Anne Shepler - US grants

This POWRE award supports a Visiting Professorship at the Department of Mathematics at Northeastern University to study differentials of isolated hypersurface singularities. This award will allow the PI
to interact with several leading researchers in her field. As part of the human resource development activities she will also sponsor the local graduate student seminar at Northeastern.

This research project combines theoretical expertise from cyclic homology and K-theory and computer algebra techniques to gather information about invariants of isolated hypersurface singularities. The most salient invariant of isolated hypersurface singularities is the so-called Tjurina number, or the dimension of the torsion module of differentials. She has identified the torsion module of differentials both as a Hodge-component of cyclic homology and as an ideal quotient. The latter identification has led to an efficient algorithm for computation of the number of generators exploiting Matlis duality in Gorenstein Artin Algebras.

As part of her research activities the PI will undertake the following three-part project:


1. The structure of isolated singularities:
The anticipated outcome of this research will be an upper bound for the number of generators and the length of the module of differentials for 3 dimensional hypersurface singularities.

2. Residues and Duality for isolated hypersurface singularities:
The goal will be an explicit description of residues and an investigation of the connection with inverse systems.

3. Description of the module of logarithmic differentials and connection with hyperplane/surface arrangements:
This part of the project will be computational in nature and will focus on the use and development of algorithms for Groebner basis computations in exterior algebras.

This POWRE project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

This award supports research in combinatorics and geometry with
applications to representation theory. The project investigates the
geometry and invariant theory of reflection groups. The PI explores
relations between complex polytopes, Coxeter-like complexes, Hecke
algebras, and the coinvariant algebra. Techniques may help develop a
Kazhdan-Lusztig theory of cells for complex reflection groups.

Reflection groups arise in nature as symmetry groups. For example, the
symmetries of the cube or dodecahedron (or any Platonic solid) form
a reflection group. These groups are generated by mirror reflections
(about hyperplanes) and appear in physics, biology, chemistry, and
computer science. Reflection groups also have a rich history of
connecting many different areas of mathematics: discrete geometry,
topology, singularity theory, arrangements of hyperplanes, Lie
theory, combinatorics, and invariant theory.

Shepler and Witherspoon will develop a theory of deformations expanding that for graded Hecke algebras. They will study deformations of skew group algebras that combine groups of symmetries with algebras of functions. Particular deformations of these algebras arose independently in work by many prominent mathematicians in representation theory and noncommutative geometry, but many open questions remain. Shepler and Witherspoon will answer some of these questions using new tools created by blending methods from invariant theory, combinatorics, homological algebra, and representation theory. They also will solve some basic open problems about the structure and cohomology of Hopf algebras and prove several conjectures on modular reflection groups, invariant theory, and arrangements of hyperplanes.

Objects throughout the natural world reveal themselves through their symmetries, for example, crystals, molecules, DNA, and quantum systems. When we deform an object, we alter or even break the symmetry. Remarkably often, we discover new attributes of the object after studying its deformations. Shepler and Witherspoon's research program on graded Hecke algebras and related deformations addresses a variety of mathematical fields and grows from the exploding interest the mathematical community shows in graded Hecke algebras. Hecke algebras are pervasive throughout mathematics, appearing in algebra, geometry, number theory, combinatorics, topology, statistics, harmonic analysis, mathematical physics, special functions, quantum groups, knot theory, and conformal field theory. Shepler and Witherspoon are active in the mathematical community, mentoring students and postdocs, collaborating with international experts, and organizing conferences and workshops. Their research program supports these broader activities.

The mathematical community shows expanding interest in the deformations Shepler and Witherspoon study, for example, in graded Hecke algebras and symplectic reflection algebras. These algebras pervade mathematics, appearing in algebra, geometry, number theory, combinatorics, topology, statistics, harmonic analysis, mathematical physics, and the theory of special functions (for example, they appear in the Langlands program, the theory of quantum groups, knot theory, and conformal field theory). Shepler and Witherspoon work towards a theory of skew group algebras and Hochschild cohomology that connects work by various prominent mathematicians working in different mathematical fields.

Their research leads to international talks and collaborations and results in publications in selective international journals. Shepler and Witherspoon dynamically engage the wider community by collaborating with international experts,organizing conferences and workshops, delivering invited talks in various countries, and working with outreach programs. They actively prepare junior colleagues and students (at all stages) for engaging careers which use mathematics. Their work supports research of several graduate students and summer research experiences for undergraduates.

This award provides support for two weekend conferences, one at the University of North Texas in Fall 2011 and the other at Oklahoma State University in Spring 2012. Each conference will feature two or three prominent guest speakers from outside the Texas-Oklahoma region, in addition to other participants including students, post-doctoral researchers, and junior faculty. Regional graduate students and researchers will also give talks describing their work. It is expected that the conferences will facilitate collaborations and interactions among the students and researchers in the region who work in the areas of Automorphic Forms, Representation Theory, and Number Theory. These conferences are planned as the first two of the Texas-Oklahoma Representations and Automorphic (TORA) conference series, pending continued funding. The TORA conference series is planned to consists of semi-annual meetings, hosted by University of North Texas, Oklahoma State University, and University of Oklahoma, on a rotating basis.

Over the last century, the theories of automorphic forms and representations have grown enormously. Important applications impact various fields of research, ranging from coding theory, algebraic geometry, and topology to Kac-Moody algebras and quantum field theory. The interplay of automorphic forms and representation theory has been especially fruitful, and many surprising and deep results have emerged. The TORA series will emphasize the interplay between automorphic forms and representations, both in the classical and adelic languages, and other related topics.

This award supports participants in the Texas-Oklahoma Representations and Automorphic Forms (TORA) conference series will take place at the University of North Texas (UNT) in Denton, TX, April 8-10, 2016; at Oklahoma State University (OSU) in Stillwater, OK in Fall 2017; and at the University of Oklahoma (OU) in Norman, OK in Fall 2018. Each TORA conference will feature three plenary guest speakers consisting of two outstanding leading research mathematicians from outside the Oklahoma-Texas region as well as a recent Ph.D. or a graduate student nearing completion of a Ph.D. In addition, regional graduate students and researchers will present their work. These weekend conferences will provide ample opportunities for collaboration and interactions among the students and researchers in the region, specializing in number theory, automorphic forms and representation theory.

Automorphic forms are basic objects of study in modern number theory. The interplay between automorphic forms and representation theory has seen a number of very exciting new developments recently which also fuel applications to several other fields. J. Arthur's work on endoscopic classification of representations of the orthogonal and symplectic groups is an example of such new developments. Research faculty from over a half dozen universities in the Oklahoma and North Texas region work in automorphic forms and representation theory. The TORA conference series has been successful in bringing together researchers from the region and beyond, as well as their students, and propel innovation and exchange. TORA VII-IX will continue this success in collaborations and exchanges not only between institutions, but also among mathematical areas. The webpage for the 2016 conference can be found here http://www.math.unt.edu/~richter/TORA/TORA7.html.