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Speaker: Pavel GalashinTitle: 10/19/2021 Combinatorics, Physics and Probability SeminarVenue: VirtualTitle: Ising model, total positivity, and criticality Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality. The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes. I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with… |
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Speaker: Lisa SauermannTitle: 10/12/2021 Combinatorics, Physics and Probability SeminarVenue: VirtualTitle: On counting algebraically defined graphs Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions… |
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Speaker: Daniel CizmaTitle: 10/5/2021 Combinatorics, Physics and Probability SeminarVenue: VirtualTitle: Geodesic Geometry on Graphs Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such… |
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Speaker: Melissa Sherman-BennettTitle: 9/28/2021 Combinatorics, Physics and Probability SeminarVenue: VirtualTitle: The hypersimplex and the m=2 amplituhedron Abstract: I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the… |
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Speaker: Nima Arkani-HamedTitle: Surfacehedra and the Binary Positive Geometry of Particle and “String” AmplitudesVenue: virtualSpeaker: Nima Arkani-Hamed, IAS Title: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes |