Fall 2014

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*

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Last updated Thu 12/11/14