Speaker: Christos Athanasiadis
Title: Combinatorics of uniform triangulations of simplicial complexes
Abstract: A triangulation of a simplicial complex \Delta is said to be uniform if the face vector of its restriction to a face of \Delta depends only on the dimension of that face. Barycentric subdivisions, edgewise subdivisions and antiprism triangulations are prototypical examples. This talk aims to show that uniform triangulations provide a convenient, general framework to study questions of the following type:
How does the h-vector of a simplicial complex \Delta transform after simplicial subdivision? Which triangulations of \Delta have a real-rooted h-polynomial? When does the latter have a nice symmetric decomposition?
Which triangulations of a simplex have a real-rooted local h-polynomial?
Some answers to these questions, obtained over the past few years, as well as related conjectures, will be presented.
Speaker: Andrii Bondarenko
Title: Small volume bodies of constant width
Abstract: For every large enough n, we explicitly construct a body of constant width 2 that has volume less than 0.9^n Vol(B^{n}), where \B^{n} is the unit ball in R^{n}. We will explain our construction and discuss related results.
Speaker: Chara Charalambous
Title: Syzygies of toric and monomial ideals.
Abstract: TbA
Speaker: Alicia Dickenstein
Title: Sparse systems with high local multiplicity
Abstract: Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). Together with Frédéric Bihan and Jens Forsgård, we explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound based on the computation of covolumes that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.
Speaker: Alexey Garber
Title: On Spheres with k Points Inside
Abstract: A classical result of Delone claims that for a finite and generic point set A in R^d, every generic point in the convex hull of A belongs to exactly one simplex with empty circumsphere. The collection of all these simplices is called the Delaunay traingulation of A. In the talk I will discuss a generalization of Delaunay’s result to the case of simplices with k points inside their circumspheres. I will also talk about possible extensions to the case of weighted points sets and point sets in S^d, and sketch a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers. The talk is based on a joint work with Herbert Edelsbrunner and Morteza Saghafian.
Speaker: Gil Kalai
Title: Some of my favorite problems
Abstract:
I will review some of my favorite problems in combinatorial geometry and some possible connections with algebra and topology.
Speaker: Eric Katz
Title: Invariants of subdivisons of polytopes and matroids
Abstract: We will discuss invariants of subdivisions of lattice polytopes (and conjecturally matroids) arising from Hodge structures. We will recount old work with Stapledon and explain how it might be extended in new directions.
Speaker: Gaku Liu
Title: Unimodular triangulations of matroid polytopes
Abstract: A unimodular triangulation of a lattice polytope is a lattice triangulation in which all simplices have the minimum possible volume. We give a construction of a regular unimodular triangulation of any matroid polytope. We also discuss connections to toric ideals of matroids and White's conjecture. Joint with Spencer Backman.
Speaker: Fedor Manin
Title: Quantitative PL bordism and other simplicial isoperimetric problems
Abstract: Given a triangulation of the n-sphere with K n-dimensional faces, how many (n+1)-dimensional faces do you need to extend this triangulation to the (n+1)-disk? The problem as stated is trivial (just take the cone!) but many variations on it are more subtle, or open problems. For example, it becomes an open problem if one only considers triangulations with vertex degree bounded by some (sufficiently large) constant L. Most of the questions I will discuss are due to Gromov, and to the extent we have answers they are joint with Shmuel Weinberger.
Speaker: Pavel Patak
Title: Shellability
Abstract: A simplicial d-complex is shellable,if it can be built by gradually gluing d-simplices together in such a way that each new d-simplex is glued to the already constructed complex along a union of (d-1)-faces.
In the first part of the talk we give a brief overview of theorems and ideas related to shellability: Balloon flights, Dehn-Sommerville relations, Upper and Lower Bound Theorems, Homotopy Cohen-Macaulayness. In the second part we turn our attention to non-shellable complexes and show the construction of non-shellable spheres and balls. This part culminates with our recent proof that shellability of 3-balls is an NP-complete problem.
Speaker: Zuzana Patáková
Title: On Radon and Helly-type theorems
Abstract: Are the classical convexity theorems really theorems about standard convex sets? To what extent do they go beyond the classical notion of convexity? Such questions - from a combinatorial viewpoint - led to the notion of convexity spaces. Here we focus on specific yet very general convexity spaces and show that we can obtain Helly-type results as soon as they satisfy some weak conditions.
Speaker: Miles Reid
Title: Codimension 4 Gorenstein rings and construction of algebraic varieties.
Abstract: Godeaux surfaces with 3-torsion were my entry point into codimension 4 Gorenstein rings in my 1978 paper. Our Tom and Jerry unprojection methods now give several hundred constructions of new families of algebraic surfaces, 3-folds and higher dimensional varieties. A current challenge is to apply these methods to 3-torsion Godeaux surfaces in mixed characteristic at 3.
Speaker: Christos Tatakis
Title: Robust and generalized robust toric ideals which are generated by quadrics and MG-toric ideals of graphs.
(joint works with Ignacio Garc\'ia-Marco and Irene M\'arquez-Corbella)
Abstract: On the first part of this talk, we study robust and generalized robust toric ideals of graphs which are generated by quadrics. A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, while it is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials.
On the second part of this talk, we introduce the notion of MG-ideals. An ideal I is called an MG-ideal if it is minimally generated by a reduced Gr\"obner basis of I. Ohsugi and Hibi proved that a toric ideal I_G of a bipartite graph G is generated by quadrics if and only if I_G has a Gr\"obner basis consisting of quadrics, i.e. all its minimal generators have degree two, if and only if the polynomial ring K[G] is Koszul. We will see that for a bipartite graph G such that all minimal generators of I_G have the same degree \mu for any \mu\geq 2, the ideal I_G is an MG-ideal, while this is not true for non bipartite graphs.
Speaker: Apostolos Thoma
Title: Strongly robust toric ideals
Abstract: A positively graded toric ideal is called strongly robust if it is minimally generated by its Graver basis. It follows that for a strongly robust toric ideal I_A the following sets are identical: the set of indispensable elements, any minimal system of generators, any reduced Gr{\"o}bner basis,the Universal Gr{\"o}bner basis and the Graver basis. We discuss a simplicial complex that determines the strongly robust property for toric ideals.
This is joint work with Dimitra Kosta and Marius Vladoiu.
Speaker: Volkmar Welker
Title: Posets of Decompositions and the Common Basis Complex
(joint works with K. Piterman and B. Br\"uck)
Abstract: In the literature, a variety of posets and simplicial complexes have been associated to a finite dimensional vectorspace. Among them the lattice of subspaces, the independence complex, the poset of partial and full decompositions (ordered versions of the preceding two constructions), the common basis complex. Analogous constructions appear in the literature for matroids, free groups, free modules, vectorspaces with non-degenerate forms. In this talk we provide a general framework unifying the constructions and establishing general homotopy equivalences between them as well as concrete results for example classes. In the talk we focus on posets of partial decompositions and the common basis complex.
Speaker: Pavel Paták
Title: Topological Helly theorems
Abstract: One way how to prove topological Helly theorem is by using some variant of the nerve theorem. However, the limitation of this approach is that it requires that the considered homology groups have to be trivial. In this talk we focus on the second way, which combines non-embeddability results and Ramsey type techniques and allows us to lift the condition of trivial homology. We also list major open problems in the area and show how their solution would lead to better bounds and more applicable results.
Speaker: Gangyotri Sorcar
Title: Arbitrarily high dimensional hyperbolic right-angled Coxeter groups that virtually fiber.
Abstract: A group is said to virtually fiber if it has a finite index subgroup that surjects onto Z with a finitely generated kernel. In this talk, we give an iterative procedure that produces infinitely many isomorphism classes of hyperbolic right-angled Coxeter groups (RACGs) in arbitrarily high virtual cohomological dimension > 2 that virtually fiber. Our methods involve a novel generalization of a simplicial thickening construction introduced by Osajda (to produce hyperbolic RACGs with arbitrarily high vcd) which allows us to then apply a combinatorial criterion given by Jankiewicz, Norin, Wise that implies virtual fibering of RACGs. This is joint work with Jean-Francois Lafont, Matthew Stover, Barry Minemyer, and Joseph Wells.
Speaker: Eleni Tzanaki
Title: Symmetric decompositions, triangulations and real-rootedness
Abstract: A triangulation of a simplicial complex Δ is said to be uniform if the f-vector of its restriction to a face of Δ depends only on the dimension of that face. The notion of uniform triangulation was introduced by Christos Athanasiadis in order to conveniently unify several well known types of triangulations such as barycentric, r-colored barycentric, r-fold edgewise etc. These triangulations have the common feature that, for certain "nice" classes of simplicial complexes Δ, the h-polynomial of the triangulation Δ′ of Δ, is real rooted with nonnegative coefficients. Athanasiadis proved that, uniform triangulations having the so called stong interlacing property, have real rooted h-polynomials with nonnegative coefficients. We continue this line of research and we study under which conditions the h-polynomial of a uniform triangulation Δ′ of Δ has a nonnegative real rooted symmetric decomposition. We also provide conditions under which this decomposition is also interlacing. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in h-vector theory arise from this work.
This is joint work with Christos Athanasiadis.
Speaker: Gökhan Yıldırım
Title: Pattern-avoiding Inversion Sequences
Abstract: The study of enumerative and probabilistic aspects of pattern-avoiding combinatorial structures, such as permutations, words, and sequences, has been a vibrant area of research. Over the past fifty years, extensive literature has emerged on pattern-avoiding permutations. Recently, researchers have shifted their focus to pattern-avoiding inversion sequences, uncovering intriguing insights in this domain. After reviewing recent results and making comparisons with pattern-avoiding permutations, we will discuss an enumeration method based on generating trees and the kernel method. This method offers a systematic approach to enumerating various pattern classes for inversion sequences. Next, we will address the sampling problem for pattern-avoiding inversion sequences and propose sampling algorithms capable of generating approximately uniform samples. The sampled data provide insights into the typical structure of pattern-restricted inversion sequences and allow us to compare distinct patterns. Based on these observations, we will present some explicit bijections within the same Wilf class for pattern-avoiding inversion sequences, preserving statistics such as the number of zeros, distinct elements, and repeated elements. We will conclude with some open problems and discuss future research directions. The talk is based on joint work with Ilias Kotsireas and Toufik Mansour, as well as another work with Melis Gezer.
Short Talks
Speaker: Thiago Holleben, Dalhousie University, Canada
Title: Lefschetz properties and analytic spread
Abstract: When studying Lefschetz properties of algebras defined by monomial ideals, one may take the sum of the variables in the ring to bethe linear form. This fact makes the computation of the matrices appearing as multiplication maps practical. In Commutative Algebra, more specifically in the study of Rees rings, an important invariant called analytic spread can, in some cases, be computed by checking the rank of a matrix. In this talk, we will go through examples where these two theories intersect, and look at consequences such as studying the failure of the Weak Lefschetz Property in positive characteristics using mixed multiplicities, and examples of level algebras failing the Strong Lefschetz properties that arise from the theory of symbolic powers.
Speaker: Geunho Lim, Hebrew University of Jerusalem,Israel
Title: Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
Abstract: Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on manifolds with arbitrary fundamental groups, as a generalization of the Atiyah-Singer rho-invariant. There are many interesting applications in geometry and topology. In this talk, we show linear bounds on the rho-invariants in terms of simplicial complexity of manifolds. First, we obtain linear bounds on Cheeger-Gromov invariants, using hyperbolizations. Next, we give linear bounds on Atiyah-Singer invariants, employing a combinatorial concept of G-colored polyhedra. As applications, we give new concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces.
This is a joint work with Shmuel Weinberger.
Speaker: Danai Deligeorgaki, KTH Royal Institute of Technology, Sweden
Title: Distributional properties of colored multiset Eulerian polynomials
Abstract: The central objects in this talk are the descent polynomials of colored permutations on multisets, referred to as colored multiset Eulerian polynomials. These polynomials generalize the colored Eulerian polynomials that appear frequently in algebraic combinatorics and are known to admit desirable distributional properties, including real-rootedness, log-concavity, unimodality and the alternatingly increasing property. In joint work with Bin Han and Liam Solus, symmetric colored multiset Eulerian polynomials are identified and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be interlaced by its own reciprocal. This property implies that the polynomial obtains all of the aforementioned distributional properties as well as others, including bi-gamma-positivity. To derive these results, multivariate generalizations of a generating function identity due to MacMahon are deduced. We will discuss this identity and end with open questions and some connections to s-Eulerian polynomials.
Speaker: Rodica Dinu, University of Konstanz and IMAR, Germany
Title: Algebraic degrees of phylogenetic varieties
Abstract: Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in studying the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this talk, we present concrete results on the phylogenetic degrees of the variety X_{G, n} with G\in {Z_2, Z_2\times Z_2, Z_3} and any n-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes P_{G,n}.
We present combinatorial methods used in our work and we give concrete formulas for these volumes.
This talk is based on a joint work with Martin Vodi\v{c}ka.
Speaker: Lazar Guterman, Hebrew University of Jerusalem, Israel
Title: Electrical networks via concordance of non-crossing partitions
Abstract: The theory of electrical networks is a subject of extensive research for the last thirty years. An embedding of electrical networks into the totally nonnegative Grassmannian was constructed by Lam. He provided a combinatorial description of the image of this embedding using a concordance of non-crossing partitions. Later it was independently established by Chepuri, George, Speyer and Bychkov, Gorbounov, Kazakov, Talalaev that the image in fact lies in the Lagrangian Grassmannian. We show that the combinatorics of concordance implies the existence of a unique symplectic form with respect to which the points of the Grassmannian corresponding to electrical networks are Lagrangian. This result provides a direct construction of the compactification of electrical networks on Lagrangian Grassmannian by passing Lam’s construction.
The talk is based on the joint work with Boris Bychkov, Vassily Gorbounov and Anton Kazakov.
Speaker: Lilja Metsälampi, Aalto University, Finland
Title: Uniqueness of psd factorizations using rigidity theory
Abstract: A positive semidefinite (PSD) factorization of size-k of a nonnegative p x q matrix M is a collection of k x k symmetric PSD matrices A_1, ..., A_p, B_1, ..., B_q, such that the (i,j)-th entry of M is given by the trace product of A_i and B_j. The smallest k for which M admits a PSD factorization of size-k is called the PSD rank of M. A matrix is of minimal PSD rank, if it is a rank- k(k+1)/2 and PSD rank-k matrix. PSD factorizations originally appeared in semidefinite programming where PSD rank is related to the complexity of a semidefinite program over a convex set. Using notions from rigidity theory we study the uniqueness of PSD factorizations. We introduce infinitesimal motions of PSD factorizations and characterize them for matrices of minimal PSD rank. Trivial motions are infinitesimal motions of any PSD factorization, and they are characterized by diagonal matrices with all entries equal. The factorizations which have only trivial infinitesimal motions are called infinitesimally rigid. In the case of rank 3 and PSD rank 2 matrices infinitesimal rigidity is dependent on the rank of the factor matrices; for infinitesimal rigidity the factorization needs to contain enough rank-1 matrices.
This is a joint work with Kristen Dawson (SFSU), Serkan Hoşten (SFSU) and Kaie Kubjas (Aalto University).