February 9 
Sophie Morel
Princeton University 
KazhdanLusztig polynomials: an arithmetic geometer's view
The goal is to explain some ways KazhdanLusztig polynomials show up in arithmetic geometry and number theory. 
February 18 
Richard Ehrenborg
Princeton University and University of Kentucky 
Weighted enumeration of consecutive 123avoiding permutations
and the Hurwitz zeta function
A permutation π =(π_{1},...,π_{n}) is consecutive 123avoiding if there is no index i such that π_{i} < π_{i+1} < π_{i+2}. Similarly, a permutation π is cyclically consecutive 123avoiding if the indices are viewed modulo n. We consider a weighted enumeration problem of consecutive 123avoiding permutations where we able to to determine an asymptotic expansion of the weighted enumeration. For the cyclically weighted enumeration we are able to determine an exact expression in terms of the Hurwitz zeta function. This yields an explicit combinatorial expression for the higher derivatives of the cotangent function. 
February 25 
Greta Panova
University of Pennsylvania 
Kronecker coefficients  combinatorics, complexity and beyond.
The Kronecker coefficients $g(\lambda,\mu,\nu)$ are defined as the multiplicity of an irreducible representation $S_\lambda$ of the symmetric group $S_n$ in the tensor product of two other irreducibles, $S_\mu \otimes S_\nu$. Finding a positive combinatorial formula for these nonnegative integers or even criteria for their positivity has been a 75+ year old problem in representation theory and algebraic combinatorics. Recently, the Kronecker coefficients appeared as central objects in the field of Geometric Complexity Theory and more questions about their computational complexity emerged. In this talk we will discuss a few problems of different characters involving these coefficients  the Saxl conjecture on the tensor square $S_\delta \otimes S_\delta$ where $\delta$ is the staircase partition, the combinatorial side with the new proof of Sylvester's theorem on the unimodality of the qbinomial coefficients as polynomials in q giving effective bounds on both, and some complexity results. 
March 4

Gábor Hetyei
University of North Carolina at Charlotte 
Counting genus one partitions and permutations
The study of counting rooted maps was initiated by Tutte, who was motivated by the four color conjecture, and the study of hypermaps grew out of this initiative. Hypermaps are pairs of permutations that can be topologically represented by labeled maps, the genus of the underlying surface may be expressed purely algebraically in terms of these permutations. Looking at the special case of rooted hypermonopoles leads to the definition of the genus of a permutation. In this talk we prove a conjecture of Martha Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a fourcolored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations. This is joint work with Robert Cori. 
March 11 
Mark McConnell
Princeton University 
Voronoi neighbors of D_{n} lattices
Voronoi defined a tiling of the space of positive definite quadratic forms on R^{n}. Each tile is determined by a perfect lattice. Examples of perfect lattices include the root lattices A_{n} and D_{n}. We will focus on the tiles that are neighbors of a D_{n} tile. These are determined by some simple combinatorics, but there are exponentially many different D_{n} neighbors, and only a few families of them have been understood for all n. I will report on two recent results, one due to myself, and one joint work with Noam Elkies. 
March 18 
No meeting

Spring Break

March 25 
Sinai Robins
Brown University and Nanyang Technological University 
Multiply tiling Euclidean space by translations
of a convex object
We study the problem of covering Euclidean space R^{d} by possibly overlapping translates of a convex body P such that almost every point is covered exactly k times for a fixed integer k. Such a covering of Euclidean space by translations of P is called a ktiling. Classical tilings by translations (which are 1tilings in this context) began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1tile Euclidean space. Today we know that ktilings can be tackled by methods from Fourier analysis, though some of their aspects can also be studied using purely combinatorial means. For many of our results there is both a combinatorial proof and a Harmonic analysis proof. For k larger than 1 the collection of convex objects that ktile is much wider than the collection of objects that 1tile. So it's a more diverse subject with plenty (infinite families) of examples in R^{2} as well. There is currently no complete knowledge of the polytopes that ktile in dimension 3 or larger, and even in dimension 2 it is still challenging. We will cover both ``ancient'' as well as very recent results concerning 1tilings and other ktilings. This is based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis. 
April 10 
Richard Stanley
MIT 
Smith normal form and combinatorics
If A is an m x n matrix over a PID R (and sometimes more general rings), then there exists an m x m matrix P and an n x n matrix Q, both invertible over R, such that PAQ is a matrix that vanishes off the main diagonal, and whose main diagonal elements e_{1}, e_{2}, … , e_{m} satisfy e_{i}  e_{i+1} in R. The matrix PAQ is called a Smith normal form (SNF) of A. The SNF is unique up to multiplication of the e_{i}'s by units in R. We will discuss some aspects of SNF related to combinatorics. In particular, we will give examples of SNF for some combinatorially interesting matrices. We also discuss a theory of SNF for random matrices over the integers recently developed by Yinghui Wang.
Room: 224 Fine

April 29 
Dennis Stanton
University of Minnesota 
Another (q,t) world
A well studied (q,t)analogue of symmetric functions are the Macdonald polynomials. In this talk I will survey another (q,t)analogue, where q is a prime power from a finite field and t is an indeterminate. Analogues of facts about the symmetric group S_{n} are given for GL_{n}(F_{q}), including (1) counting factorizations of certain elements into reflections, (2) combinatorial properties of appropriate (q,t)binomial coefficients, (3) Hilbert series for invariants on polynomial rings. Some new conjectured explicit Hilbert series of rings of invariants over finite fields are given. This is joint work with Joel Lewis and Vic Reiner. 
May 8 
Yue Cai
University of Kentucky 
qCombinatorics: A new view
The idea of qanalogues can be traced back to Euler in the 1700's who was studying qseries, especially specializations of theta functions. Recall a qanalogue is a method to enumerate a set of objects by keeping track of its mathematical structure. For example, a combinatorial interpretation of the qanalogue of the Gaussian polynomial due to MacMahon in 1916 is given by summing over all 01 bit strings consisting of nk zeroes and k ones the statistic q to the inversion number. Setting q=1 returns to the familiar binomial coefficient. In this talk we show the classical qStirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1 + q. We extend this enumerative result via a decomposition of a new poset whose rank generating function is the qStirling number S_{q}[n,k] which we call the Stirling poset of the second kind. This poset supports an algebraic complex and a basis for integer homology is determined. This is another instance of Hersh, Shareshian and Stanton's homological version of the Stembridge q = 1 phenomenon. A parallel enumerative, poset theoretic and homological study for the qStirling numbers of the first kind is done beginning with de Médicis and Leroux's rook placement formulation. Time permitting, we will indicate a bijective argument à la Viennot showing the (q,t)Stirling numbers of the first and second kind are orthogonal. This is joint work with Margaret Readdy.
Room: 214 Fine
