**University
of Miami Combinatorics Seminar**

Organizers: Bruno Benedetti, José Alejandro Samper, and Michelle Wachs

When not specified, we meet at** 5pm** on **Mondays**,
in **Ungar Building, Room 402.**

December 3, 2018 (4pm in UB 506):

**Zvi Rosen (Florida Atlantic University)**

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. I will discuss research with Curto et al and Itskov et al, characterizing combinatorial properties of convex neural codes as well as a restrictive variant -- neural codes defined by dissections of a convex set.

November 12, 2018

**Alejandro Ginory (Rutgers)**

In the course of investigating a statistical problem involving estimators for a parameter matrix, Donald Richards and Siddhartha Sahi have recently formulated certain positivity conjectures involving Jack polynomials. In this talk, I will present a strengthened version of the Richards-Sahi conjectures, which depends on a pair of partitions, and sketch a proof in a number of cases. This strengthened conjectures suggests new combinatorial identities involving Jack analogues of Kostka numbers and hook-length formulas.

October 29, 2018

**Joseph Doolittle (Kansas U)**

We review the historical progress of the problem of determining all faces of a sphere from partial information, starting in 1916 through the modern day. We culminate in a counterexample which disproves the strongest possible version of a conjecture made by Perles in 1960. This strengthened conjecture would imply that simplicial 3-spheres are reconstructible from their facet-ridge graph. While this conjecture fails, in its failure it leaves behind a new technique which may yet solve the problem of reconstructiblity of simplicial spheres.

October 15, 2018

**Amzi Jeffs (U Washington)**

Given a collection of convex open sets, one can form an associated simplicial complex that records their intersection patterns. This complex retains important topological information about the sets; for example, Borsuk's Nerve Lemma states that it is homotopy equivalent to the union of the sets. I will discuss what happens when the union of the sets is convex. In this case, the associated simplicial complex has a number of rich combinatorial properties. I will also describe an application to the theory of convex codes.

October 8, 2018

**Alex Lazar (UM)**

In 2017, Hetyei introduced the homogenized Linial arrangement and showed that the number of regions is equal to a median Genocchi number. In this talk, I will discuss joint work with Wachs, in which we refine Hetyie’s result by computing the M\"obius function of the lattice of intersections of the arrangement. We show that the M\"obius invariant of the intersection lattice is a Genocchi number. Our techniques also yield a type B analog of Hetyei’s result and more generally a Dowling arrangement analog involving a new q-analog of the median Genocchi numbers.

October 1, 2018

**Bruno Benedetti (UM)**

Computational topology aims at understanding the `shape' (=homotopy type, or sometimes just homology) of big data. In 2014 with Frank Lutz we introduced Random Discrete Morse theory as an experimental measure for the complicatedness of a triangulation. This measure depends both on the homotopy type of the space, and on how nicely the space is triangulated. Our approach was elementary, but sometimes successful even for huge inputs. I'll discuss some variants, drawbacks, and possible new ideas that were figured out in the meantime. At the same time, these approaches reveal that the existing libraries of examples in computational topology are all `too easy' for testing algorithms. So let's build a new one!

Sept. 24, 2018

** José Alejandro Samper (UM)**

It is known that the space of polytopes is dense in the space of closed bounded convex sets endowed with various different metrics. Given that polytopes come equipped with combinatorial structure it is reasonable to ask about the combinatorial structure of a polytope that is a good approximation to a given convex body K. We will discuss theorems about simplicial polytopes approximating convex bodies whose boundary is smooth (e.g an Euclidean ball of radius one).

April 23, 2018

** Alex Lazar (UM)**

The homogenized Linial arrangement was introduced by Hetyei in 2017 to prove enumerative results in graph theory. In this talk we present preliminary results of a deeper study of the combinatorics of this hyperplane arrangement.

April 16, 2018

** Felix Gotti (UC Berkeley)**

The set of lattice points T(n,d) inside the regular simplex obtained by intersecting the nonnegative cone of R^d with the affine hyperplane x_1 + ... + x_d = n-1 is the ground set of a matroid M(n,d) whose independent sets are precisely those subsets S of T(n,d) satisfying that the intersection of S and T has at most k elements for each parallel translate T of the regular simplex T(k,d). We will present some matroidal properties of M(n,3) in connection to certain tilings of holey triangular regions associated to the subsets of T(n,3). In particular, we will provide characterizations for the independent sets and circuits of M(n,3) related to certain tilings of their holey triangular regions, extending a characterization of the bases of M(n,3) already given by Ardila and Billey. If time permits, we will also exhibit connections between tilings and the flats and connectivity of the matroids M(n,3).

April 9, 2018

** Bruno Benedetti (UM)**

We survey four cute ways to apply the combinatorial concept of parity to
other fields. Namely:

- in algebra, the parity distinction for permutations (dating back at
least to Cauchy, 1815);

- in topology, the combinatorial proof of Brouwer's fixed point theorem
(Sperner, 1928);

- in geometry, the neighborlyness of cyclic polytopes (Gale, 1963);

- and in number theory, the `one-sentence proof' of the sum-of-squares
theorem (Zagier, 1990).

This talk is intended as didactical, rather than research-oriented; it
does not assume expertise in any of the four fields above.

Tuesday, April 3, 2018 5PM - COLLOQUIUM

** Fu Liu (UC Davis)**

The Ehrhart polynomial counts the number of lattice points inside dilation of an integral
polytope, that is, a polytope whose vertices are lattice points. We say a polytope is
Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature,
different families of polytopes have been shown to be Ehrhart positive using different
techniques. We will survey these results in the first part of the talk, after giving a
brief introduction to polytopes and Ehrhart polynomails.

Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients
relating to the normalized volumes of faces. In the second part of the talk, I will
discuss joint work with Castillo in which we try to make this relation more explicit
in the case of regular permutohedra. The motivation is to prove Ehrhart positivity
for generalized permutohedra. If time permits, I will also discuss some other related
questions.

Tuesday, March 27, 2018

** Brittney Ellzey (UM)**

Dissertation defense

Monday, March 19, 2018

** José Alejandro Samper (UM)**

Threshold graphs were introduced by Chvatal and Hammer(1974) as tools in optimization. They coincide with the class of shifted graphs and can be described and studied in three different ways: purely combinatorial, slicing the second hypersimplex or slicing a cube. A question of Golumbic(1978) answered in the negative by Reiterman, Rodl, Sinajova and Tuma(1985), asks for higher dimensional analogues. We will give a geometric explanation for the negative answer to such question and propose a corrected version of Golumbic's question. We will then highlight the relevance of this question in the theory matroid polytopes.

Monday, March 5, 2018

** Jessica Striker (North Dakota State U)**

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.

Monday, Feb.19, 2018

** Michelle Wachs (UM)**

Smirnov words are words over the alphabet of positive integers with no adjacent equal letters. The enumerator of these words by descent number is a symmetric function, which arose in work with Shareshian on q-Eulerian polynomials, on Rees products of posets, and on chromatic quasisymmetric functions. In this talk I will discuss this work with Shareshian and recent work with Ellzey on the enumerator of Smirnov words by cyclic descents.

Monday, Feb. 5, 2018

** Bruno Benedetti (UM)**

We discuss the problem of whether all contractible d-complexes can be drawn in R^{2d}. This is clear only for d=1 (in which case the answer is: “yes, all trees are planar graphs”.) We also look at combinatorial strengthenings of contractibility, like collapsibility and non-evasiveness. This is work in progress with Karim Adiprasito.

Monday, Jan. 29, 2018

** Jai Aslam (Northeastern)**

We present Kneser's conjecture and its reformulation into a graph coloring problem. We then introduce the generalized Erdos-Kneser conjecture partially proven by Sarkaria in 1990 and its associated hypergraph coloring problem. We prove this conjecture for r-uniform hypergraphs with the size of intersection s, not too close to r. We discuss what's still open related to this conjecture and possible methods for further proofs.

Monday, Jan. 22, 2018

** Richard Stanley (UM & MIT)**

A finite graded partially ordered set P has the * Sperner property * if the largest
level of P is an antichain of maximum size. Most of the talk will be a survey of the
Sperner property, beginning with Sperner's result that the boolean algebra of all
subsets of a finite set has the Sperner property. (Of course Sperner did not use this
terminology.) We will focus our attention on the use of linear algebra. We conclude
with a discussion of the weak Bruhat order of the symmetric group. It is an open
problem whether this poset has the Sperner property. We will discuss a determinantal
conjecture which would imply the Sperner property.

Monday, Nov. 13, 2017

** James McKeown (UM)**

It is quite ordinary to consider how a group acts on an object. What if instead, one fixes a representation and lets the set of linear transformations (id-g) act on the object? In 2005, Waldspurger showed that, for the regular representation of a finite reflection group, the action of (id-g) on the cone over the fundamental weights gives a tiling of the cone over the positive roots. Shortly thereafter, Meinrenken considered the case of affine Weyl groups, and showed that the action of (id-g) on a fundamental alcove gives a tiling of the whole vector space. Bibikov and Zhgoon then proved analogous results for all cocompact hyperbolic reflection groups. We will look at some combinatorial consequences of these theorems for finite and affine types A and B. In particular, we will investigate the Dedekind-MacNeille completion or Bruhat order--- the smallest lattice containing Bruhat order as a subposet.

Monday, Nov. 06, 2017

** Michelle Wachs (UM)**

I will discuss an algebro-geometric approach to proving the longstanding Stanley-Stembridge e-positivity conjecture for chromatic symmetric functions that was proposed by Shareshian and myself several years ago. Our approach to this conjecture involves a refinement of Stanley’s chromatic symmetric functions. We conjectured a certain relationship between our refinement and Hessenberg varieties. Our conjecture was recently proved by Brosnan and Chow using techniques from algebraic geometry, and more recently by Guay-Paquet using Hopf algebras. I will describe this result, some of its consequences, and what still needs to be done to prove the Stanley-Stembridge conjecture.

Monday, Oct. 30, 2017

**Brittney Ellzey (UM)**

I will be presenting my work on expansions (in various bases for the ring of symmetric and quasisymmetric functions) of chromatic quasisymmetric functions for digraphs. This is a version of the talk I will be giving at the Combinatorics Seminar at Brandeis.

Monday, Oct. 23, 2017

**Manuel Rivera (UM)**

To any topological space we may associate a topological monoid called the based loop space: as a set it consists of all loops in the space based at a fixed point and the multiplication is given by concatenation of loops. The homology of the based loop space has the structure of a Hopf algebra: the product is induced by concatenation of loops, the coproduct by the Alexander-Whitney diagonal, and the antipode by the map sending a loop to its inverse. From a classical result of homotopy theory we know that sufficiently nice topological spaces may be modeled by combinatorial objects called simplicial sets. I will explain how to model the above construction in purely combinatorial terms, namely, to any connected simplicial set S I will construct a natural differential graded Hopf algebra, based on the combinatorics of S, having the property that its homology is isomorphic to the homology Hopf algebra of the based loop space of the geometric realization of S. This is joint work with Samson Saneblidze and generalizes classical results of Adams and Baues.

Monday, Sept. 25, 2017

**Tiago Novello de Brito (Pontifícia Universidade Católica do Rio de Janeiro)**

The discrete line field is our proposal for a possible discretization of the theory of line fields. The discrete object will be a Morse matching just between the vertices and edges of a cellular complex. The objective is to define the critical objects and their indices, and then show that the complex is homotopy equivalent to a cellular complex with just the critical objects.

Monday, August 21, 28, Sept. 18, Oct. 16, 2017

**José Alejandro Samper (U Miami)**

This is a continuing seminar that started during the summer.

Monday, April 24, 2017

**Vasu Tewari (U Washington)**

Gessel introduced a multivariate formal power series tracking the
distribution of ascents and descents in labeled binary trees. In addition
to showing that it was a symmetric function, he conjectured that it was
Schur-positive.

In this talk, I will present a proof of this conjecture which utilizes an
extension of a beautiful bijection of Preville-Ratelle and Viennot
concerning generalized Tamari lattices. I will subsequently discuss
connections between specializations of Gessel's symmetric function and
Frobenius characteristics of symmetric group actions on certain Coxeter
deformations, focusing in particular on semiorder and Linial arrangements.
Finally, I will discuss some potential avenues to pursue.

This is joint work with Ira Gessel and Sean Griffin.

Monday, April 17, 2017

**Michelle Wachs (UM)**

The r-inversion number is a statistic on words of length n (over the positive integers), which interpolates between the descent number (r=2) and the inversion number (r=n). We consider a symmetric function U_{n,r} that enumerates words of length n by this statistic. The symmetric function U_{n,r} is an example of an LLT polynomial. The LLT polynomials were shown to be Schur-positive by Grojnowski and Haiman by means of Kazhdan-Lusztig theory. It is an open question to give a combinatorial description of the coefficients in the Schur basis expansion. For r = 2 and r=n, such descriptions are well known. For r = 3, a description (in a more general setting) was conjectured by Haglund and was proved by Blasiak using noncommutative Schur functions and Lam's algebra of ribbon Schur operators. In this talk I will describe a more elementary proof for the r = 3 case, which uses classical RSK theory. I will also discuss results for some other cases, and a consequence involving an r-analog of the q-binomial coefficients. This is joint work with Yuval Roichman.

Monday, April 3, 2017

**Felix Gotti (UC Berkeley)**

It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Combining work of Skandera and Reed and work of Postnikov, we will assign a rank n positroid on [2n] to each unit interval order on [n]. We call such positroids "unit interval positroids." Then we will give a characterization of the unit interval positroids by describing their associated decorated permutations, showing that each one must be a 2n-cycle encoding a Dyck path of length 2n.

Monday, March 27, 2017

**Tewodros Amdeberhan (Tulane)**

Determinants are found everywhere in mathematics and other scientific endeavors. Their particular role in Combinatorics does not need any cynical introduction or special advertisement. In this talk, we will illustrate certain techniques which proved to be useful in the evaluation of several class of determinantal evaluations. We conclude this seminar with an open problem. The content of our discussion is accessible to anyone with "an intellectual appetite".

Monday, March 20, 2017

**Fabrizio Zanello (Michigan Tech)**

We discuss the (non)unimodality of the rank-generating function, $F_{\lambda}$, of the poset of partitions with distinct parts contained inside a given partition $\lambda$. This work, in collaboration with Richard Stanley (European J. Combin., 2015), is in part motivated by an attempt to place into a broader context the unimodality of $F_{\lambda}(q)=\prod_{i=1}^n(1+q^i)$, the rank-generating function of the ``staircase'' partition $\lambda=(n,n-1,\dots,1)$, for which determining a combinatorial proof remains an outstanding open problem. We will present a number of results on the polynomials $F_{\lambda}$. Surprisingly, these results carry a remarkable similarity to those proven in 1990 by Dennis Stanton. His work extended, to any partition $\lambda$, the study of the unimodality of $q$-binomial coefficients --- that is, the rank-generating functions of the \emph{arbitrary} partitions contained inside given rectangular partitions. We will also discuss some open problems and recent developments. These include a (prize-winning) paper by Levent Alpoge, who solved our conjecture on the unimodality of $F_{\lambda}$ when $\lambda$ is the ``truncated staircase'' $(n,n-1, \dots,n-c)$, for $n\gg c$.

Tuesday, March 7, 2017, 5PM - COLLOQUIUM

**Anders Björner (KTH, Sweden) **

Being drawable in the plane without intersecting
edges is a very important and much studied graph
property. Euler observed in 1752 that planarity implies a linear
upper bound on the number of edges of a graph (which
otherwise is quadratic in the number of vertices). Several
ways of characterizing planar graphs have been given during
the previous century.

Planarity is, of course, a special case of a general notion of
embedding a simplicial $d$-complex into real $k$-space.
The $k=d+1$ and $k=2d$ cases are of particular interest
in higher dimensions, since they both generalize planarity.
Embedding a space into some manifold is a much studied
question in geometry/topology. For instance, van Kampen
showed that in the $k=2d$ case there is a very useful
cohomological obstruction to embeddability.

Higher-dimensional embeddability has been studied also
from the combinatorial point of view, in a tradition inspired
by Euler. In this talk I will survey a few topics from the combinatorial
study of embeddings, such as bounds for the number of
maximal faces and algorithmic questions. I will end with
mention of some joint work with A.Goodarzi concerning an
obstruction to $k=d+1$ embeddings.

The talk will not presuppose previous familiarity with the topic.

Monday, March 6, 2017

**Laura Escobar Vega (Urbana-Champaign)**

Elnitsky gave an elegant bijection between rhombic tilings of 2n-gons and commutation classes of reduced words in the symmetric group on n letters. We explain a natural connection between Elnitsky’s and Magyar’s construction of the Bott-Samelson resolution of Schubert varieties. This suggests using tilings to encapsulate Bott-Samelson data and indicates a geometric perspective on Elnitsky’s combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by Tenner. This is based on joint work with Pechenik, Tenner and Yong.

Tuesday, February 28, 2017, 5PM - COLLOQUIUM

**Ron Adin (Bar Ilan U, Israel)**

Descents of permutations have been studied since Euler. This notion has been vastly generalized in several directions, and in particular to the context of standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding notion for SYT, Rhoades found a very elegant solution for rectangular shapes.

In an attempt to extend this concept, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter $n$ is ignored. Consequently, the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones.

This talk will report on the surprising resolution of this conjecture: Cyclic descent sets do exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness.

Joint with Sergi Elizalde, Vic Reiner and Yuval Roichman.

Monday, February 27, 2017

**Yuval Roichman (Bar Ilan U, Israel)**

Permutations in the symmetric groups, as well as standard Young tableaux,
are equipped with a well-established notion of descent set.
The cyclic descent set of permutations was introduced by Cellini and further studied by Dilks, Petersen and Stembridge,
while cyclic descents on standard Young tableaux (SYT) of rectangular shapes were introduced by Rhoades.

The existence of cyclic descent maps for SYT of all non-ribbon skew shapes was recently proved,
using nonnegativity properties of Postnikov's toric Schur polynomials.
The proof and its implications will be explained by Ron Adin in tomorrow's colloquium talk.

In this talk we will focus on explicit combinatorial interpretations of the concept,
applications to Schur-positivity and open problems.

Based on joint works with Ron Adin, Sergi Elizalde and Vic Reiner.

Monday, February 20, 2017

**Hai Long Dao (Kansas U)**

Let I be an homogenous ideal in a polynomial ring S over a field. The Betti table of I describes the graded minimal free resolution of I over S. When I is a Stanley-Reisner ideal of a simplicial complex C, the Betti table can be used to compute the h- and f-vectors of C. In this talk I will describe several recent results about what I call local-global phenomena in the Betti tables. Namely, information on a small part of the table forces strong result on the whole resolution, and give structural information about C such as its depth, regularity or chordality. If time permits, I will also explain the connection of these results to classical commutative algebra, and some new connections to group cohomological dimensions. The talk will be based on various joint works with Schweig-Huneke, Schweig, and Vu.

Friday, February 17, 2017 - 4 pm

**Sebi Cioaba (U Delaware)**

A few years ago, Jeremy L. Martin and Jennifer D. Wagner introduced the simplicial rook graphs SR(d,n) as the graph whose vertices are the lattice points in the n-th dilate of the standard simplex in Rd, with two vertices adjacent if they differ in exactly two coordinates. Martin and Wagner proved that SR(3,n) has integral eigenvalues and determined other interesting properties of these graphs. In this talk, I will describe our work proving some conjectures made by Martin and Wagner as well as determining other algebraic and combinatorial facts about these graphs. This is joint work with Andries Brouwer (TU Eindhoven, The Netherlands), Willem Haemers (Tilburg University, The Netherlands) and Jason Vermette (Missouri Baptist Univ., USA).

Monday, February 13, 2017

**Brittney Ellzey (UM)**

Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this talk, we present an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We show that the chromatic quasisymmetric functions of proper circular arc digraphs are symmetric functions, which generalizes a result of Shareshian and Wachs on natural unit interval graphs. The directed cycle on n vertices is contained in the class of proper circular arc digraphs, and we give a generating function for the e-basis expansion of the chromatic quasisymmetric function of the directed cycle, refining a result of Stanley for the undirected cycle. We discuss a generalization of the Shareshian-Wachs refinement of the Stanley-Stembridge e-positivity conjecture. We present our F-basis expansion of the chromatic quasisymmetric functions of all digraphs and our p-basis expansion for all *symmetric* chromatic quasisymmetric functions of digraphs, which extends work of Shareshian-Wachs and Athanasiadis.

Monday, February 6, 2017

**Jose' Samper (UM)**

We will present various curious similarities between shifted simplicial complexes and matroid independence complexes and provide evidence that all these similarities should be proved geometrically by extending the theory of matroid polytopes. Along the way we will pose questions, conjectures and explain some of the final goals. Based on joint work with Jeremy Martin.

Monday, January 30, 2017

**Bruno Benedetti (UM)**

A 3-ball is a simplicial complex homeomorphic to the unit ball in R^3. A ``tree of tetrahedra'' is a 3-ball whose dual graph is a tree. It is easy to see that every (connected) 3-manifold can be obtained from some tree of tetrahedra by recursively gluing together two boundary triangles. The quantum physicist Tsugui Mogami has studied ``Mogami manifolds'', that is, those manifolds that can be obtained from a tree of tetrahedra by recursively gluing together two *incident* boundary triangles. In 1995 he conjectured that all 3-balls are Mogami. Mogami's conjecture would imply a much desired exponential bound (crucial for the convergence of certain models in quantum gravity) for the number of 3-balls with N tetrahedra. Unfortunately, we show that Mogami's conjecture does not hold.

Monday, January 23, 2017

**Richard Stanley (UM & MIT)**

The archetypal result is the theorem of Lucas that the number of
coefficients of the polynomial (1+x)^{n} not divisible by a prime p is
_{i}), where n = _{i}p^{i} is the base p expansion of n. We
will discuss numerous generalizations and analogues of this result. For
example, the number of partitions of n for which the number of standard
Young tableaux of shape λ is odd is equal to 2^{Σ bi }, where
n=^{bi } is the binary expansion of n (due to I. G. Macdonald).

Monday, November 21, 2016

**Jim Haglund (U Penn)**

Shareshian and Wachs have conjectured that a certain symmetric function, which depends on a Dyck path and a parameter t, has positive coefficients when expressed as a polynomial in the elementary symmetric functions. Their conjecture implies an earlier conjecture of Stanley and Stembridge. We show how some elements of the preprint of Carlsson and Mellit "A proof of the shuffle conjecture" imply that the Shareshian-Wachs symmetric function can be expressed, via a plethystic substitution, in terms of LLT polynomials, specifically LLT products of single cells. As corollaries we obtain combinatorial formulas for the expansion of Jack polynomials into the Schur basis, and also the power-sum basis. These formulas are signed, not always positive, but perhaps could be simplified. Other applications include a quick method for computing the chromatic symmetric function using plethystic operators. Based on joint work with Per Alexandersson, Greta Panova, and Andy Wilson.

Monday, November 14, 2016

**John Shareshian (Washington University)**

Let G be a finite group. A subset S of G is called a subrack if S is closed under conjugation. The set R(G) of all subracks of G is partially ordered by inclusion. With Istvan Heckenberger and Volkmar Welker of Philipps-University\"at Marburg, we have studied relations between the combinatorial structure of R(G) and the algebraic structure of G. I will discuss our results.

Monday, November 7, 2016

**Alex Lazar (UM)**

In order to address some questions in tropical geometry, Mikhalkin and Ziegler introduced the notion of a filtered geometric lattice. These posets can be seen as generalizations of geometric semilattices (introduced by Wachs and Walker), which are themselves generalizations of geometric lattices.

In this talk, we will discuss some topological results of Adiprasito and Bjoerner about filtered geometric lattices, as well as some open questions about these posets.

Monday, October 31, 2016

**Katharina Jochemko (TU Vienna, Austria)**

The prototypical valuation is presumably the volume. It has various favorable properties such as homogeneity, monotonicity and translation-invariance. In the continuous setting, valuations are well-studied and the volume plays a prominent role in many classical and structural results. In the less examined discrete setting, the number of lattice points in a polytope - its discrete volume - takes a central role. Although homogeneity and continuity are lost, some striking parallels can be drawn. In this talk, I will discuss some similarities, analogies and differences between the continuous and discrete world of translation-invariant valuations.

Monday, October 24, 2016

**Rainer Sinn (Georgia Tech)**

I will discuss the positive semidefinite matrix completion problem arising e.g. in combinatorial statistics and explain how we can use results in algebraic geometry to understand it better. The object linking the two different areas is the cone of sums of squares and its properties as a convex cone.

Monday, October 17, 2016

**Anastasia Chavez (UC Berkeley, California)**

The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is a simplicial polytope the Dehn--Sommerville relations condense the f-vector into the g-vector, which has length $\lceil{\frac{d+1}{2}}\rceil$. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which $(\lceil{\frac{d+1}{2}}\rceil)$-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.

Thursday, October 6, 2016 - COLLOQUIUM

**Frank H. Lutz (TU Berlin, Germany)**

Canceled due to hurricane Matthew

Monday, October 3, 2016

**Nancy Abdallah (Linkoeping U, Sweden)**

We study the Bruhat order on the sets of twisted involution and twisted identities in a Coxeter group W equipped by an involutive automorphism. When W is the symmetric group of odd rank, we define the Kazhdan-Lusztig-Vogan polynomials indexed by elements in the set of twisted identities and we prove that they are combinatorially invariant for intervals that start with the identity. This generalizes the combinatorial invariance of the classical Kazhdan-Lusztig polynomials for lower bound intervals in a symmetric group. This is joint work with Axel Hultman.

Monday, September 19, 2016

**Alessio Sammartano (Purdue U)**

The Rees ring and the special fiber ring of a polynomial ideal I, also known as the blowup algebras of I, play an important role in commutative algebra and algebraic geometry. A central problem is to describe the defining equations of these algebras. I will discuss the solution to this problem when I is the homogeneous ideal of a rational normal scroll.

Monday, September 12, 2016

**Jay Yang (U Wisconsin-Madison)**

I will present a construction of a random toric surface inspired by the construction of a random graph. With this construction we show a threshold result for smoothness of the surface. The hope is that this inspires further application of randomness to Algebraic Geometry. This talk does not require any background in Algebraic Geometry or Toric Geometry.

Monday, August 29, 2016

**Hailung Zheng (U Washington)**

The celebrated low bound theorem states that any simplicial
manifold of dimension ≥ 3 satisfies g_{2} ≥ 0, and equality holds
if and only if it is a stacked sphere. Furthermore, more recently, the
class of all simplicial spheres with g_{2} = 1 was characterized by Nevo and
Novinsky, by an argument based on rigidity theory for graphs. In this talk,
I will first define three different retriangulations of simplicial
complexes that preserve the homeomorphism type. Then I will show that all
simplicial manifolds with g_{2} ≤ 2 can be obtained by retriangulating
a polytopal sphere with a smaller g_{2}. This implies Nevo and Novinsky’s
result for simplicial spheres of dimension ≥ 4. More surprisingly, it
also implies that all simplicial manifolds with g_{2} = 2 are polytopal
spheres.

Monday, April 25, 2016

**Miriam Farber (MIT)**

Following the proof of the purity conjecture for weakly
separated sets, recent years have revealed a variety of wider classes of
pure domains in different settings. In this paper we prove the purity for
domains consisting of sets that are weakly separated from a pair of
“generic” sets I and J. Our proof also gives a simple formula for the rank
of these domains in terms of I and J. This is a new instance of the purity
phenomenon which essentially differs from all previously known pure
domains. We apply our result to calculate the cluster distance and to give
lower bounds on the mutation distance between cluster variables in the
cluster algebra structure on the coordinate ring of the Grassmannian.
Using a linear projection that relates weak separation to the octahedron
recurrence, we also find the exact mutation distances and cluster
distances for a family of cluster variables.

This is a joint work with Pavel Galashin.

Monday, April 18, 2016

**Brittney Ellzey (UM)**

Chromatic quasisymmetric functions were introduced by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric functions. The results of Shareshian and Wachs focus primarily on incomparability graphs of natural unit interval orders. In this talk I will present my recent work on the chromatic quasisymmetric functions of other graphs, specifically the n-cycle, as well as a generalization of the n-cycle. I will give expansions of the chromatic quasisymmetric functions for these graphs in terms of Gessel’s fundamental quasisymmetric basis and in terms of the power sum basis and see how these expansions compare to those obtained by Shareshian-Wachs and Athanasiadis.

Monday, April 11, 2016, 4:00 pm

**Efrat Engel Shaposhnik (MIT)**

Efrat Engel Shaposhnik will defend her Ph.D. thesis.

Monday, April 4, 2016

**Martin Charles Golumbic (The Caesarea Rothschild Institute for
Computer Science, U Haifa, Israel)**

In this talk, we will explore various intersection and
containment based representations of graphs and posets along with their
associated parameters. Among these are the boxicity and cubicity of
graphs, the dimension and interval dimension of posets and their
comparability graphs, the bending number of intersecting paths on a grid,
and the grid dimension of a graph.

We will also present recent work on the new notions of the separation
dimension of a graph and the induced separation dimension of a graph. One
of our main aims has been to find significant interconnections between such
dimensional parameters. For example, we establish bounds relating the
bending number to the partial order dimension for co-comparability graphs,
and relating the induced separation dimension with the separation dimension
and boxicity.

Monday, March 28, 2016

**James McKeown (UM)**

In 2005 J.L. Waldspurger proved a remarkable theorem. Given a
finite reflection group G, the closed cone over the positive roots is
equal to the disjoint union of images of the open weight cone under the
action of 1-g. When G is taken to be the symmetric group the decomposition
is related to the familiar combinatorics of permutations but also has some
surprising features. To see this, we give a nice combinatorial
description of the decomposition.

The decomposition is not a simplicial, or even CW complex and attempts to
complete it to one are problematic. It does, however, define a dual graph
on n-cycles. We prove some basic facts about this graph and state a few
conjectures and open problems relating to it.

Monday, March 21, 2016

**Luca Moci (Paris 7 - Inst. Math. de Jussieu, France) **

In a recent series of papers by various authors, the theory of
colorings and flows on graphs has been extended to the higher-dimensional
case of CW complexes. We will survey this theory and show how the
arithmetic Tutte polynomial naturally comes into play. (Joint work with E.
Delucchi).

After recalling the basic properties of this polynomial, we will show some
convolution formulae and their applications to the case of CW complexes
(ongoing joint work with S. Backman, A. Fink and M. Lenz).

Thursday(!), March 17, 2016

**Emanuele Delucchi (U Fribourg, Switzerland)**

Recent work of De Concini, Procesi and Vergne on vector partition
functions gave a fresh impulse to the study of toric arrangements from an algebraic,
topological and combinatorial point of view.
In this context, many new combinatorial structures have recently appeared in the
literature, each tailored to one of the different facets of the subject. Yet, a
comprehensive combinatorial framework is lacking.

As a unifying structure, in this talk I will propose the study of group actions on
semimatroids and of related polynomial invariants, recently introduced in joint work
with Sonja Riedel. In particular, I will outline some new open problems brought to
the fore by this new point of view.

Monday, March 14, 2016, 4:00 PM

**Mark Skandera (Lehigh U)**

In 1993, Haiman studied certain functions from the Hecke algebra to Z[q] called monomial traces. He conjectured that the evaluation of these at Kazhdan-Lusztig basis elements resulted in polynomials in N[q]. A weakening of this conjecture is that the evaluations of other traces, called power sum traces, results in polynomials in N[q]. We will discuss several combinatorial interpretations of these polynomials for Kazhdan-Lusztig basis elements indexed by permutations which avoid the patterns 3412 and 4231. These interpretations come from joint work with Matthew Hyatt, and results of Athanasiadis, Shareshian, and Wachs.

Monday, February 29, 2016

**Cynthia Vinzant (North Carolina State U)**

Real stable polynomials define real hypersurfaces that are maximally nested ovaloids. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Branden recently used this theory and the Vamos matroid to disprove the generalized Lax conjecture, which concerns representing polynomials as determinants. I will discuss the fascinating connections between these fields and some extensions to some varieties associated to hyperplane arrangements.

Monday, February 22, 2016

**Russ Woodroofe (Mississippi State U)**

Jay Schweig and I recently discovered a large class of shellable lattices. Our original motivation were the order congruence lattices of finite posets.
Afterwards, we noticed that the subgroup lattices of solvable groups are also
contained in the class, and indeed, the definition may be seen as a
lattice-theoretic abstraction of solvable groups.

In this talk, I'll review some of the theory of supersolvable lattices, and show how to extend similar ideas to our class of lattices.

Monday, February 15, 2016

**Matteo Varbaro (U Genova, Italy)**

In the talk I will discuss an ongoing joint work with Bruno Benedetti and Michela Di Marca on the Castelnuovo-Mumford regularity of subspace arrangements. The Castelnuovo-Mumford regularity of an embedded projective variety is an important invariant measuring its complexity. For a Gorenstein subspace arrangement, it turns out that this invariant has an amazingly simple description in terms of the dual graph of the arrangement. The goal of the talk is to discuss the concepts of dual graph (keeping in mind the motivating case of simplicial complexes) and Castelnuovo-Mumford regularity, to explain the connection we found out for Gorenstein subspace arrangements, and to provide several examples.

Tuesday (!), February 9, 2016

**Yuval Roichman (Bar-Ilan U, Israel)**

Characterizing sets of permutations whose associated
quasisymmetric function is symmetric is a long-standing problem in
algebraic combinatorics. We present a general method to construct
symmetric and Schur-positive sets and multisets, based on
algebraic and geometric operations on grid classes. This approach
produces new instances of Schur-positive sets and explains the
existence of known such sets that until now were sporadic cases.

Joint with Sergi Elizalde.

Monday, February 1, 2016, 4pm

**Richard Stanley (UM, MIT)**

A parking
function is a sequence (a_{1}, ..., a_{n}) of positive integers whose
increasing rearrangement (b_{1}, ... , b_{n}) satisfies b_{i}
≤ i. We first explain the connection with parking cars and give the
elegant proof of
Pollak that the number of parking functions of length n is (n+1)^{n-1}.
The symmetric group S_{n} acts on parking functions of length n by
permuting coordinates. This action corresponds to a symmetric
function F_{n} with many interesting properties. We discuss properties of
the generating function ∑ F_{n} z^{n} and show its connection
with some variants of parking functions. Finally we consider a
q-analogue of the previous theory.

Monday, January 25, 2016

**Alex Engstrom (Aalto U, Finland)**

We introduce a new function b(x) associated to simplicial
complexes that measures how versatile you can decompose it into boolean
lattices. The function interpolates between three important cases: b(2) is
the number of faces; b(1) is a good upper bound on the number of maximal
faces; and b(0) is the minimal number of critical cells in a discrete
Morse matching, which is a natural upper bound on the total dimension of
cohomology. In contemporary graph theory b(1) is central, and in
topological combinatorics b(2) is. We try to unify some approaches in
these branches of discrete mathematics by this setup and manage for
example to prove some new theorems regarding the structure of maximal
independent sets in triangle-free graphs, improving on recent work by
Balogh et al.

In the talk we will also explain why all of the bounds mentioned above are
optimal for shellable simplicial complexes.

Monday, December 7, 2015

**Carolina Benedetti (Fields Institute & York U,
Toronto)**

Inspired by the theory pioneered by Stanley's chromatic symmetric
function and its connection to Hopf algebras, we will see how abstract
simplicial complexes can be endowed naturally with a combinatorial Hopf
structure that gives rise to chromatic generating functions. Using
principal specializations on these generating functions we derive certain
combinatorial identities involving acyclic orientations related to the
complexes.
We will also discuss work in progress aiming to derive the analogous of
Shareshian-Wachs "chromatic quasisymmetric functions" for simplicial
complexes.

This is joint work with J. Hallam and J. Machacek. No familiarity with Hopf
algebras is required, we will give the necessary background.

Monday, November 30, 2015

**Barbara Bolognese (Northeastern U, Boston)**

In 1962, Hartshorne proved that the dual graphs of an
arithmetically Cohen-Macaulay scheme is connected. After establishing a
correspondence between the languages of algebraic geometry, commutative
algebra and combinatorics, we are going to refine Hartshorne's result and
measure the connectedness of the dual graphs of certain projective schemes
in terms of an algebro-geometric invariant of the projective schemes
themselves, namely their Castelnuovo-Mumford regularity. Time permitting,
we are also going to address briefly the inverse problem of Hartshorne's
result, by showing that any connected graph is the dual graph of a
projective curve with nice geometric properties.

This is joint work with Bruno Benedetti and Matteo Varbaro.

Monday, November 16, 2015

**Bruno Benedetti (UM)**

In classical Morse theory, for any given manifold there is always a unique
optimal Morse vector (=the vector counting the
number of critical points of index 0,1,..., up to the dimension). It turns
out that in Forman's discrete version of Morse
theory, this is no longer the case. I will sketch how to construct a
contractible 3-complex on which the 'best' discrete
Morse vectors are (1,0,1,1) and (1,1,1,0), because (1,0,0,0) is out of reach.

This is joint work with
Karim Adiprasito and Frank Lutz.

Tuesday (!), November 10, 2015

**Jose Alejandro Samper Casas (U Washington, Seattle)**

Motivated by a question of Duval and Reiner about eigenvalues of combinatorial Laplacians, we develop various generalisations of (ordered) matroid theory to wider classes of simplicial complexes. In addition to all independence complexes of matroids, each such class contains all pure shifted simplicial complexes, and it retains a little piece of matroidal structure. To achieve this, we relax many cryptomorphic definitions of a matroid. In contrast to the matroid setting, these relaxations are independent of each other, i.e., they produce different extensions. Imposing various combinations of these new axioms allows us to provide analogues of many classical matroid structures and properties. Examples of such properties include the Tutte polynomial, lexicographic shellability of the complex, the existence of a meaningful nbc-complex and its shellability, the Billera-Jia-Reiner quasisymmetric function, and many others. We then discuss the h-vectors of complexes that satisfy our relaxed version of the exchange axiom, extend Stanley's pure O-sequence conjecture about the h-vector of a matroid, solve this conjecture for the special case of shifted complexes, and speculate a bit about the general case. Based on joint works with Jeremy Martin, Ernest Chong and Steven Klee.

Monday, November 2, 2015

**Brittney Ellzey (UM)**

Shareshian and Wachs introduced the chromatic quasisymmetric function of a graph as a refinement of Stanley's chromatic symmetric function. In their paper, Shareshian and Wachs conjecture a formula for the expansion of the chromatic quasisymmetric function of incomparability graphs of natural unit interval orders in the power sum basis. Recently, Athanasiadis proved the conjecture by using a formula of Roichman for the irreducible characters of the symmetric group. In this talk, we will present Athanasiadis' work.

Monday, October 26, 2015

**Bennet Goeckner (U Kansas)**

In joint work with Art Duval, Caroline Klivans, and Jeremy Martin, we construct a non-partitionable Cohen-Macaulay simplicial complex. This construction disproves a longstanding conjecture by Stanley that would have provided an interpretation of h-vectors of Cohen-Macaulay complexes. Due to an earlier result of Herzog, Jahan, and Yassemi, this construction also disproves the conjecture that Stanley depth is always greater than or equal to depth. Time permitting, we will also discuss Garsia?s open conjecture that every Cohen-Macaulay poset has a partitionable order complex.

Monday, October 12, 2015

**Ivan Martino (Freibourg, switzerland)**

Given a faithful representation V of a group G one can consider the
partially ordered set of conjugacy classes of stabilizer
subgroups. Using this combinatorial object we proved that the motivic
class of the classifying stack of every finite linear (or projective)
reflection group is trivial.

This poset is a key combinatorial tool also in the study of the motivic
class of the quotient variety U/G, where U is the open set of V where
the group acts trivially. We discuss the study of such classes by
starting from a theorem of Aluffi in the reflection groups case
and we conclude by showing that a similar result holds for finite
subgroups of GL_3(k) for an algebraically closed field of characteristic
zero.

These results relate naturally to Noether's Problem and to its
obstruction, the Bogomolov multiplier.

(Part of this is joint work with Emanuele Delucchi.)

Monday, October 5, 2015

**Florian Frick (Cornell)**

Tverberg-type
theorems are concerned with the intersection pattern of faces in a simplicial
complex when mapped to Euclidean space. One has to distinguish between results
for affine maps (with straight faces) and continuous maps: In the topological
case, number-theoretic conditions on the multiplicity of intersections play a
role. We will show that most Tverberg-type results, which were believed to
require proofs using involved techniques from algebraic topology, follow from a
simple combinatorial reduction via the pigeonhole principle. We will construct counterexamples
to the topological Tverberg conjecture by Bárány from 1976 building on recent
work of Mabillard and Wagner, and we will apply similar ideas to investigate
zero-sum problems in Euclidean space.

Joint work
with Pavle Blagojevic and Günter M. Ziegler.

Monday, September 28, 2015

**Michelle Wachs (UM)**

This talk is a continuation of the Sept. 21 talk.

Monday, September 21, 2015

**Michelle Wachs (UM)**

We consider a weighted version of the bond lattice of a graph. This generalizes the poset of weighted partitions introduced by Dotsenko and Khoroshkin and studied in a previous paper of the authors. We show that for chordal graphs, each interval of the weighted bond poset has the homotopy type of a wedge of spheres, and we present an intriguing connection with h-vectors of graph associahedra studied by Postnikov, Reiner and Williams, and others. This is joint work with Rafael Gonzalez D'Leon.