Research Training Group in Combinatorics, Geometry, Representation Theory, and Topology
University of Oregon Department of Mathematics
Supported by NSF grant DMS-2039316.
This RTG is dedicated to the advancement of training opportunities for young mathematicians at the University of Oregon.
The intellectual focus of the group is concentrated in the following areas:
Below is a list of personnel affiliated with this group.
- Derived categories in algebra and geometry,
including as one example the development of planar diagrammatics for accessing tilting modules in modular representation theory.
- Equivariant topology, including the topology of configuration spaces and
the computation of equivariant cohomology groups of spaces with ties to combinatorics.
- The geometry and topology of posets, including computations and analysis of generalized Kazhdan-Lusztig polynomials as well as the study of the Bruhat order on Coxeter groups and its generalizations.
- Cohomology and representation theory in positive characteristic, including an attempt to adapt discrete Morse theory to obtain characteristic-sensitive results regarding group cohomology.
- Nicolas Addington:
I work in algebraic geometry, mainly using derived categories of coherent sheaves. My interests include compact hyperkähler manifolds, rationality questions, and classical algebraic geometry.
- Ben Elias:
I study categorical representation theory, a relatively new field that takes representation theory to the next level. The categorification of a ring is a monoidal category whose Grothendieck group is that ring. A categorified module is an action of the monoidal category on another category, whose Grothendieck group is that module. Categorical representation theory has an astounding amount of structure, making it an interesting topic of study in its own right, but it also can be used to study ordinary representation theory, especially in finite characteristic.
In recent years most of my work has been in equivariant algebraic topology, and particularly on understanding cohomology theories indexed on group representations rather than integers. But I also maintain interests in K-theory, motivic homotopy theory (applications of homotopical methods to algebraic geometry), and homological commutative algebra.
- Patricia Hersh:
I work in algebraic and topological combinatorics. This includes studying combinatorial structure of partially ordered sets and topological structure of associated simplicial complexes, studying stratified spaces arising out of areas such as representation theory by playing combinatorial and topological structure off of each other, group actions on posets, the development and refinement of combinatorial-topological techniques such as shellability and discrete Morse theory, total positivity theory, Coxeter groups, and Bruhat order.
- Nicholas Proudfoot:
My work is somewhere in between algebraic geometry, combinatorics, representation theory, and algebraic topology. I work mostly with algebraic varieties that are built using the data of a hyperplane arrangement; examples include hypertoric varieties and various generalizations of (partial compactifications of) configuration spaces. I then study the relationship between the algebraic invariants of these spaces and the combinatorial invariants of the input data. The flow of information goes in both directions. Sometimes I use combinatorics to compute objects of intrinsic geometric interest (categories of sheaves, cohomology rings, etc.) and sometimes I use geometric machinery to prove purely combinatorial theorems.
I work in a range of areas in topology, mostly in algebraic topology
but some in geometric topology as well. I like to see the
geometry which underlies homotopical structures. My interests
are broad, with my most recent projects being in cohomology
operations on manifolds, cohomology of groups, group theory
related to rational homotopy theory, knot theory, and in neural
networks. Configuration spaces are a part of much of what I do,
and I like to study topology, geometry, algebra and
combinatorics related to them. Some of my research along with
other topics of interest in algebraic topology are the subject
of a lecture series
- Benjamin Young:
I work in enumerative, bijective and algebraic combinatorics. Most of what I am working on at the moment is related to the dimer model, or to Schubert calculus and the combinatorics of reduced words. Lately I've also been spending a lot of time thinking about Kazhdan-Lusztig polynomials for matroids. I use computers heavily in my work.
- Anna Cepek
- George Nasr (primarily supported by DMS-2053243)
Former PhD students
- Adam Bamforth (Young)
- Elijah Bodish (Elias)
- Corey Brooke (Addington)
- Ross Casebolt (Dugger)
- Jay Hathaway (Elias)
- Hanna Hoffman
- Yang Hu (Sinha)
- Dana Hunter (Sinha)
- Sebastian Jaramillo Diaz (Sinha)
- Nicolás Jaramillo Torres (Elias)
- Chris Keane (Addington)
- Stephen Lacina (Hersh)
- Jacob Lebovic (Sinha)
- Aurel Malapani (Dugger)
- Diego Manco Berrio (Dugger)
- Marissa Masden (Sinha)
- Stewart McGinnis (Dugger)
- Dane Miyata (Proudfoot)
- Jeff Monroe (Sinha)
- Aydin Ozbek (Sinha)
- Bo Phillips (Dugger)
- Kelly Pohland (Dugger)
- Jake Potter (Young)
- Arya Yae