Oct 09, 2017
Monday
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09:15 AM - 09:30 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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09:30 AM - 10:30 AM
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Negative correlation and Hodge-Riemann relations
June Huh (Institute for Advanced Study)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 non-technical: Nothing will be assumed beyond basic linear algebra.
- Supplements
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10:30 AM - 11:00 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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11:00 AM - 12:00 PM
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A counterexample to the extension space conjecture for realizable oriented matroids
Xue (Gaku) Liu (Max-Planck-Institut für Mathematik in den Naturwissenschaften)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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12:00 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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Recent developments on chromatic quasisymmetric functions
Michelle Wachs (University of Miami)
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- Location
- SLMath:
- Video
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- 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 e-positivity conjecture, and formulated an algebro-geometric approach to proving this refined conjecture involving Hessenberg varieties. Significant progress in this direction has recently been made by Brosnan and Chow and by Guay-Paquet. 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
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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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Intersections of finite sets: geometry and topology
Florian Frick (Carnegie Mellon University)
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- Location
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- Video
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- Abstract
Given a collection of finite sets, Kneser-type problems aim to partition the collection into parts with well-understood intersection pattern, such as in each part any two sets intersect. Since Lovász' solution of Kneser's conjecture, concerning intersections of all k-subsets of an n-set, 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
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Oct 10, 2017
Tuesday
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09:30 AM - 10:30 AM
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Combinatorial positive valuations
Katharina Jochemko (Royal Institute of Technology (KTH))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In the continuous setting, valuations are well-studied and the volume plays a prominent role in many classical and structural results. It has various desirable properties such as homogeneity, monotonicity and translation-invariance. 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 translation-invariant valuations as well as applications to Ehrhart theory
- Supplements
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10:30 AM - 11:00 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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11:00 AM - 12:00 PM
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Symmetric Sums of Squares over k-Subset Hypercubes
Annie Raymond (University of Massachusetts Amherst)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Polynomial optimization over hypercubes has important applications in combinatorial optimization. We develop a symmetry-reduction method that finds sums of squares certificates for non-negative symmetric polynomials over k-subset 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
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12:00 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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Mogami triangulations
Bruno Benedetti (University of Miami)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 3-manifolds (with or without boundary) that can be obtained this way. Mogami, a quantum physicist, conjectured in 1995 that all triangulated 3-balls are Mogami. This conjecture implied a much more important one, namely, that "there are only exponentially many triangulation of the 3-sphere with N tetrahedra".
We study this Mogami property in relation other notions, like simply-connectedness, shellability, and collapsibility. With a topological trick we show that Mogami's conjecture is false. The more important conjecture remains unfortunately wide open.
- 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 geometry of scheduling
Caroline Klivans (Brown University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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, h-vectors and other favorites of the audience
- Supplements
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04:30 PM - 06:15 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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Oct 11, 2017
Wednesday
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09:00 AM - 10:00 AM
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Around the lower bound theorem for polytope
Eran Nevo (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 sub-classes and super-classes of interest. For example, we'll consider centrally-symmetric simplicial polytopes, flag polytopes, cubical polytopes, polytopes with one non-simplex facet, general polytopes, and analogues for polyhedral spheres and manifolds.
- 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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Generic toric varieties, one-sided pairings and the Lefschetz theorem
Karim Adiprasito (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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11:30 AM - 12:30 PM
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Computing simplicial representatives of homotopy group elements
Uli Wagner (Institute of Science and Technology Austria)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A central problem of algebraic topology is to understand the homotopy groups πd(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 π1(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 π1(X) trivial), compute
the higher homotopy group πd(X) for any given d ≥ 2.
However, these algorithms come with a caveat: They compute the isomorphism type of πd(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 πd(X).
We present an algorithm that, given a simply connected simplicial complex X, computes πd(X) and represents its elements
as simplicial maps from a suitable triangulation of the d-sphere 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 πd(X), the size of the triangulation of Sd 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 Tverberg-type
problems, and towards quantitative bounds on the complexity of such maps.
Joint work with Marek Filakovský, Peter Franek, and Stephan Zhechev
- Supplements
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Oct 12, 2017
Thursday
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09:30 AM - 10:30 AM
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Combinatorics of the Tree Amplituhedron
Lauren Williams (Harvard University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The tree amplituhedron A(n, k, m) is a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed andTrnka in 2013 in order to give a geometric basis for the computationof scattering amplitudes in N = 4 supersymmetric Yang-Mills 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 non-intersecting lattice paths and plane partitions.
This is joint work with Steven Karp, and part of it is additionally joint work with Yan Zhang
- Supplements
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10:30 AM - 11:00 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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11:00 AM - 12:00 PM
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Unwinding The Amplituhedron
Hugh Thomas (Université du Québec à Montréal)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Arkani-Hamed and Trnka recently defined an object, which they dubbed the amplituhedron, which encodes the scattering amplitudes for planar N=4 super Yang-Mills. 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 non-negative 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 Arkani-Hamed and Trnka in which we give a new and simpler definition of the amplituhedron
- Supplements
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12:00 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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Hyperbolicity, determinants, and reciprocal linear spaces
Cynthia Vinzant (University of Washington)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A reciprocal linear space is the image of a linear space under coordinate-wise 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
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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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04:10 PM - 05:00 PM
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The first hundred years of Helly’s theorem
Jesus De Loera (University of California, Davis)
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Oct 13, 2017
Friday
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09:30 AM - 10:30 AM
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Complexity of counting integer points
Igor Pak (University of California, Los Angeles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:30 AM - 11:00 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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11:00 AM - 12:00 PM
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Bott-Samelson varieties and combinatorics
Laura Escobar (University of Illinois at Urbana-Champaign)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 Bott-Samelson varieties. In this talk I will describe their relation with associahedra. I will also discuss joint work with Pechenick-Tenner-Yong linking Magyar’s construction of these varieties as configuration spaces with Elnitsky’s rhombic tilings. Finally, based on joint work with Wyser-Yong, I will give a parallel for the Barbasch-Evens desingularizations of certain families of linear spaces which are constructed using symmetric subgroups of the general linear group.
- Supplements
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12:00 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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Moving a robotic arm in a tunnel
Federico Ardila (San Francisco State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Triangulations of root polytopes and Tutte polynomials
Alexander Postnikov (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Supplements
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