Sep 05, 2017
Tuesday
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09:00 AM - 09:15 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:15 AM - 10:15 AM
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Matroids and Tropical Geometry
Federico Ardila (San Francisco State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Tropical algebraic geometry studies varieties by `tropicalizing' them into polyhedral complexes. The hope, which is realized surprisingly often, is that difficult algebro-geometric questions will reduce to simpler polyhedral questions. The resulting questions are usually not easy, and that’s good news for us: they have led to several new and interesting research directions in geometric combinatorics. Matroids often play a fundamental role because they are equivalent to tropical linear spaces. This talk will survey a few central aspects of the interplay between matroids and tropical geometry, and discuss some recent developments.
- Supplements
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10:15 AM - 10:45 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:45 AM - 11:45 AM
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Polynomial equations and polyhedra
Josephine Yu (Georgia Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss how Newton polyhedra and their generalizations are used to study solutions of systems of polynomial equations. For polynomials with generic coefficients, we can tell from the Newton polytopes whether the solution set is empty, how many solutions there are if finite, and whether the solution set is irreducible. We will also see tropical varieties, resultants, and polyhedral homotopies.
- Supplements
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11:45 AM - 12:30 PM
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Introductions by junior participants
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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12:30 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Unimodualr triangulations of lattice polytopes
Francisco Santos Leal (University of Cantabria)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Lattice polytopes (that is, polytopes with integer-coordiante vertices) are important both in Algebraic Geometry (toric geometry, commutative algebra, singularities) and Optimization (integer programming). In particular some attention has been devoted to triangulations of them and, most particularly, to the existence or not of unimodular triangulations for various families of them. In this talk I’ll try to survey what is known about triangulations of lattice polytopes, with an excursion into the classification of low-dimensional empty simplices, that is, lattice simplices with no lattice points other than their vertices. Empty simplices are important since they are the ``building blocks’’ in to which every lattice polytope can be triangulated.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Order, geometrically
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Geometric combinatorics is the art of studying discrete structures by way of geometry. True gems in this area are Stanley’s “two poset polytopes”. The order polytope and the chain polytope reflect much of the combinatorics of partially ordered sets (or, posets) in their boundary structures, their volumes, and their Ehrhart polynomials. In this talk I will discuss four more such polytopes associated to partial orders with applications to permutation statistics, increasing/alternating sequences, and valuations on distributive lattices. On the geometric side, these polytopes make interesting connections to anti-blocking polytopes from combinatorial optimization, compressed and equidissectable polytopes from discrete geometry, and arrangements of tropical min- and max-hyperplanes.
- Supplements
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Sep 06, 2017
Wednesday
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09:00 AM - 10:00 AM
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Shape of Associativity
Satyan Devadoss (University of San Diego)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Associativity is ubiquitous in mathematics. Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance. But over the past few decades, this concept has blossomed and matured. We start with a brief look at how this has transpired, and then explore the visualization of associativity in the form of the associahedron polytope and its generalizations.
- Supplements
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10:00 AM - 10:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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f-vectors of polytopes and simplicial complexes, part I: results and problems
Eran Nevo (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
What are the possible face numbers of d-dimensional: polytopes? simplicial polytopes (namely those whose proper faces are all simplices)? flag polytopes (namely simplicial polytopes all whose minimal nonfaces have size 2)? centrally symmetric polytopes (namely those invariant under the map x-->-x)?...
We'll survey results and open problems concerning face numbers (f-vectors) of polytopes and simplicial complexes, including subfamilies of particular interest.
As the area is vast and fast growing, this talk will cover only a portion of it, representing some of the main methods used. These methods range from algebraic to combinatorial, topological and geometric, and will form Part II given by Martina Juhnke-Kubitzke.
- Supplements
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11:30 AM - 12:30 PM
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f-vectors of simplicial complexes and polytopes, part II: methods
Martina Juhnke-Kubitzke (Universität Osnabrück)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
This talk builds on Eran Nevo's talk, where he presented results and problems on f-vectors of simplicial complexes and polytopes. We will provide an overview of some of the most important tools used to prove this kind of results. This includes combinatorial methods such as shellings, as well as algebraic and geometric methods such as Stanley-Reisner rings, sigma modules, toric varieties, Lefschetz properties and rigidity of frameworks. We will also hint at specific examples, where those techniques have been applied successfully.
- Supplements
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12:30 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Ehrhart positivity
Fu Liu (University of California, Davis)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. There are different examples of polytopes shown to be Ehrhart positive using different techniques. We will survey some of these results. Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. We try to make this relation more explicit in the particular case of the regular permutohedron. The goal is to prove Ehrhart positivity for generalized permutohedra. If time permits, I will also discuss some related questions. This is joint work with Federico Castillo.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Ehrhart Unimodality and Simplices for Numeral Systems
Liam Solus (Royal Institute of Technology (KTH))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In the field of Ehrhart theory, identification of lattice polytopes with unimodal Ehrhart h*-polynomials is a cornerstone investigation. The study of h*-unimodality is home to numerous long-standing conjectures within the field, and proofs thereof often reveal interesting algebra and combinatorics intrinsic to the associated lattice polytopes. Proof techniques for h*-unimodality are plentiful, and some are apparently more dependent on the lattice geometry of the polytope than others. In recent years, proving a polynomial has only real-roots has gained traction as a technique for verifying unimodality of h-polynomials in general. However, the geometric underpinnings of the real-rooted phenomena for h*-unimodality are not well-understood. As such, more examples of this property are always noteworthy. In this talk, we will discuss a family of lattice n-simplices that associate via their normalized volumes to the n^th-place values of positional numeral systems. The h*-polynomials for simplices associated to special systems such as the factoradics and the binary numerals recover ubiquitous h-polynomials, namely the Eulerian polynomials and binomial coefficients, respectively. Simplices associated to any base-r numeral system are also provably real-rooted. We will put the h*-real-rootedness of the simplices for numeral systems in context with that of their cousins, the s-lecture hall simplices, and discuss their admittance of this phenomena as it relates to other, more intrinsically geometric, reasons for h*-unimodality.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Sep 07, 2017
Thursday
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09:00 AM - 10:00 AM
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Equivariant Methods in Discrete Geometry: Problems and Progress
Günter Ziegler (Freie Universität Berlin)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In this lecture, I will discuss three different problems from Discrete Geometry,
- the Topological Tverberg Problem,
- the Colored Tverberg Problem, and
- the Grünbaum Hyperplane Problem.
These problems have many things in common:
- They are easy to state, and may look harmless,
- They have very nice and classical configuration spaces,
- they may be attacked by ``Equivariant Obstruction Theory'',
- this solves the problems --- but only partially,
- which leads us to ask more questions, look for new tools ...
- and this yields surprising new results.
(Joint work with Pavle Blagojevic, Florian Frick, Albert Haase, and Benjamin Matschke)
- Supplements
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10:00 AM - 10:30 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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Linear versions of Tverberg's theorem: Progress and problems
Pablo Soberón (Northeastern University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will discuss different versions of Tverberg's theorem and the linear-algebraic methods that can be used to to prove them. In particular, we will talk about the malleability of Sarkaria's trick, and about quantitative versions of Tverberg's theorem.
- Supplements
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11:30 AM - 12:30 PM
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Hypertrees
Nathan Linial (Hebrew University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In a seminal paper Kalai (1983) extended the notion of a tree to higher dimensions. Formally, an n-vertex d-dimensional hypertee is a Q-acyclic simplicial complex with a full (d-1) dimensional skeleton and {n-1 \choose d} d-dimensional faces. We will use instead an equivalent intuitive definition that relies only on elementary linear algebra. In this talk I will try to give a flavor of these exciting concepts. I will discuss several of the many open problems that arise here and describe some of our new discoveries.
- Supplements
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12:30 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Random Simplicial Complexes
Matthew Kahle (Ohio State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will survey the rapidly developing field of stochastic topology, especially from a combinatorial point of view. In particular, we will discuss a number of results for the 2-dimensional random simplicial complex introduced by Linial and Meshulam. We will pay particularly attention to thresholds for various topological properties.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Cutting Cakes with Combinatorial Fixed Point Theorems
Kathryn Nyman (Willamette University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Sperner's Lemma, a combinatorial analogue to Brower's Fixed Point Theorem, guarantees a "fully-labeled cell" in any labeled triangulation of the n-simplex meeting suitable conditions. These fully-labeled cells provide crisp solutions to envy-free cake-cutting questions. We look at equivalences between combinatorial results and topological fixed-point theorems, along with applications to fair division problems, including the division of multiple cakes and consensus halving.
- Supplements
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Sep 08, 2017
Friday
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09:00 AM - 10:00 AM
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Linear lifts and nonnegative factorizations
Samuel Fiorini (Université Libre de Bruxelles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In the first part of this introductory talk, I will give you a hint of how and why the area of linear lifts of polytopes (AKA linear extended formulations) came into being. Starting with Yannakakis' fundamental contribution, which laid dormant for too long, I will take you to the results that revived the interest in lifts of polytopes. In the second part, I will explain some of the many connections that sustain the area, the amazing results that have been obtained recently, the open problems that remain, and the barriers that stand in our way.
- Supplements
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10:00 AM - 10:30 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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Positive semidefinite lifts and factorizations
João Gouveia (University of Coimbra)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Representing polytopes by means of linear matrix inequalities as been a highly successful strategy in combinatorial optimization. Geometrically it corresponds to writing a polytope as the projection of an affine slice of the cone of positive semidefinite (psd) matrices i.e., a spectrahedron. Efforts to understand the theoretical limits of such techniques have connected the existance of such representations to a particular type of matrix factorization, the psd factorization of a nonnegative matrix, and its corresponding notion of psd rank. In this talk we will do a brief survey of the main results in the area, its connections to matrix theory and combinatorics and some of the open problems that remain.
- Supplements
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11:30 AM - 12:30 PM
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Working Time
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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12:30 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Spectrahedra
Cynthia Vinzant (University of Washington)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A spectrahedron is an affine slice of the cone of positive semidefinite matrices. Spectrahedra form a rich class of convex bodies that are computationally tractable and appear in many areas of mathematics. Examples include polytopes, ellipsoids, and more exotic convex sets, like the convex hull of some curves. I will introduce the theory of spectrahedra with many examples and discuss some applications in distance geometry and combinatorial optimization.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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The Crossing Lemma Revisited
János Pach (Renyi Institute of Mathematics)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The Crossing Lemma of Ajtai, Chvatal, Newborn, Szemeredi (1982) and Leighton (1983)states that if a graph of n vertices and e>4n edges is drawn in the plane, then the number of crossings between its edges must be at least constant times e^3/n^2. This statement, which is asymptotically tight, has found many applications in combinatorial geometry and in additive combinatorics. However, most results obtained using the Crossing Lemma do not appear to be optimal, and there is a quest for improved versions of the lemma for graphs satisfying certain special properties. In this talk, I describe some recent extensions of the lemma to multigraphs (joint work with G. Toth) and to families of continuous arcs in the plane (joint work with N. Rubin and G. Tardos, and with G. Tardos).
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