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In this talk, I'll discuss a journey taken by my research interests leading from dynamic graphical models to image segmentation. I'll start out discussing dynamic graphical models (DGMs) and their efficient inference. Entailed in performing inference in such models is the problem of identifying a vertex cut that determines the boundary of a critical periodic graphical component. This component is used when the model is expanded to an arbitrary length, and also determines a local junction tree segment used as part of this expansion. Depending on the boundary of this component, inference costs can change by orders of magnitude. There are many ways of judging the quality of the component without performing inference itself, including the standard vertex cut criterion. Most interestingly, however, is the case where the DGM may be sparse, and the quality is determined by an objective that takes into account this sparsity. It turns out that such an objective is submodular. Considering this, and when vertex cut is seen as standard graph cut, a new problem is naturally introduced that involves finding a minimal cut in a graph where edges are judged not by a sum of edge weights but instead by a submodular function, a problem we call cooperative cut. The remainder of the talk will provide a brief background on submodularity, cooperative cut, and various hardness and approximate solutions to this problem, and how it can be applied to image segmentation. In this latter application, results show significant improvements over the standard (node submodular) graph cut approaches, in particular for images that exhibit a \"shrinking bias\" problem and for those images with smooth illumination gradients. Joint work with Stefanie Jegelka, with earlier contributions from Chris Bartels.
Spring, 2021Speaker Lisa Carbone, Rutgers University and Institute for Advanced Study, School of Natural SciencesTitle Complete pro-unipotent automorphism group for the monster Lie algebraTime 1/22/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract We construct a complete pro-unipotent group of automorphisms for a completion of the monster Lie algebra. We also construct an analog of the exponential map and Adjoint representation. This gives rise to some useful identities involving imaginary root vectors.Speaker Hao Li, SUNY-AlbanyTitle Arc spaces, vertex algebras and principal subspacesTime 1/29/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract Arc spaces were originally introduced in algebraic geometry to study singularities. More recently they show in connections to vertex algebras. There is a closed embedding from the singular support of a vertex algebra V into the arc space of associated scheme of V. We call a vertex algebra \"classically free\" if this embedding is an isomorphism. In this introductory survey talk, we will first introduce arc spaces and some of its backgrounds. Then we will provide several examples of classically free vertex algebras including Feigin-Stoyanovsky principal subspaces, and explain their applications in differential algebras, $q$-series identities, etc. In particular, we will show the classically freeness of principal subspaces of type A at level 1 by using a method of filtrations and identities from quantum dilogarithm or quiver representations. As a result, we obtain new presentations and graded dimensions of the principal subspaces of type A at level 1, which can be thought of as the continuation of previous works by Calinescu, Lepowsky and Milas. The classically freeness of some principal subspaces which possess free fields realisation will also be discussed. Most of the talk is based on the joint work with A. Milas.Speaker Lilit Martirosyan, University of North Carolina, WilmingtonTitle Braided rigidity for path algebras (joint work with Hans Wenzl)Time 2/5/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations ofthe braid groups Bn for all n in N. We say that such representations are rigid if they are determined by the path algebra and the representations of B2. We show that besides theknown classical cases also the braid representations for the path algebra for the 7-dimensional representation of G2 satisfies the rigidity condition, provided B3 generates End(V^{3}). Weobtain a complete classification of ribbon tensor categories with the fusion rules of g(G2) if this condition is satisfied.Slides pdf file. Archive paper arXiv:2001.11440 and arXiv:1609.08440 YouTube video
Speaker Jason Saied, Rutgers UniversityTitle Combinatorial formula for SSV polynomialsTime 3/26/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract Macdonald polynomials are homogeneous polynomials that generalize many important representation-theoretic families of polynomials, such as Jack polynomials, Hall-Littlewood polynomials, affine Demazure characters, and Whittaker functions of GL_r(F) (where F is a non-Archimedean field). They may be constructed using the basic representation of the corresponding double affine Hecke algebra (DAHA): a particular commutative subalgebra of the DAHA acts semisimply on the space of polynomials, and the (nonsymmetric) Macdonald polynomials are the simultaneous eigenfunctions. In 2018, Sahi, Stokman, and Venkateswaran constructed a generalization of this DAHA action, recovering the metaplectic Weyl group action of Chinta and Gunnells. As a consequence, they discovered a new family of polynomials, called SSV polynomials, that generalize both Macdonald polynomials and Whittaker functions of metaplectic covers of GL_r(F). We will give a combinatorial formula for these SSV polynomials in terms of alcove walks, generalizing Ram and Yip's formula for Macdonald polynomials. YouTube video Geometric properties of sheaves of coinvariants and conformal blocks.Speaker Haisheng Li, Rutgers University at CamdenTitle Deforming vertex algebras by module and comodule actions of vertex bialgebrasTime 4/9/2021, Friday, 12:00-12:45 pm (Eastern Time)Location Zoom link above Abstract Previously, we introduced a notion of vertex bialgebra and a notion of module vertex algebra for a vertex bialgebra, and gave a smash product construction of nonlocal vertex algebras. Here, we introduce a notion of right comodule vertex algebra for a vertex bialgebra. Then we give a construction of quantum vertex algebras from vertex algebras witha right comodule vertex algebra structure and a compatible (left) module vertex algebra structure for a vertex bialgebra.As an application, we obtain a family of deformations of the lattice vertex algebras. This is based on a joint work with Naihuan Jing, Fei Kong, and Shaobin Tan.Slides pdf file. Speaker Corina Calinescu, New York City College of Technology and CUNY Graduate CenterTitle Principal subspaces of standard modules for twisted affine Lie algebrasTime 4/16/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract This talk is an overview of results about principal subspaces for twisted affine Lie algebras. In joint works with J. Lepowsky, A. Milas, M. Penn and C. Sadowski, we studied the principal subspaces of certain standard modules for the twisted affine Lie algebra of type A, by using vertex algebraic methods. Their graded dimensions are given in connection with q-series identities. Speaker Christopher Sadowski, Ursinus CollegeTitle Permutation orbifold subalgebras of Virasoro vertex algebrasTime 4/23/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract In this talk, we examine certain permutation orbifold subalgebras (fixed-point subalgebras) of tensor powers of Virasoro vertex operator algebras under certain group actions. In particular, we determine the strong generators of these subalgebras and point out isomorphisms between these subalgebras at certain central charges and W-algebras. The search for these strong generators makes heavy use of Mathematica. In this talk, we will demonstrate exactly how such computations are performed with live examples. One of the primary aims of this talk is to demonstrate the usefulness of computers in the study of examples of vertex operator algebras, and to promote the use of computer algebra software to derive results that would be difficult to obtain by hand.This talk is based on joint work with Antun Milas and Michael Penn.Speaker Abid Ali, Rutgers UniversityTitle Eisenstein Series on Arithmetic Quotients of Rank 2 Kac--Moody Groups Over Finite FieldsTime 4/30/2021, Friday, 12:00 (Eastern Time)Location Zoom link above Abstract Let G be a rank 2 Kac-Moody group over a finite field. The group G comes equipped with a data (X, K), where X is the Tits building of G and K is the standard parabolic subgroup of the negative BN-pair. By using this data and the Iwasawa decomposition of G, we define Eisenstein series on the quotient graph K/X, which is induced by a character defined on the set of vertices of X.
Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound 2^{O(sqrt(k) log^2(k))} n^{O(1)}. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph; examples of such problems include Directed k-Path, Weighted k-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface. We also provide a similar statement for graph classes of polynomial growth. 153554b96e
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