Oct 09, 2017
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Negative correlation and HodgeRiemann relations
June Huh (Institute for Advanced Study)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections. This talk will be nontechnical: Nothing will be assumed beyond basic linear algebra.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


A counterexample to the extension space conjecture for realizable oriented matroids
Xue (Gaku) Liu (MaxPlanckInstitut für Mathematik in den Naturwissenschaften)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The extension space conjecture, proposed by Sturmfels and Ziegler in 1993, is a conjecture about the topology of a realizable oriented matroid's "extension space", which is a topological model for the set of all extensions of the oriented matroid by a single element. Equivalently, it is a conjecture about the poset of proper zonotopal tilings of a zonotope, namely that this poset is homotopy equivalent to a sphere. In this talk we describe a counterexample to this conjecture in three dimensions.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Recent developments on chromatic quasisymmetric functions
Michelle Wachs (University of Miami)

 Location
 MSRI:
 Video

 Abstract
The chromatic quasisymmetric function of a labeled graph was introduced by Shareshian and myself as a refinement of Stanley’s chromatic symmetric function. We conjectured a refinement of the long standing Stanley Stembridge epositivity conjecture, and formulated an algebrogeometric approach to proving this refined conjecture involving Hessenberg varieties. Significant progress in this direction has recently been made by Brosnan and Chow and by GuayPaquet. In this talk, I will discuss the connection with Hessenberg varieties, and also present some new directions, including results on generalizations to directed graphs obtained by my student Brittney Ellzey
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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03:30 PM  04:30 PM


Intersections of finite sets: geometry and topology
Florian Frick (Carnegie Mellon University)

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Given a collection of finite sets, Knesertype problems aim to partition the collection into parts with wellunderstood intersection pattern, such as in each part any two sets intersect. Since Lovász' solution of Kneser's conjecture, concerning intersections of all ksubsets of an nset, topological methods have been a central tool in understanding intersection patterns of finite sets. We will develop a method that in addition to using topological machinery takes the topology of the collection of finite sets into account via a translation to a problem in Euclidean geometry. This leads to simple proofs of old and new results.
 Supplements



Oct 10, 2017
Tuesday

09:30 AM  10:30 AM


Combinatorial positive valuations
Katharina Jochemko (Royal Institute of Technology (KTH))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In the continuous setting, valuations are wellstudied and the volume plays a prominent role in many classical and structural results. It has various desirable properties such as homogeneity, monotonicity and translationinvariance. In the less examined discrete setting, the number of lattice points in a polytope  its discrete volume  takes a fundamental role. Although homogeneity and continuity are lost, some striking parallels to the continuous setting can be drawn. The central notion here is that of combinatorial positivity. In this talk, I will discuss similarities, analogies and differences between the continuous and discrete world of translationinvariant valuations as well as applications to Ehrhart theory
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Symmetric Sums of Squares over kSubset Hypercubes
Annie Raymond (University of Massachusetts Amherst)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Polynomial optimization over hypercubes has important applications in combinatorial optimization. We develop a symmetryreduction method that finds sums of squares certificates for nonnegative symmetric polynomials over ksubset hypercubes that improves on a technique due to Gatermann and Parrilo. For every symmetric polynomial that has a sos expression of a fixed degree, our method finds a succinct sos expression whose size depends only on the degree and not on the number of variables. Our results relate naturally to Razborov's flag algebra calculus for solving problems in extremal combinatorics. This leads to new results involving flags and their power in finding sos certificates.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Mogami triangulations
Bruno Benedetti (University of Miami)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Starting with a tree of tetrahedra, suppose that you are allowed to recursively glue together two boundary triangles that have nonempty intersection.
You may perform this type of move as many times you want. Let us call "Mogami manifolds" the triangulated 3manifolds (with or without boundary) that can be obtained this way. Mogami, a quantum physicist, conjectured in 1995 that all triangulated 3balls are Mogami. This conjecture implied a much more important one, namely, that "there are only exponentially many triangulation of the 3sphere with N tetrahedra".
We study this Mogami property in relation other notions, like simplyconnectedness, shellability, and collapsibility. With a topological trick we show that Mogami's conjecture is false. The more important conjecture remains unfortunately wide open.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


The geometry of scheduling
Caroline Klivans (Brown University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
By considering a simple form of a scheduling problem, we explore the geometric combinatorics behind a host of polynomial counting functions. Along the way we will encounter Coxeter complexes, quasisymmetric functions, inside out polytopes, Hilbert series, hvectors and other favorites of the audience
 Supplements


04:30 PM  06:15 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
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Oct 11, 2017
Wednesday

09:00 AM  10:00 AM


Around the lower bound theorem for polytope
Eran Nevo (The Hebrew University of Jerusalem)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Starting from Barnette's classical lower bound theorem for simplicial polytopes (LBT), and its proof via framework rigidity by Kalai, I'll then discuss corresponding theorems and conjectures, as well as proof techniques, for subclasses and superclasses of interest. For example, we'll consider centrallysymmetric simplicial polytopes, flag polytopes, cubical polytopes, polytopes with one nonsimplex facet, general polytopes, and analogues for polyhedral spheres and manifolds.
 Supplements


10:00 AM  10:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Generic toric varieties, onesided pairings and the Lefschetz theorem
Karim Adiprasito (The Hebrew University of Jerusalem)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements


11:30 AM  12:30 PM


Computing simplicial representatives of homotopy group elements
Uli Wagner (Institute of Science and Technology Austria)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
A central problem of algebraic topology is to understand the homotopy groups π_{d}(X) of a topological space X.
For the computational version of the problem, it is well known that there is no algorithm to decide whether the
fundamental group π_{1}(X) of a given finite simplicial complex X is trivial. On the other hand, there are several
algorithms that, given a finite simplicial complex X that is simply connected (i.e., with π_{1}(X) trivial), compute
the higher homotopy group π_{d}(X) for any given d ≥ 2.
However, these algorithms come with a caveat: They compute the isomorphism type of π_{d}(X), d ≥ 2, as an abstract
finitely generated abelian group given by generators and relations, but they work with very implicit representations
of the elements of π_{d}(X).
We present an algorithm that, given a simply connected simplicial complex X, computes π_{d}(X) and represents its elements
as simplicial maps from a suitable triangulation of the dsphere to X. For fixed d, the algorithm runs in time singly exponential
in size(X), the number of simplices of X.
Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected simplicial complexes X
such that for any simplicial map representing a generator of π_{d}(X), the size of the triangulation of S^{d} on which the map is defined is
exponential in size(X).
Apart from the intrinsic importance of homotopy groups, we view this as a first step
towards the more general goal of computing explicit maps with specific topological properties,
e.g., computing explicit embeddings of simplicial complexes or counterexamples to Tverbergtype
problems, and towards quantitative bounds on the complexity of such maps.
Joint work with Marek Filakovský, Peter Franek, and Stephan Zhechev
 Supplements



Oct 12, 2017
Thursday

09:30 AM  10:30 AM


Combinatorics of the Tree Amplituhedron
Lauren Williams (Harvard University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The tree amplituhedron A(n, k, m) is a geometric object generalizing the positive Grassmannian, which was introduced by ArkaniHamed andTrnka in 2013 in order to give a geometric basis for the computationof scattering amplitudes in N = 4 supersymmetric YangMills theory. Iwill give an elementary introduction to the amplituhedron, and then describe what it looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron A(n, k, 1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement (and hence is homeomorphic to a closed ball). I will also present some conjectures relating the amplituhedron A(n, k, m) to combinatorial objects such as nonintersecting lattice paths and plane partitions.
This is joint work with Steven Karp, and part of it is additionally joint work with Yan Zhang
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Unwinding The Amplituhedron
Hugh Thomas (Université du Québec à Montréal)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
ArkaniHamed and Trnka recently defined an object, which they dubbed the amplituhedron, which encodes the scattering amplitudes for planar N=4 super YangMills. This object feels polytopal, and indeed, in simple examples, it is a cyclic polytope inside projective space. However, in general, it lives in a Grassmannian rather than in a projective space. Amplituhedra are closely linked to the geometry of total positivity; indeed, the totally nonnegative part of a Grassmannian is also an example of an amplituhedron. I will try to explain some of what this means, and report on joint work with ArkaniHamed and Trnka in which we give a new and simpler definition of the amplituhedron
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Hyperbolicity, determinants, and reciprocal linear spaces
Cynthia Vinzant (North Carolina State University; University of Washington)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A reciprocal linear space is the image of a linear space under coordinatewise inversion. This nice algebraic variety appears in many contexts and its structure is governed by the combinatorics of the underlying hyperplane arrangement. A reciprocal linear space is also an example of a hyperbolic variety, meaning that there is a family of linear spaces all of whose intersections with it are real. This special real structure is witnessed by a determinantal representation of its Chow form in the Grassmannian. In this talk, I will introduce reciprocal linear spaces and discuss the relation of their algebraic properties to their combinatorial and real structure
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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04:10 PM  05:00 PM


The first hundred years of Helly’s theorem
Jesus De Loera (University of California, Davis)



Oct 13, 2017
Friday

09:30 AM  10:30 AM


Complexity of counting integer points
Igor Pak (University of California, Los Angeles)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will survey some recent progress on complexity of counting integer points in convex polytopes of fixed dimension. A few our results will have a seductively easy yet counterintuitive statements. The talk will assume no background and will be very accessible. Joint work with Danny Nguyen.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


BottSamelson varieties and combinatorics
Laura Escobar (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Schubert varieties parametrize families of linear spaces intersecting certain hyperplanes of C^n in a predetermined way. In the 1970’s Hansen and Demazure independently constructed resolutions of singularities for Schubert varieties: the BottSamelson varieties. In this talk I will describe their relation with associahedra. I will also discuss joint work with PechenickTennerYong linking Magyar’s construction of these varieties as configuration spaces with Elnitsky’s rhombic tilings. Finally, based on joint work with WyserYong, I will give a parallel for the BarbaschEvens desingularizations of certain families of linear spaces which are constructed using symmetric subgroups of the general linear group.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Moving a robotic arm in a tunnel
Federico Ardila (San Francisco State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We study the motion of a robotic arm inside a rectangular tunnel. We prove that the configuration space of all possible positions of the robot is a CAT(0) cubical complex. To do this we use a bijection between rooted CAT(0) cubical complexes and a family of combinatorial objects that we call “posets with inconsistent pairs”. This bijection allows us to use techniques from geometric group theory and poset theory to find the optimal way of moving the arm from one position to another. We also compute the diameter of the configuration space, that is, the longest distance between two positions of the robot. This talk will include joint work with Tia Baker, Hanner Bastidas, Cesar Ceballos, John Guo, Megan Owen, Seth Sullivant, and Rika Yatchak, and will assume no previous knowledge of the subject.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Triangulations of root polytopes and Tutte polynomials
Alexander Postnikov (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
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