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.

Schedule (Fall 2018)

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

Zvi Rosen (Florida Atlantic University)

#### Combinatorial Neural Codes

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)

#### Positivity Conjectures for Jack Polynomials

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)

#### Reconstructing Spheres and Polytopes

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)

#### Convex Union Representability of Simplicial Complexes

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)

#### On the Intersection Lattice of the Homogenized Linial Arrangement

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)

#### Random preprocessing in computational topology

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)

#### Round polytopes

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

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)

#### A connection between tilings and matroids on the lattice points of a regular simplex

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)

#### Parity arguments in combinatorics and beyond

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)

#### Ehrhart positivity

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)

#### The chromatic quasisymmetric function of directed graphs

Dissertation defense

Monday, March 19, 2018

José Alejandro Samper (UM)

#### Threshold hypergraphs revisited

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)

#### Sign matrix polytopes

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)

#### On enumerators of Smirnov words by descents and by cyclic descents

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)

#### Some contractible 2-complexes do not embed in R^4

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)

#### Intersection patterns of sets

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)

#### The Sperner property

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)

##### TILINGS OF SPACE AND THE DEDEKIND-MACNEILLE COMPLETION OF BRUHAT ORDER

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)

##### CHROMATIC QUASISYMMETRIC FUNCTIONS AND HESSENBERG VARIETIES

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)

##### CHROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTED GRAPHS

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)

##### A COMBINATORIAL MODEL FOR THE BASED LOOP SPACE

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)

##### DISCRETE LINE FIELD

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)

##### HOPF ALGEBRAS IN COMBINATORICS

This is a continuing seminar that started during the summer.

Monday, April 24, 2017

Vasu Tewari (U Washington)

##### LABELED BINARY TREES, SCHUR-POSITIVITY AND GENERALIZED TAMARI LATTICES

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)

##### ON r-INVERSIONS AND SYMMETRIC FUNCTIONS

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)

##### DYCK PATHS AND POSITROIDS FROM UNIT INTERVAL ORDERS

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 IN WONDERLAND

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)

##### PARTITION INTO DISTINCT PARTS AND UNIMODALITY

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)

##### AROUND CODIMENSION ONE EMBEDDINGS

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)

##### RHOMBIC TILINGS AND BOTT-SAMELSON VARIETIES

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)

##### CYCLIC DESCENTS, TORIC SCHUR FUNCTIONS AND GROMOV-WITTEN INVARIANTS

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)

##### CYCLIC DESCENTS OF STANDARD YOUNG TABLEAUX

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)

##### ON LOCAL GLOBAL PHENOMENA IN THE BETTI TABLES OF STANLEY-REISNER IDEALS

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)

##### SIMPLICIAL ROOK GRAPHS: ALGEBRAIC AND COMBINATORIAL PROPERTIES

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)

##### A DIRECTED GRAPH GENERALIZATION OF CHROMATIC QUASI-SYMMETRIC FUNCTIONS

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)

##### ON CONJECTURAL RELATIVES OF MATROID POLYTOPES

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)

##### MOGAMI CONSTRUCTIONS OF MANIFOLDS FROM TREES OF TETRAHEDRA

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)

##### COUNTING WITH CONGRUENCE CONDITIONS

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 Π(1+ai), where n = Σ aipi 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 2Σ bi , where n=Σ 2bi is the binary expansion of n (due to I. G. Macdonald).

Monday, November 21, 2016

Jim Haglund (U Penn)

##### LLT POLYNOMIALS AND THE CHROMATIC SYMMETRIC FUNCTION OF UNIT INTERVAL ORDERS

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)

##### SUBRACK LATTICES OF GROUP RACKS

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)

##### FILTERED GEOMETRIC LATTICES

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)

##### DISCRETE VS CONTINUOUS VALUATIONS: SIMILARITIES AND DIFFERENCES

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)

##### POSITIVE SEMIDEFINITE MATRIX COMPLETION AND FREE RESOLUTIONS

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 DEHN-SOMMERVILLE RELATIONS AND THE CATALAN MATROIDS

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)

##### ON THE TOPOLOGY OF STEEL

Canceled due to hurricane Matthew

Monday, October 3, 2016

##### BRUHAT ORDER ON TWISTED IDENTITIES AND KLV POLYNOMIALS

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)

##### BLOWUP ALGEBRAS OF RATIONAL NORMAL SCROLLS

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

##### RANDOM TORIC SURFACES AND A THRESHOLD FOR SMOOTHNESS

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)

##### A CHARACTERIZATION OF SIMPLICIAL MANIFOLDS WITH g2 ≤ 2

The celebrated low bound theorem states that any simplicial manifold of dimension ≥ 3 satisfies g2 ≥ 0, and equality holds if and only if it is a stacked sphere. Furthermore, more recently, the class of all simplicial spheres with g2 = 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 g2 ≤ 2 can be obtained by retriangulating a polytopal sphere with a smaller g2. This implies Nevo and Novinsky’s result for simplicial spheres of dimension ≥ 4. More surprisingly, it also implies that all simplicial manifolds with g2 = 2 are polytopal spheres.

Monday, April 25, 2016

Miriam Farber (MIT)

##### WEAK SEPARATION, PURE DOMAINS, AND CLUSTER DISTANCE

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)

##### THE CHROMATIC QUASISYMMETRIC FUNCTION OF THE CYCLE

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)

##### MIT DISSERTATION DEFENSE: Antichains of Interval Orders and Semiorders, and Dilworth Lattices of Maximum Size Antichains

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)

##### NEW AND OLD GRAPH DIMENSION PARAMETERS

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)

##### THE COMBINATORICS OF THE WALDSPURGER DECOMPOSITION

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)

##### HIGHER DIMENSIONAL COLORINGS AND FLOWS, ARITHMETIC TUTTE POLYNOMIALS, AND CONVOLUTION FORMULAE

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)

##### TORIC ARRANGEMENTS AND GROUP ACTIONS ON SEMIMATROIDS

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)

##### EVALUATIONS OF THE POWER SUM TRACES AT KAZHDAN-LUSZTIG BASIS ELEMENTS OF THE HECKE ALGEBRA

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, DETERMINANTS, AND MATROIDS

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)

##### A BROAD CLASS OF SHELLABLE LATTICES

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)

##### DUAL GRAPHS AND THE CASTELNUOVO-MUMFORD REGULARITY OF SUBSPACE ARRANGEMENTS

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)

##### GEOMETRIC GRID CLASSES AND SYMMETRIC SETS OF PERMUTATIONS

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)

##### PARKING FUNCTIONS AND THEIR GENERATING FUNCTION

A parking function is a sequence (a1, ..., an) of positive integers whose increasing rearrangement (b1, ... , bn) satisfies bi ≤ 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 Sn acts on parking functions of length n by permuting coordinates. This action corresponds to a symmetric function Fn with many interesting properties. We discuss properties of the generating function ∑ Fn zn 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)

##### BOOLEAN STATISTICS

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)

##### CHROMATIC SYMMETRIC FUNCTIONS ON SIMPLICIAL COMPLEXES

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)

##### DUAL GRAPHS OF PROJECTIVE CURVES

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)

##### OPTIMAL DISCRETE MORSE VECTORS ARE NOT UNIQUE

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)

##### RELAXATIONS OF THE MATROID AXIOMS

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)

##### POWER SUM EXPANSION OF THE CHROMATIC QUASISYMMETRIC FUNCTIONS

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)

##### A NON-PARTITIONABLE COHEN-MACAULAY COMPLEX

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)

##### ARRANGEMENTS OF SUBSPACES FOR FINITE GROUPS AND THEIR GEOMETRICAL APPLICATIONS

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 AND ZERO-SUM PROBLEMS

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)

##### WEIGHTED BOND POSETS AND GRAPH ASSOCIAHEDRA

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

Monday, September 21, 2015

Michelle Wachs (UM)

##### WEIGHTED BOND POSETS AND GRAPH ASSOCIAHEDRA

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.