in fields as diverse as commutative algebra, topological data analysis, fair division, voting theory, statistics, algebraic topology, simplicial and differential geometry, medical imaging, the history of mathematics, and more.
Please read below for detailed descriptions of faculty research interests.
simplicial geometry, topology
My mathematical research centers around polyhedral geometry in three and four dimensions. I have studied immersed surfaces in space that have the fewest possible vertices, and surfaces that have a special cutting property called “tightness”, where any plane will divide the surface into at most two pieces. My most significant result represents one of the few cases in low dimensions where the polyhedral theory differs in a significant way from the smooth case. Much of my work involves computer software that I have developed, and I contribute to a number of open source projects, including MathJax (for displaying math notation on the web) and WeBWorK (an on-line homework system used at Union). I have been active in exploring how to communicate mathematics in new ways since the earliest days of the internet.
representation theory, Lie groups
My research focused on the representation theory of Lie groups. In particular, I looked at how certain representations produced by cohomological induction fit into the Langlands classification scheme.
computational geometry and topology, differential topology
I use techniques from algebraic and differential topology to address questions in image analysis and high-dimensional data analysis. On the image and shape analysis side, one of the main tools I use is Blum’s medial axis, a skeleton-like topological structure that captures the shape and geometric properties of a region. On the data analysis side, the techniques I utilize fall under the heading of topological data analysis, a relatively new branch of mathematics that focuses on discovering global and local structure within a dataset and finding meaning in the “shape” of the data. I am interested in applications of topology to problems arising in image recognition, machine learning, and geometric data modeling.
number theory, arithmetic geometry
I work in number theory, a classical branch of mathematics which is primarily focused on understanding two things: the properties of prime numbers and solutions to polynomial equations. Using special geometric objects like elliptic curves and modular forms, for each prime number p we can construct p-adic Galois representations. Each of these Galois representations allows us to gain a little bit of information about all polynomial equations at once. I study various properties of these Galois representations. I’m especially interested in families of Galois representations which arise from geometric objects which are congruent mod p.
Big Data analytics, statistical engineering, experimental design
Much of my early research was in the area of regression analysis, especially shrinkage estimators. In the private sector I developed a greater appreciation for, and interest in, experimental design methods. I have recently investigated Big Data analytics, particularly how and why things can go wrong when analyzing massive data sets. This ties to the discipline of statistical engineering, which emphasizes effective integration of multiple statistical and non-statistical methods in an overall approach to scientific inquiry. I am currently conducting research into how statistical engineering can provide effective strategies for attacking Big Data problems.
geometric analysis, general relativity
I work in geometric analysis, emphasizing connections with general relativity. Einstein’s theory of general relativity describes the universe as a spacetime, which is a four-dimensional continuum containing all points and events, past, present and future. Gravitational effects (for instance, due to a black hole) manifest through the curvature of spacetime, and thus geometry plays an important role in the theory. My research typically involves scalar curvature and explores connections between mass and geometry, including “quasi-local” mass and the total “ADM” mass of a spacetime. My current interests include convergence of sequences of asymptotically flat manifolds, Bartnik’s quasi-local mass conjectures, and codimension-two geometric flows within a spacetime.
homotopy theory, homological algebra
I am interested in algebraic topology, especially homotopy theory. My work has focused primarily on the calculus of homotopy functors, and, in particular, on developing alternative models for the calculus of homotopy functors for use in algebraic settings and more general homotopy-theoretic contexts. Recent work involves developing a general framework for understanding functor calculus that includes the calculus of homotopy functors and orthogonal calculus and provides a means for developing new calculus theories as well. I have also worked with undergraduates on problems in knot theory, especially questions about intrinsically linked and knotted graphs.
commutative algebra, homological algebra
My main research interests lie in the field of commutative algebra, with connections to algebraic geometry and homological algebra. Early in my career, my research was focused on the use of homological methods in commutative algebra. More specifically, I studied Gorenstein dimensions of modules over commutative local rings. In recent years, my research has shifted directions to include problems concerning pairs of commuting nilpotent matrices. My interest in this project originates from my commutative algebra background and my algebraic geometry interests, while the nature of the study has invited the use of additional tools from algebraic combinatorics and representation theory.
algebraic topology, unstable homotopy theory
I study algebraic topology, specifically unstable homotopy theory. I started out by studying the unstable Adams spectral sequence, in a problem related to the Sullivan Conjecture on maps from projective spaces. I returned to the unstable Adams spectral sequence in two later papers on the infinite orthogonal group SO. Most of my recent work has to do with connections between the Goodwillie calculus and the Whitehead Conjecture (proved by Kuhn and Priddy), along with the analogous connections between the orthogonal calculus and an unproved version of the Whitehead Conjecture for connective complex K-theory.
history of mathematics
I study the historical development of math, astronomy and related subjects, mostly in Sanskrit, Arabic and Latin texts from before the twentieth century. Research travel takes me most often to India, where there are tens of thousands of manuscripts of little-known early scientific works that I use in piecing together this history. Topics I’ve published on include the early history of numerical approximation methods in Sanskrit texts, different approaches to spherical trigonometry in Islamic and Indian astronomy, and Euler’s study of Indian calendar computations. These days I’m collaborating with a colleague in New Zealand on a study of algorithms in Indian astronomical tables; we’re hoping to break out some of the work into undergraduate research projects (no Sanskrit knowledge required!).
category theory, algebra, logic
I have been teaching mathematics at Union College for over three decades. During that time I have published two books and numerous articles on various aspects of category theory. From 2000-2008 I served as Dean of Studies at the college, overseeing the curriculum and the academic lives of the student body. After resigning as dean, I returned to the mathematics classroom with a renewed vigor and enthusiasm and have taught the First-Year Seminar on critical reading and writing. My attention has turned from mathematical research to writing fiction, with a story recently appearing in Prime Number.
set theory, logic, simple games, social choice, mathematical political science
My graduate training was in the field of mathematical logic, and I spent the first fifteen years of my career doing infinitary combinatorics. Most of my work involved ultrafilters on omega, ideals on uncountable cardinals, and partition theory (including a bit of work with finite Ramsey theory). I spent the following fifteen years with a number of questions from the area of “fair division” and with some topics arising from the theory of voting. Here, I was primarily studying simple games. For the past decade I have returned to set theory with somewhat of a focus on coordinated inference as captured by so-called hat problems.
number theory, function fields
I work in number theory and function fields. While number theory is primarily concerned with integers, the area of function fields is primarily concerned with polynomials. One can add, subtract, and multiply polynomials just as one can with integers and this analogy can be extended in many directions. By exploiting this analogy we can better understand both worlds. Of particular interest to me are multizeta values. First studied by Euler in an attempt to understand zeta values at odd positive integers, multizeta values are now studied due to their connection with mixed Tate motives. I study relations between multizeta values in function fields of a positive characteristic and the algebraic structures that arise from them.
differential geometry, Kähler geometry
The overarching motivation in my research is the classification and explicit construction of Riemannian metrics with properties that generalize the Einstein property. For the first many years of my career I studied this in the realm of Kähler Geometry. I mostly focused on extremal Kähler metrics but I also considered constant scalar curvature metrics, (generalized) Ricci solitons, weakly Bochner-flat metrics, and other Kähler metrics with special geometric properties.
In the last few years I have expanded this interest to include the odd-dimensional sibling of Kähler geometry, namely Sasakian Geometry.
fluid dynamics and turbulence, medical image processing and analysis, mathematics in health care
I have always been motivated to devote my research to developing effective methods and models, that can help increase the accessibility and accuracy of clinically-meaningful information embedded in the complex data, in order to improve the health care outcome. I have worked on 3D vascular segmentation using a sequential Monte-Carlo approach, flat field correction of digital X-ray detectors, and ultrasound modulated optical tomography assessment of osteoporosis. My current work includes attenuation compensation in medical ultrasound images, developing a 3D ultrasound image acquisition protocol, and fusion with CT images for treatment planning.
mathematical logic, set theory, game theory, voting and political power, social choice theory, fair division
My early research was in combinatorial set theory, especially large cardinals and ideals and filters on Pκλ. Today, I work on applications of mathematics to the social sciences: voting and social choice, fair division, and cooperative game theory. I’ve enjoyed co-authoring papers with mathematicians, political scientists, economists, and undergraduates from Canada, Catalonia, France, Ireland, Italy, Turkey, Venezuela, and the U.S. I’m on the editorial board of Mathematical Social Sciences and co-author, with Alan D. Taylor, of the monograph Simple Games (Princeton, 1999). I’m most attracted to fundamental issues in the social sciences that lead to questions of independent combinatorial or geometric interest. For a taste, try our online interactive rubber band voting simulator (with Davide Cervone).
Research descriptions for faculty emeriti
logic, set theory, fair division
I began my research career in set theory. In particular, my interests were in large cardinal theory, which is the study of very large infinite sets. After about fifteen years in this field, I moved into game theory, focusing specifically on fair division. This involves the allocation of goods among a collection of players, where the goals include both fairness and efficiency. I worked on both abstract existence results and on algorithms in this area. After about fifteen years in this field, I became interested in the ancient Greek foundations of modern mathematics. I developed a general education course on this subject, called “Ancient Greek Mathematics”.
exponentiability, double categories, toposes, locales, quantales
My research involves using category theory, especially adjoint functors, to draw analogies between different areas of mathematics. Much of this work concerns characterizing exponentiable morphisms in non-locally cartesian closed categories (including topological spaces, locales, toposes, posets, and affine schemes), and finding relationships between these characterizations. I am also interested in structures which capture similarities between different mathematical objects, e.g., quantales (which relate the lattice of ideals of a ring to that of the open subsets of a space) and double categories (which relate topological spaces, locales, quantales, toposes, posets, modules, and small categories).
number theory, formal groups
Most of my research is in the area of arithmetic geometry and in particular, formal groups, but more recently, I’ve become interested in a problem related to both formal groups and local (p-adic) dynamical systems. While thinking about a conjecture of Lubin involving power series that commute under composition, I started working with commuting polynomials that have coefficients in the complex numbers. A complete classification of these polynomials was given around 1920 by Ritt and Julia, working independently. That result is easily stated, but the proof is topological and fairly deep. My current work involves looking at commuting polynomials from an algebraic point of view and trying to apply formal group theoretic techniques, when possible, to gain a better understanding of the problem.