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Ryan Kinser Contact information: Department of Mathematics University of Connecticut Office: MSB M233
Office phone: (860)486-2362 Office Hours: M 12-1, Th 5-6, F 9-10 This semester I am teaching Math 1070, Math for Business and Economics, and Math 3230, Abstract Algebra. | 
Me in Sonoma Valley, March 2008 |
I recently completed my Ph.D. in mathematics at the University of Michigan, studying quiver representations with Harm Derksen. I defended my thesis on May 1, 2009. The project employeed combinatorial constructions to study quiver representations, especially tensor products of quiver representations and representation rings of quivers. I am generally interested interactions among combinatorics, representation theory, and algebraic geometry.
My newest interest is the possibility of computing Euler characteristics of quiver Grassmannians, using some of my previous work, with the aim of getting explicit expressions for F-polynomials in cluster algebras using the formula of Derksen-Weyman-Zelevinsky. I am also trying to get into some project with cluster categories, (cluster-)tilting theory, and things along those lines.
I have at times worked on subspace arrangements and realizability of polymatroids. I've also been thinking about analogous things in information theory, namely Shannon entropy of random vectors and information inequalities. This never quite came together in my head the way I had hoped, but I still find it interesting.
I also have some projects in mind dealing with geometric applications of the "rank functors" developed in my thesis, specifically their relation to orbit closures in representation spaces of quivers (quiver loci). I am also interested in a variety of other tangentially related topics, for example connections between cluster algebras and quivers, quivers with potentials, cluster categories, Hall algebras....
Papers
- "Constructibility of Rank Functions" (or something like that), in preparation.
It will be shown that the rank functions for quivers studied below give constructible functions on representation spaces of quivers, quiver Grassmannians, and also on various moduli spaces (e.g. King's of semi-stable reps or Kac's of indecomposables). This should lead to some interesting interaction of rank functions and algebraic geometry.
- "New Inequalities for Subspace Arrangements", preprint available at arXiv:0905.1519. To appear in Journal of Combinatorial Theory, Series A.
An infinite family of inequalities that hold for the rank function of a subspace arrangement is given. Previously, only one non-elementary inequality for subspace arrangements was known. Could be interesting to matroid theorists since it gives criteria for realizability.
- Ph.D. thesis, University of Michigan, 2009.
This essentially a concatenation of the two papers below, improved with the benefit of hindsight, more readers, and lack of space limitations (more examples, more background, and remarks on generalizations).
- "Representation Rings of Rooted Tree Quivers", preprint available at arXiv:0807.4496. To appear in Duke Mathematical Journal.
The tools constructed in the paper below are combined with combinatorial methods, involving the category of quivers over a given quiver, to find structure in representation rings.
- "The Rank of a Quiver Representation", J. Algebra, 320(6):2363-2387, 2008. Preprint available at arXiv:0711.1135.
A functor is constructed that generalizes the rank of a linear map to the setting of an arbitrary diagram of vector spaces and linear maps (a quiver representation). Using maps of directed graphs, we get more, similar functors. These can be used to construct numerical invariants of a quiver representation which include, as the simplest cases, its dimension vector and the ranks of all maps appearing in the representation.
- Slides from my talk at the Spring 2009 AMS Central Sectional Meeting. Basically a summary of my thesis.
- Here is a "wordle" made from the LaTeX code for my first paper! (It's not intended as a substitute for reading the original paper.)
- This maple script computes certain combinatorial objects associated to a rooted tree quiver. Namely, the (disconnected version of) the poset LQ, and the dimension vectors of the "reduced representations", labeled by LQ. (requires Stembridge's posets package)
- Notes for a talk at the International Conference on Representations of Algebras and Related Topics, Woods Hole, MA, 2008-04-26
- Notes for a talk on Gabriel's theorem in the UM Student Representation Theory seminar, 2008-02-20.
- Notes from my talk on rank functors at the University of Michigan Algebra seminar, 2007-11-20.
We may have met at.... (math related travel)
- AMS Fall Western Sectional Meeting, to speak at Special Session on Representations of Finite Dimensional Algebras, Riverside, California, Nov. 2009.
- XXIe rencontre de théorie des représentations des algèbres, Sherbrooke, Québec, Oct. 2009. Gave talk.
- Maurice Auslander Lectures and International Conference, Woods Hole, Massachussets, April 2009. Attended.
- AMS Spring Central Sectional Meeting, spoke at Special Session on Algebra, Geometry and Combinatorics, Urbana, Illinois, March 2009.
- Depaul University, spoke at Algebra seminar, Sept. 17, 2008.
- ICRA XIII, August 2008, São Paulo, Brazil. Attended workshop, gave talk at conference.
- Summer School on Algebraic Geometry and Quiver Representations, Grenoble, France June 2008. Attended 3 week summer school, gave talk in grad student seminar.
- Maurice Auslander Lectures and International Conference, Woods Hole, Massachussets, April 2008. Attended Auslander Lectures, gave talk at conference.
- Topics in Combinatorial Representation Theory, March 2008, MSRI. Attended.
- ICRA XII, July 2007, Toruń, Poland. Attended.
- Universidad Católica de Chile, spoke in Complex Geometry seminar, June 2007.
- Symmetry and Spaces; A Conference on the Occasion of Gerry Schwarz's 60th Birthday. The Fields Institute, Toronto, July 2006. Attended.
Friends in math who have homepages that I know of:
Jasun Gong
Dave Anderson
Craig Spencer
Sue Sierra
