Events
(Polytop)ics: Recent advances on polytopes

MPI für Mathematik in den Naturwissenschaften Leipzig

Live Stream

conference

06.04.21 09.04.21

(Polytop)ics: Recent advances on polytopes

Polytopes have been present since the dawn of civilization, in the pyramids of Egypt and the platonic solids of Greece. They are fundamental objects in modern mathematics and naturally appear in many different fields, from optimization to algebraic geometry.

In recent years we have seen solutions to old problems as well as new exciting directions emerge. The aim of this conference is to provide an overview of the recent advances as well as showcasing some important questions for the future. Along with invited and contributed talks, we will have open problem sessions where participants can present a problem that they find interesting.

Confirmed Speakers are:

Federico Ardila (San Francisco State University)
Christos A. Athanasiadis (University of Athens)
Matthias Beck (San Francisco State University & Freie Universität Berlin)
Nick Early (Institute for Advanced Study, Princeton)
Laura Escobar (Washington University in St. Louis)
Katharina Jochemko (Royal Institute of Technology (KTH))
Michael Joswig (Technical University Berlin & MPI MiS)
Martina Juhnke-Kubitzke (University of Osnabrück)
Fu Liu (University of California, Davis)
Karola Mészáros (Cornell University)
Arnau Padrol (Institut de Mathématiques de Jussieu)
Igor Pak (University of California, Los Angeles)
Victor Reiner (University of Minnesota)
Francisco Santos (Universidad de Cantabria)
Raman Sanyal (Goethe University, Frankfurt)
Günter Ziegler (Freie Universität Berlin)

The deadline for submitting contributed talks is March 9, 2021. General registration deadline is March 30, 2021.

The link to the virtual event will be made available via email to the registered participants in due time.

Program

09:00 - 10:00	Bullet Intro
10:00 - 10:30	Coffee Break
10:30 - 11:30	Michael Joswig (TU Berlin & MPI MiS) Geometric disentanglement by random convex polytopes We propose a new geometric method for measuring the quality of representations obtained from deep learning. Our approach, called "Random Polytope Descriptor", provides an efficient description of data points based on the construction of random convex polytopes. We demonstrate the use of our technique by qualitatively comparing the behavior of classic and regularized autoencoders. This reveals that applying regularization to autoencoder networks may decrease the out-of-distribution detection performance in latent space. While our technique is similar in spirit to k-means clustering, we achieve significantly better false positive/negative balance in clustering tasks on autoencoded datasets. Joint work with Marek Kaluba and Lukas Ruff. Video (720p) Video (1080p) Slides
11:30 - 12:00	Coffee Break
12:00 - 13:00	Francisco Santos (Universidad de Cantabria) Empty simplices of large width An empty simplex is a lattice simplex in which vertices are the only lattice points. After reviewing what is known about width of lattice polytopes and convex bodies, we show two constructions leading to the first known empty simplices of width larger than their dimension:(1) We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$ . Among them we find five empty ones of width $11$ , and none of larger width.(2) Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and of width growing asymptotically as $d /$ arcsinh $(1) \sim 1.1346 d$ .The width in part (2) is (asymptotically) only $3 %$ lower than the widest convex bodies known.This is joint work with Joseph Doolittle, Lukas Katthän, and Benjamin Nill. Video (720p) Video (1080p) Slides
16:00 - 17:00	Nick Early (Institute for Advanced Study) Generalized scattering amplitudes, matroidal blade arrangements and the positive tropical Grassmannian Seventy three years ago, Richard Feynman introduced a diagrammatic formalism in Quantum Field Theory (QFT) to organize the expansion of a scattering amplitude as a sum of elementary rational functions, labeled by graphs. These graphs, now called Feynman Diagrams, revolutionized theoretical physics and are still widely used today.In using the scattering equations formalism to compute scattering amplitudes in terms of a Morse theoretic question posed on the Riemann sphere ${CP}^{1}$ , as introduced by Cachazo-He-Yuan in 2013, one can't help but feel that Feynman Diagrams should be living, breathing beings in their own right. Taking this idea seriously, in 2019, in joint work with Cachazo, Guevara and Mizera, we formulated an analog of the scattering equations on ${CP}^{k - 1}$ , and introduced a generalized notion of scattering amplitudes: the generalized biadjoint scalar $m_{n}^{(k)}$ . We found that, indeed, the set of Generalized Feynman Diagrams for the first new generalized amplitude $m_{6}^{(3)}$ should be understood as the set of maximal cones of the positive tropical Grassmannians ${Trop}^{+} G (3, 6)$ ! This observation was quickly generalized to all ${Trop}^{+} G (k, n)$ .In this talk, we will explore how a novel structure in combinatorial geometry, the matroidal blade arrangement, can be used to study the rays of the positive tropical Grassmannian by selecting, essentially canonically, a very special basis of height functions over the hypersimplex $Δ_{k, n}$ , and, in particular, to organize and represent the poles appearing in Generalized Feynman Diagrams for $m_{n}^{(k)}$ . Applications include an appearance of weak separation as a compatibility condition for certain positroidal subdivisions of $Δ_{k, n}$ . Video (720p) Video (1080p) Slides
17:00 - 17:30	Coffee Break
17:30 - 18:30	Contributed Talks Session 2
17:30 - 18:30	Contributed Talks Session 1
	Christoph Hertrich (TU Berlin) Tackling neural network expressivity via polytopes This talk deals with polytope theory in the context of an open problem concerning the expressivity of feedforward neural networks (NNs) with rectified linear unit (ReLU) activations. While it has been shown that NNs with logarithmic depth (in the input dimension) can represent any continuous, piecewise linear (CPWL) function, it is still unknown whether there exists any CPWL function that cannot be represented with three layers. In the first part of the talk, we present first progress towards depth lower bounds: based on mixed-integer linear programming we show that a 3-layer NN cannot compute the maximum of five scalar inputs under a natural, yet unproven assumption. In the second part, we explain how polytope theory might help to tackle the open problem via Newton polytopes of convex CPWL functions (aka tropical signomials). This is joint work in progress with Amitabh Basu, Marco Di Summa, and Martin Skutella. Video (720p) Video (1080p) Slides
	Sophie Rehberg (Freie Universität Berlin) Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned inside-out polytopes Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart--Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar--Ardila (2017) and Billera--Jia--Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval--Karaboghossian--Tanasa (2020). Video (720p) Video (1080p) Slides
	Jose Bastidas (Cornell University) Generators for type B permutahedra via McMullen's polytope algebra Ardila-Benedetti-Doker showed that any generalized permutahedron is a signed Minkowski sum of the faces of the standard simplex. In contrast, Ardila-Castillo-Eur-Postnikov observed that the faces of the cross-polytope only generate a subspace of roughly half the dimension in the space of deformations of the type B permutahedron. In this talk, I will use McMullen's polytope algebra to help explain this phenomenon. Concretely, I will consider the subalgebra generated by (type B) generalized permutahedra and endow it with the structure of a module over the Tits algebra of the corresponding Coxeter arrangement. The module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least $2^{d - 1}$ full-dimensional polytopes. I will present a family of generators that shows that this lower bound is sharp. Video (720p) Video (1080p) Slides
	Jonathan Leake (TU Berlin) Flow polytope volume bounds via polynomial capacity Polynomial capacity has been used in the past 20 years to obtain lower bounds on various combinatorial quantities, including the permanent, the mixed discriminant, matchings of a graph, and the intersection of two matroids (to name a few). In joint work with Petter Brändén and Igor Pak, we have recently used capacity to obtain improved lower bounds on the number of contingency tables with given marginals. Contingency tables can be viewed as the lattice points of transportation polytopes and flow polytopes more generally, and so these lower bounds imply volume lower bounds for such polytopes. In this talk, we state these bounds and discuss a few key points of their proofs. Video (720p) Video (1080p) Slides
18:30 - 19:00	Coffee Break
19:00 - 20:00	Bullet Intro

09:00 - 10:00	Katharina Jochemko (KTH Royal Institute of Technology) Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity Generalized permutahedra form a combinatorially rich class of polytopes that appear naturally in various areas of mathematics. They include many interesting and significant classes of polytopes such as matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We present classification results and discuss how these can be applied to prove positivity of the linear coefficient of the Ehrhart polynomial of generalized permutahedra. This is joint work with Mohan Ravichandran. Video (720p) Video (1080p) Slides
10:00 - 10:30	Coffee Break
10:30 - 11:30	Matthias Beck (San Francisco State & FU Berlin) Rational Ehrhart Theory The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results with classical Ehrhart quasipolynomials, as well as generalized reflexive polytopes studied by Fiset-Kasprzyk (2008) and Kasprzyk-Nill (2012). This is joint work with Sophia Elia and Sophie Rehberg. Video (720p) Video (1080p) Slides
11:30 - 12:00	Coffee Break
12:00 - 13:00	Contributed Talks Session 2
12:00 - 13:00	Contributed Talks Session 1
	Theo Douvropoulos (University of Massachusetts, Amherst) Volumes of root zonotopes via the W-Laplacian The normalized volume of the classical permutahedron is given by the formula $n^{n - 2}$ which, ask any combinatorialist and they will tell you, agrees with the number of trees on n vertices.This coincidence is well understood; the classical permutahedron is a unimodular zonotope, on the set of positive roots of the Symmetric group $S_{n}$ , and its bases are indexed by trees and hence enumerated by the determinant of a Laplacian matrix. This description of the permutahedron lends itself to a natural generalization: for a Weyl group W, the zonotope associated to the collection of positive roots of W will be called the root zonotope $Z_{W}$ . These zonotopes $Z_{W}$ are not unimodular, but their bases can be differentiated with respect to the reflection subgroup they generate (and its connection index). In joint work with Guillaume Chapuy (arxiv.2012.04519) we have introduced for any Weyl group W a (nxn) Laplacian matrix $L_{W}$ whose spectrum encodes (nontrivially) many enumerative properties of W. In particular we will present new short formulas (ibid: Section 8.3) for the volumes of the root zonotopes $Z_{W}$ involving only the Coxeter numbers of W and its reflection subgroups. Our approach is uniform and does not rely on the classification. Video (720p) Video (1080p) Slides
	Doriann Albertin (Université Gustave Eiffel) Type cones of permutree fans Permutreehedra are polytopes that interpolate between the permutahedron, the associahedron and the cube. They were constructed as removahedra, i.e. by deleting inequalities in the facet description of the classical permutahedron. We investigate the type cone (the space of polytopal realizations) of permutree fans and prove that this removahedral construction works starting from any realization of the braid fan. Video (720p) Video (1080p) Slides
	Martin Winter (TU Chemnitz) Capturing polytopal symmetries in the edge-graph In general dimension the edge-graph of a polytope carries very little information about the polytope itself. For example, the edge-graph can have many more symmetries than the polytope or any realization thereof. Using techniques from spectral graph theory and convex geometry we show that the edge-graph of every polytope can be colored so that every combinatorial symmetry of the colored edge-graph extends to a geometric symmetry of the polytope. Up to now, the only proof known of this fact makes use of spectral graph theory and demonstrates the usefulnes of this seemingly unrelated subject in the study of polytopes. Video (720p) Video (1080p) Slides
	Luis Ferroni (University of Bologna) The Erhart polynomials of matroids and hypersimplices The Ehrhart polynomial of a polytope counts the number of lattice points in every integer dilation of the polytope. A conjecture of De Loera et al. states that if the polytope is a matroid polytope, then these polynomials have positive coefficients. For uniform matroids it is indeed true, because in fact in that case the polytopes are hypersimplices. We will also discuss some purely matroidal operations that behave nicely with the Ehrhart polynomial of a matroid and discuss further conjectures and results on the matter. Video (720p) Video (1080p) Slides
16:00 - 17:00	Open Problems
17:00 - 17:30	Coffee Break
17:30 - 18:30	Laura Escobar (Washington University in St. Louis) Determining the complexity of Kazhdan-Lusztig varieties Kazhdan-Lusztig varieties are defined by ideals generated by certain minors of a matrix, which are chosen by a combinatorial rule. These varieties are of interest in commutative algebra and Schubert varieties. Each Kazhdan-Lusztig variety has a natural torus action from which one can construct a polytope. The complexity of this torus action can be computed from the dimension of the polytope and, in some sense, indicates how close the geometry of the variety is to the combinatorics of the associated polytope. In joint work with Maria Donten-Bury and Irem Portakal we address the problem of classifying which Kazhdan-Lusztig varieties have a given complexity. We do so by utilizing the rich combinatorics of Kazhdan-Lusztig varieties, which is reflected on the polytopes.
18:30 - 19:00	Coffee Break
19:00 - 20:00	Igor Pak (UCLA) Domes over curves A closed PL-curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve $γ$ in the $3$ -dimensional euclidean space, there is a dome over $γ$ , i.e. whether $γ$ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. I will survey both positive and negative results on the subject. Joint work with Alexey Glazyrin. Video (720p) Video (1080p) Slides

09:00 - 10:00	Günter M. Ziegler (Freie Universität Berlin) Problems and methods, examples and counterexamples My plan is to talk about the value of examplesthe value and the limits of enumerationour limited tool-box of construction methodsthe challenges of explicit constructionsthe value of counter-examples the value and challenges of numerical computationin polytope theory and to illustrate this with some of my favourite (open and solved) polytope problems from the last 35 years. Video (720p) Video (1080p) Slides
10:00 - 10:30	Coffee Break
10:30 - 11:30	Raman Sanyal (Goethe-Universität Frankfurt) Normally inscribable polytopes A polytope $P$ is inscribable if there is a combinatorially equivalent polytope $P^{'}$ with all vertices contained in a sphere. This notion relates to the combinatorics of ideal hyperbolic polytopes and Delaunay subdivisions. Steinitz showed that not every polytope is inscribable and Rivin gave a complete and effective characterization for $3$ -dimensional polytopes. There is no such characterization in dimensions $4$ and up. I will discuss the related notion of normally inscribable polytopes: A polytope $P$ is normally inscribable if there is a normally equivalent polytope $P^{'}$ with all vertices on a sphere. Normal inscribability, it turns out, reveals a fascinating interplay of algebra, geometry, and combinatorics. I will explain connections to a deformation theory of ideal hyperbolic polytopes and Delaunay subdivisions, routed particle trajectories, and reflection groupoids. In particular, for zonotopes, this reveals new connections to reflection groups and Grünbaum's quest for simplicial arrangements. This is based on joint work with Sebastian Manecke. Video (720p) Video (1080p) Slides
11:30 - 12:00	Coffee Break
12:00 - 13:00	Open Problems
16:00 - 17:00	Karola Meszaros (Cornell University) Schubert polynomials from a polytopal point of view Schubert polynomials are multivariate polynomials representing cohomology classes on the flag manifold. Despite the beautiful formulas developed for them over the past three decades, the coefficients of these polynomials remained mysterious. I will explain Schubert polynomials from a polytopal point of view, answering, at least partially, the questions: Which coefficients are nonzero? How do the coefficients compare to each other in size? Are the Newton polytopes of these polynomials saturated? Are their coefficients log-concave along lines? Is there a polytope whose integer point transform specializes to Schubert polynomials? As the questions themselves suggest, we will find that polytopes play an outsized role in our understanding. The talk is based on joint works with Alex Fink, June Huh, Ricky Liu, Jacob Matherne and Avery St. Dizier.
17:00 - 17:30	Coffee Break
17:30 - 18:30	Contributed Talks Session 2
17:30 - 18:30	Contributed Talks Session 1
	Rida Ait El Manssour (MPI MiS) Stable set polytopes in differential algebra We present joint work with Anna-Laura Sattelberger at the interface of geometric combinatorics and differential algebra. We show how stable set polytopes of certain regular triangulations arise naturally in the study of jet schemes of monomials. Video (720p) Video (1080p) Slides
	Aniket Shah (Ohio State University) A q-analogue of Brion's identity from quantum K-theory We give a Brion-type formula for the weighted enumeration of lattice points in a polytope. The weighting has combinatorial significance, and also arises from the arc scheme of the associated toric variety. We obtain some new and old identities of q-series as consequences. Video (720p) Video (1080p) Slides
	Alex Black (UC Davis) Monotone paths on cross-polytopes In the early 1990s, Billera and Sturmfels introduced monotone path polytopes (MPPs). MPPs encode the combinatorial structure of paths potentially chosen by the simplex method to solve a linear program on a given polytope for a fixed linear functional. In their original paper, they showed that MPPs of simplices and cubes for any generic linear functional are combinatorial cubes and permutahedra respectively. The natural next question to ask is: What are the MPPs of cross-polytopes for generic linear functionals? I will present a complete characterization of MPPs of cross-polytopes for generic linear functionals and show how this result informs our understanding of MPPs of centrally symmetric polytopes. This talk is based on joint work with Jesús De Loera. Video (720p) Video (1080p) Slides
	Mariel Supina (University of California, Berkeley) The universal valuation of Coxeter matroids Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective. Video (720p) Video (1080p) Slides
18:30 - 19:00	Coffee Break
19:00 - 20:00	Fu Liu (University of California, Davis) Permuto-associahedra as deformations of nested permutohedra A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset (with a reasonable structure), determine whether there exists a polytope whose face lattice it the poset. In 1990s, Kapranov defined a poset called permuto-associhedron as a hybrid between the face poset of the permutohedron and the associahedron, and he asked whether this poset is realizable. Shortly after his question was posed, Reiner and Ziegler provided a realization. In this talk, I will discuss a different construction we obtained as a deformations of nested permutohedra. Video (720p) Video (1080p) Slides

09:00 - 10:00	Arnau Padrol (Sorbonne Université) Shard polytopes For any lattice congruence of the weak order on permutations, N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called quotientope. In this talk, I will present an alternative simpler approach to realize this quotient fan based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, this Minkowski sum approach extends to type B. This is joint work with Vincent Pilaud and Julian Ritter. Video (720p) Video (1080p) Slides
10:00 - 10:30	Coffee Break
10:30 - 11:30	Contributed Talks Session 2
10:30 - 11:30	Contributed Talks Session 1
	Sophia Elia (Freie Universität Berlin) Congruence normality for simplicial hyperplane arrangements The normal fan of a simple zonotope is a simplicial hyperplane arrangement. In rank 3 there is still much to learn about these arrangements. We provide an updated catalogue of the currently known simplicial hyperplane arrangements of rank 3, and compute normals and invariants of the arrangements. Additionally, we determine whether the associated posets of regions possess the combinatorial property of "congruence normality," which hints at potential geometric interpretations. We use methods from oriented matroids that make the computations possible. This refines the structure of the catalogue, breaking it into three separate combinatorial categories. In particular, we show that arrangements stemming from finite Weyl groupoids have congruence normal posets of regions. This is joint work with Michael Cuntz and Jean-Philippe Labbé. Video (720p) Video (1080p) Slides
	Marta Panizzut (TU Berlin) Algebraic degrees of 3-dimensional polytopes Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of three dimensional polytopes with edges tangent to the unit sphere. In this talk we introduce the study of algebraic degrees of such realizations. We will show how these notions provide algebraic complexity measures reflecting the combinatorics of the polytopes. Joint work with Mara Belotti and Michael Joswig. Video (720p) Video (1080p) Slides
	Akiyoshi Tsuchiya (University of Tokyo) Castelnuovo polytopes It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieves this bound are called Castelnuovo. On the other hand, there is a one-to-one correspondence between full-dimensional lattice polytopes and polarized toric varieties and we call a lattice polytope Castelnuovo if the associated polarized toric variety is Castelnuovo. In this talk, I give a combinatorial characterization of Castelnuovo polytopes and an application of this characterization. Video (720p) Video (1080p) Slides
	Eva Philippe (Ecole Normale Supérieure de Paris) Sweeps of a point configuration Consider a configuration of n labeled points in a Euclidean space. Any linear functional gives an ordering of these points: an ordered partition that we call a sweep, because we can imagine its parts as the sets of points successively hit by a sweeping hyperplane. The set of all such sweeps forms a poset which is isomorphic to a polytope, called the sweep polytope. I will present several constructions of the sweep polytope, related to zonotopes, projections of permutahedra and monotone path polytopes of zonotopes. This structure can also be generalized in terms of oriented matroids. For oriented matroids that admit a sweep oriented matroid, we gain precision on the topological description of their poset of cellular strings, refining a particular case of the Generalized Baues Problem. This is joint work with Arnau Padrol. Video (720p) Video (1080p) Slides
11:30 - 12:00	Coffee Break
12:00 - 13:00	Christos Athanasiadis (National and Kapodistrian University of Athens) Face numbers of barycentric subdivisions of cubical polytopes About fifteen years ago F. Brenti and V. Welker asked whether the face enumerating polynomial of the barycentric subdivision of any convex polytope has only real roots and showed that this is the case for simplicial polytopes. Until very recently, no strong evidence had been provided in the literature that such a result may hold beyond the simplicial case. This talk will give an affirmative answer for another important class of convex polytopes, namely that of cubical polytopes, and will discuss some related topics. Video (720p) Video (1080p) Slides
16:00 - 17:00	Martina Juhnke (Universität Osnabrück) On the bond polytope Given a graph $G = (V, E)$ , the maximum bond problem searches for a maximum cut $δ (S) \subseteq E$ with $S \subseteq V$ such that $G [S]$ and $G [V ∖ S]$ are connected. This problem is closely related to the well-known maximum cut problem and known under a variety of names such as largest bond, maximum minimal cut and maximum connected (sides) cut. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While cut polytopes have been intensively studied, there are no results on bond polytopes. We start a structural study of the latter.We investigate the relation between cut- and bond polytopes and study the effect of graph modifications on bond polytopes and their facets. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and $3$ -connected planar $(K_{5} - e)$ --minor free graphs. This is joint work with Markus Chimani and Alexander Nover. Video (720p) Video (1080p) Slides
17:00 - 17:30	Coffee Break
17:30 - 18:30	Victor Reiner (University of Minnesota - Twin Cities) Universal parameters for Stanley-Reisner rings and a colorful Hochster formula The Stanley-Reisner ring R of any simplicial complex contains a "universal" family of elements that form an h.s.o.p. (homogeneous system of parameters): the elementary symmetric functions in the vertex variables. This talk will discuss an interesting depth-detection property of this universal h.s.o.p., as well as a conjecture on the shape of the minimal free resolution of the ring R over this universal parameter ring. This conjecture is closely connected to a generalization of Hochster's formula on resolutions of Stanley-Reisner rings, one that applies to any proper vertex-coloring of the simplicial complex. Video (720p) Video (1080p) Slides
18:30 - 19:00	Coffee Break
19:00 - 20:00	Federico Ardila (San Francisco State University / Universidad de Los Andes) A tale of two polytopes: The bipermutohedron and the harmonic polytope In our work with Graham Denham and June Huh on the Lagrangian geometry of matroids, we introduced the conormal fan of a matroid M. Using tools from combinatorial Hodge theory, we used this fan to prove Brylawski and Dawson's conjectures on the log-concavity of the h-vector of the broken circuit and the independence complex of M. Two $(2 n - 2)$ -dimensional polytopes with $3^{n} - 3$ facets arose in our investigation of conormal fans: the bipermutohedron and the harmonic polytope. The bipermutohedron has $(2 n)! / 2^{n}$ vertices. Its h-polynomial is an analog of the Eulerian polynomial. It is real-rooted, and hence its coefficients are log-concave and unimodal. The harmonic polytope has $(n!)^{2} (1 + 1 / 2 + \dots + 1 / n)$ vertices. Its volume is a combination of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges. My talk will discuss these combinatorial constructions. The work on the harmonic polytope is joint with Laura Escobar. Video (720p) Video (1080p) Slides

Participants

Ron Adin

Bar-Ilan University

Marcelo Aguiar

Cornell University

Rida Ait El Manssour

MPI MiS

Mohammed Saquib Alam

KTH

Doriann Albertin

Université Gustave Eiffel

Anne Albuquerque

Universidade Federal Fluminense

Carlos Alejandro Alfaro

Banco de Mexico

Elie Alhajjar

United States Military Academy

Ayah Almousa

Cornell University

Ulysses Alvarez

Binghamton University

Imran Anwar

LUMS, Lahore Pakistan.

Federico Ardila

San Francisco State University / Universidad de Los Andes

Ahmed Ashraf

Habib University

Jai Aslam

North Carolina State University

Christos Athanasiadis

National and Kapodistrian University of Athens

Esme Bajo

UC Berkeley

Esther Banaian

University of Minnesota

Đorđe Baralić

Mathematical Institute SANU

mary barker

Washington University in St. Louis

Jose Bastidas

Cornell University

Amitabh Basu

Johns Hopkins University

Margaret Bayer

University of Kansas

Matthias Beck

San Francisco State & FU Berlin

Mara Belotti

Technische Universität Berlin

Bruno Benedetti

University of Miami

Carolina Benedetti Velásquez

Universidad de los Andes

Aenne Benjes

Goethe University Frankfurt

Olivier Bernardi

Brandeis University

Marcelo Bianchetti

University Of Buenos Aires

Anurag Bishnoi

TU Delft

Alex Black

UC Davis

Valentin Boboc

University of Manchester

Merve Bodur

University of Toronto

Alessio Borzì

University of Warwick

Tom Braden

University of Massachusetts Amherst

Marie Brandenburg

MPI MiS

Benjamin Braun

University of Kentucky

Sarah Brauner

University of Minnesota

Paul Breiding

MPI MiS

Matthias Brugger

TU München

Taylor Brysiewicz

MPI MiS

Théophile Buffière

Université Sorbonne Paris Nord

Amanda Burcroff

Durham University

Federico Castillo

Max Planck Institute for Mathematics in the Sciences

Cesar Ceballos

TU Graz

Marcel Celaya

ETH Zürich

Apostolos Chalkis

National and Kapodistrian University of Athens

Venkat Chandrasekaran

Caltech

Carles Checa Nualart

University of Athens

Giulia Codenotti

Goethe University Frankfurt

Laura Colmenarejo

UMass Amherst

Andrei Comăneci

Technische Universität Berlin

Norbert Cot

Univesity ParisDescartes

Luis Crespo Ruiz

Universidad de Cantabria

Angelica Cueto

The Ohio State University

Mark Curiel

University of Hawaii at Manoa

Alessio D'Alì

Universität Osnabrück

Jesus De Loera

University of California, Davis

Colin Defant

Princeton University

Nadine Defoßa

Goethe-Universität

Bérénice Delcroix-Oger

Université de Paris, IRIF

Vincent Delecroix

CNRS - Université de Bordeaux

Danai Deligeorgaki

KTH

Diego Delle Donne

ESSEC Business School of Paris

Maxim Demenkov

Institute of Control Sciences

Aram Dermenjian

York University

Antoine Deza

McMaster University

Marco Di Summa

Università degli Studi di Padova

Nicholai Dimov

Washington University in St. Louis

Anton Dochtermann

Texas State University

Joseph Doolittle

TU Graz

Galen Dorpalen-Barry

University of Minnesota

Maria Dostert

Royal Institute of Technology (KTH)

Theo Douvropoulos

University of Massachusetts, Amherst

Eliana Duarte

Otto-Von-Guericke Universität Magdeburg

Mathieu Dutour Sikiric

Institut Rudjer Boskovic

Nick Early

Institute for Advanced Study

Boulos El Hilany

Johann Radon Institute for Computational and Applied Mathematics

Sophia Elia

Freie Universität Berlin

Jonas Ellwanger

Goethe-Universität-Frankfurt

Ioannis Emiris

Athena Researh Center, and U. of Athens

Chris Eppolito

Binghamton University

Laura Escobar

Washington University in St. Louis

Christopher Eur

Stanford University

Elisenda Feliu

University of Copenhagen

Antonio Fernández

Universidad de Cantabria

Luis Ferroni

University of Bologna

Alex Fink

Queen Mary University London

Christian Gaetz

MIT

Yibo Gao

MIT

Alexey Garber

The University of Texas Rio Grande Valley & IST Austria

Shoni Gilboa

The Open University of Israel

Kurt Klement Gottwald

TU Chemnitz

Adam Gregory

University of Florida

Emily Gunawan

University of Oklahoma

Christian Haase

freie universität berlin

Tom Hall

University of Nottingham

Derek Hanely

University of Kentucky

Teresa Heiss

IST Austria

Paul Alexander Helminck

Swansea University

Martin Henk

TU Berlin

Oskar Henriksson

University of Copenhagen

Milena Hering

University of Edinburgh

Christoph Hertrich

TU Berlin

Johannes Hofscheier

The University of Nottingham

Mark Hogan

Cardiff University

Sam Hopkins

University of Minnesota

Serkan Hosten

San Francisco State University

Adeli Hutton

Washington University in St. Louis

Oscar Iglesias Valiño

Universidade de Vigo

Jelena Ivanovic

University of Belgrade

Hassan Izanloo

University of Bristol

Thomas Jahn

TU Chemnitz

Aryaman Jal

Royal Institute of Technology (KTH)

Robert Amzi Jeffs

University of Washington

Marija Jelic Milutinovic

University of Belgrade, Faculty of Mathematics

Katharina Jochemko

KTH Royal Institute of Technology

Michael Joswig

TU Berlin & MPI MiS

Martina Juhnke

Universität Osnabrück

Aikaterini Kalampogia-Evangelinou

University of Athens

Lars Kastner

TU Berlin

Christina Katsamaki

Inria Paris / IMJ-PRG

Elizabeth Kelley

University of Minnesota

Sangwook Kim

Chonnam National University

Tamas Kis

Institute for Computer Science and Control (SZTAKI)

Steven Klee

Seattle University

Florian Kohl

Aalto University

Daniel Köhne

Universität Osnabrück

Sampada Kolhatkar

Freie Universität Berlin

Christos Konaxis

National and Kapodistrian University of Athens

George Kontogeorgiou

University of Warwick

Andreas Kretschmer

OvGU Magdeburg

Lukas Kühne

MPI MIS

Olga Kuznetsova

Aalto University

Jean-Philippe Labbé

Électron Libre

Evgeniya Lagoda

FU Berlin

Matt Larson

Stanford University

Alexander Lazar

KTH Royal Institute of Technology

Jonathan Leake

TU Berlin

Kang-Ju Lee

Seoul National University

Gabriel Leuenberger

University of Groningen

Hsin-Chieh Liao

University of Miami

Apostolos Lioumpas

Aristotle University of Thessaloniki

Fu Liu

University of California, Davis

Georg Loho

University of Twente

Benjamin Lorenz

TU Berlin

Dante Luber

Technische Universität Berlin

Antonio Macchia

Freie Universität Berlin

John Machacek

Hampden-Sydney College

Moira MacNeil

University of Toronto

Pritam Majumder

Tata Institute of Fundamental Research

Joseph Malkoun

Notre Dame University-Louaize

Yelena Mandelshtam

UC Berkeley

Sebastian Manecke

Goethe Universität Frankfurt

Alessio Mansutti

University of Oxford

Aida Maraj

MPI MiS

Javier Marenco

University of Buenos Aires

Alex McDonough

Brown University --> UC Davis

Jodi McWhirter

Washington University in St. Louis

Chiara Meroni

MPI MIS Leipzig

Karola Meszaros

Cornell University

Victor Miller

Institute for Defense Analyses

Sebastian Mizera

IAS Princeton

Leonid Monin

University of Bristol

Guido Montufar

UCLA / MPI MiS

Vassilis-Dionyssis Moustakas

National and Kapodistrian University of Athens

Henri Mühle

Technische Universität Dresden

Hanna Mularczyk

Harvard University

Tommi Muller

University Of British Columbia

Philippe Nadeau

CNRS & Institut Camille Jordan

Mehdi Naima

Université Sorbonne Paris Nord

Methu Nath

Bangladesh University of Engineering and Technology

Shaheen Nazir

LUMS, Lahore Pakistan.

Eran Nevo

The Hebrew University

Louis Ng

UC Davis

Lisa Nicklasson

MPI MiS

Benjamin Nill

Otto-von-Guericke-Universität Magdeburg

Alexander Nover

Universität Osnabrück

Isabella Novik

University of Washington

Nida Obatake

Texas A&M University

Timm Oertel

Cardiff University

Jorge Alberto Olarte

TU Berlin

Arnau Padrol

Sorbonne Université

Igor Pak

UCLA

Marta Panizzut

TU Berlin

Greta Panova

University of Southern California

Zuzana Patáková

Charles University

Sasha Pevzner

University of Minnesota

Julian Pfeifle

Universitat Politècnica de Catalunya

Eva Philippe

Ecole Normale Supérieure de Paris

Vincent Pilaud

CNRS & École Polytechnique

Guillermo Pineda Villavicencio

Deakin University

Daria Poliakova

University of Copenhagen

Irem Portakal

OVGU Magdeburg

Germain Poullot

Institut de Mathématiques de Jussieu

Lionel Pournin

Université Sorbonne Paris Nord

Manuel Radons

TU Berlin

Rado Rakotonarivo

Université Sorbonne Paris Nord

Fanja Rakotondrajao

Université d'Antananarivo

Ivan Rasskin

Université de Montpellier

Laith Rastanawi

Freie Universität Berlin

Dimbiniaina RATOVILEBAMBOAVISON

Faculté des Sciences - Université d'Antananarivo

Farrokh Razavinia

University of Porto

Sophie Rehberg

Freie Universität Berlin

Victor Reiner

University of Minnesota - Twin Cities

Matthias Reitzner

Universität Osnabrück

Yue Ren

Swansea University

William Reynolds

University of Edinburgh

Harry Richman

University of Washington

Günter Rote

Freie Universität Berlin

Rowan Rowlands

University of Washington

Laxman Saha

Balurghat College

Manjil Saikia

Cardiff University

Phillippe Samer

Universitetet i Bergen

Jose Alejandro Samper Casas

PUC Chile

Francisco Santos

Universidad de Cantabria

Raman Sanyal

Goethe-Universität Frankfurt

Anna-Laura Sattelberger

MPI MiS Leipzig

Leonardo Saud Maia Leite

KTH

Karin Schaller

Freie Universität Berlin

Daniel Schmidt genannt Waldschmidt

TU Berlin

Benjamin Schröter

Royal Institute of Technology (KTH)

Andreas S. Schulz

TU München

Matthias Schymura

BTU Cottbus-Senftenberg

Aniket Shah

Ohio State University

Isabelle Shankar

UC, Berkeley

Elima Shehu

Polytechnic University of Tirana

Steffania Sierra Galvis

University of Padua

Rainer Sinn

Universität Leipzig

Martin Skutella

TU Berlin

Dorian Smith

University of Minnesota Twin Cities

Athanasios Sofronis

Heriot-Watt University

Liam Solus

KTH Royal Institute of Technology

Miruna-Stefana Sorea

SISSA

Daniel Soskin

Norte Dame University

Sylvain Spitz

TU Berlin

Avery St Dizier

University of Illinois at Urbana-Champaign

Dumitru Stamate

University of Bucharest

Mima Stanojkovski

MPI MiS

Johanna Steinmeyer

Hebrew University of Jerusalem & University of Copenhagen

Eric Nathan Stucky

University of Minnesota–Twin Cities

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

John Sullivan

TU Berlin

Gizem Süngü

Gebze Technical University

Mariel Supina

University of California, Berkeley

Pedro Tamaroff

Trinity College Dublin

Daniel Tamayo Jiménez

Université Paris-Saclay

Gabrielle Tauscheck

University of South Carolina

Andrew Tawfeek

Amherst College

Máté Telek

University of Copenhagen

Rolex Teologia

Xavier University-Ateneo de Cagayan

Vasu Tewari

University of Hawaii

Warut Thawinrak

University of California, Davis

Eliana Tolosa Villarreal

San Francisco State University

Elias Tsigaridas

Inria Paris

Akiyoshi Tsuchiya

University of Tokyo

Eleni Tzanaki

University of Crete, Greece

Yannic VARGAS

Instituto Venezolano de Investigaciones Científicas

Lorenzo Venturello

Royal Institute of Technology (KTH)

Ralihe R. Villagran

CINVESTAV

Diego Villamizar

Tulane University

Andrés Vindas Meléndez

University of Kentucky

Cynthia Vinzant

North Carolina State University

Matias von Bell

University of Kentucky

Matthias Walter

University of Twente

Amy Wiebe

Freie Universität Berlin

Kaelyn Willingham

Ohio State University

Martin Winter

TU Chemnitz

Rupei Xu

The University of Texas at Dallas

Martha Yip

University of Kentucky

David Yost

Federation University

Josephine Yu

Georgia Tech

Chi Ho Yuen

Brown University

Oguzhan Yürük

TU Berlin

Zafeirakis Zafeirakopoulos

Gebze Technical University

Allan Zea

TU Berlin

Becky Zhang

Washington University in St.Louis

Hailun Zheng

University of Copenhagen

Günter M. Ziegler

Freie Universität Berlin

Charilaos Zisopoulos

Saarland University

Rade Živaljević

Serbian Academy of Sciences and Arts

Scientific Organizers

Federico Castillo

Max Planck Institute for Mathematics in the Sciences

Giulia Codenotti

Goethe University Frankfurt

Benjamin Schröter

Royal Institute of Technology (KTH)

Administrative Contact

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail