Last updated Thu 12/11/14
The Combinatorics Seminar meets on Fridays, 4-5 PM, in Snow 408.
Please contact Jeremy Martin if you are interested in speaking.
August 29
Organizational meeting
September 5
Jeremy Martin
Cyclotomic and simplicial matroids
Abstract: Let \(n\) be an integer and \(\zeta\) a primitive \(n\)th root of unity. What are the \(\mathbb{Q}\)-linear dependences among \(1,\zeta,\dots,\zeta^{n-1}\)? Equivalently, which sets of powers of \(\zeta\) are \(\mathbb{Q}\)-vector space bases for the cyclotomic field \(\mathbb{Q}(\zeta)\)? Vic Reiner and I discovered that the answer turns out to involve simplicial complexes in an unexpected way. Subsequently, Reiner and Gregg Musiker used some of these ideas to give a beautiful combinatorial interpretation of the coefficients of cyclotomic polynomials.
September 12 (joint with Algebraic Geometry/Analytic Number Theory Seminar)
Prof. Yasuyuki Kachi
Interplay between Combinatorics, Number Theory and Algebraic Geometry - Asymptotic formula, Renormalization and Anomaly
Abstract: I would like to share one particularly noteworthy formula that my co-author P. Tzermias and I stumbled across that lies at the crossroads of combinatorics and analytic number theory, of which Stirling's formula serves as an archetype. An alternate, more casual, title of the talk would be "factorial numbers m! and beyond".
This naturally grew out of the context of the recent resurgence of the study of analytic continuations of Riemann's zeta function \(\zeta(s)\), to which we made a contribution which I reported in my May 1st talk, which I will briefly revisit. Most importantly, the formula invokes a new method to re-define the pre-existing notion of "renormalization" of products a la Riemann and Lerch. This appears to be an uncharted territory. I suggest open problems in this direction, along with viable approaches to those problems, some of which I hope will potentially attract Ph.D. degree seekers. Naturally, the talk is accessible to graduate students (and apt undergraduate students).
Key words: Kurokawa tensor product; Functional identity and integral representation of Hurwitz-Lerch's zeta function.
September 19
No seminar
September 26
Jeremy Martin
Simplicial Trees
Abstract: I will talk about the theory of spanning trees in simplicial and cellular complexes, which I have been developing for the last several years in collaboration with Art Duval and Caroline Klivans (based ultimately on ideas of Gil Kalai and others). The talk will be intended as a warmup for Josh Hallam's talk on October 3 (see below), which uses many of these ideas.
October 3
Josh Hallam (Michigan State University)
Increasing Forests in Graphs and Simplicial Complexes
Abstract: Suppose that \(G\) is a finite graph with a total ordering on the vertex set. We say that a subtree \(T\) of \(G\) is increasing if the vertices along every path that starts at the minimum vertex of \(T\) increase. We say a forest is increasing if every component of the forest is increasing. The generating function for the number of increasing spanning forests has some nice properties. We discuss these properties as well the generalization of these ideas to simplicial complexes. This is joint work with Art Duval, Jeremy Martin and Bruce Sagan.
October 10
No seminar (Fall Break)
October 17
Jeremy Martin
Parking Functions and the Shi Arrangement
Abstract: Cayley's formula says that the complete graph \(K_n\) has \(n^{n-2}\) spanning trees. This number shows up a lot in combinatorics; in particular, it is also the number of ways to park \(n-1\) cars on a one-way street, and it is the number of regions in the Shi arrangement of hyperplanes in \(\mathbb{R}^n\). These objects are important in algebraic and geometric combinatorics and will feature prominently in Mikhail Mazin's talk next week.
October 24
Mikhail Mazin (Kansas State University)
Hyperplane Arrangements and Parking Functions
Abstract: Back in the nineties Pak and Stanley introduced a labeling of the regions of a \(k\)-Shi arrangement by \(k\)-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph \(G\). They introduced the \(G\)-Shi arrangement and a labeling of its regions by \(G\)-parking functions. They conjectured that their labeling is surjective, i.e. every \(G\)-parking function appears as a label of a region of the \(G\)-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the \(k=1\) case. We generalize Hopkins and Perkinson's construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary \(k\). In this talk, I will introduce necessary background and definitions and sketch the proof of the surjectivity of the labeling.
October 31
Brent Holmes
Rainbow colorings of some geometrically defined uniform hypergraphs in the plane
November 7
Open discussion
Bring your questions of the form, "What is a...?" and we'll try to answer them.
November 14
No seminar (most of us will be at an IMA workshop)
November 21
Open discussion
November 28
No seminar (Thanksgiving)
December 5
Billy Sanders
Gröbner bases of representations
December 12, 2:00-3:00pm (note time change!)
Billy Sanders
Gröbner bases of representations, II
Last updated Thu 12/11/14