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PRODID:Mathematics Seminar Calendar
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UID:20221205T022328-29409@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Symplectic groupoid and cluster algebra description of closed Riemann surfaces
DESCRIPTION:Speaker\: Leonid Chekhov, Michigan State University\r\nWe use the Fock-Goncharov higher Teichmuller space directed networks to solve the symplectic groupoid condition: parameterize pairs of $SL_n$ matrices (B,A) with A unipotent such that $BAB^T$ is also unipotent. A natural Lie-Poisson bracket on B generates the Goldman bracket on elements of A and $BAB^T$, which are simultaneously elements of the corresponding upper cluster algebras. Using this input we identify the space of X-cluster algebra elements with Teichmuller spaces of closed Riemann surfaces of genus 2 (for $n$=3) and 3 (for $n$=4) endowed with Goldman bracket structure: for $g$=2 all geodesic functions are positive Laurent polynomials and Dehn twists correspond to mutations in the corresponding quivers. This is the work in progress with Misha Shapiro.
LOCATION:C304 Wells Hall
DTSTART:20220912T163000Z
DTEND:20220912T173000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=29409
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BEGIN:VEVENT
UID:20221205T022328-29414@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Cluster structures on SL_n and the Belavin-Drinfeld classification
DESCRIPTION:Speaker\: Alexander Vainshtein, Haifa University\r\nCluster structures were discovered by S. Fomin and A. Zelevinsky about twenty years\r\nago and quickly found applications in various fields of mathematics and mathematical physics.\r\n\r\nIn the latter, several advances were made in a study of classical and quantum integrable\r\nsystems arising in the context of cluster structures. These systems "live" on Poisson-Lie\r\ngroups and their Poisson homogeneous spaces, hence it is important to understand an\r\ninterplay between cluster and Poisson structures on such objects.\r\nIn this talk I will explain a construction of a family of (generalized) cluster structures in the\r\nalgebra of regular functions on SL_n related to the Belavin-Drinfeld classification\r\nof Poisson-Lie structures on SL_n.\r\n\r\nBased on a joint work with M.~Gekhtman (Notre Dame) and M.~Shapiro (MSU).
LOCATION:C304 Wells Hall
DTSTART:20220919T163000Z
DTEND:20220919T173000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=29414
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BEGIN:VEVENT
UID:20221205T022328-29421@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:BD Schubert varieties of parahoric group schemes and global Demazure modules of twisted current algebras
DESCRIPTION:Speaker\: Jiuzu Hong, University of North Carolina at Chapel Hill \r\nIt is well-known that there is a duality between affine Demazure modules and the spaces of sections of line bundles on Schubert varieties in affine Grassmannians. This should be regarded as a local theory. In this talk, I will explain an algebraic theory of global Demazure modules of twisted current algebras. Moreover, these modules are dual to the spaces of sections of line bundles on Beilinson-Drinfeld Schubert varieties of certain parahoric groups schemes, where the factorizations of global Demazure modules are compatible with the factorizations of line bundles. This generalizes the work of Dumanski-Feigin-Finkelberg in the untwisted setting. In order to establish this duality in the twisted case, following the works of Zhu, we prove the flatness of BD Schubert varieties, and establish factorizable and equivariant structures on the rigidified line bundles over BD Grassmannians of these parahoric group schemes. This work is joint with Huanhuan Yu.
LOCATION:C304 Wells Hall
DTSTART:20220926T163000Z
DTEND:20220926T173000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=29421
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BEGIN:VEVENT
UID:20221205T022328-29422@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Sp(4) stated skein algebras
DESCRIPTION:Speaker\: Vijay Higgins, Michigan State University\r\nSkein algebras are spanned by webs or links in a thickened surface subject to skein relations. When the skein relations are the Kauffman bracket relations associated to SL(2), they provide a diagrammatic way to encode cluster algebras, as shown by Muller, and also quantum groups, as shown by Costantino and Le.\r\nIn this talk, we will explore a construction of a basis for the stated skein algebra for Sp(4) which is built from Kuperberg's web relations along with extra skein relations along the boundary of the surface. We will use the basis to obtain results about the structure of the skein algebra, relating it to the quantum group associated to Sp(4). We will also recover Kuperberg's result about the Sp(4) web category.
LOCATION:C204A Wells Hall
DTSTART:20221010T190000Z
DTEND:20221010T200000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=29422
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BEGIN:VEVENT
UID:20221205T022328-30467@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY: Cluster algebras and Nakajima's graded quiver varieties
DESCRIPTION:Speaker\: Li Li, Oakland University\r\nNakajima's graded quiver varieties are complex algebraic varieties associated with quivers. They are introduced by Nakajima in the study of representations of universal enveloping algebras of Kac-Moody Lie algebras, and can be used to study cluster algebras. In the talk, I will explain how to precisely locate the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras by applying the decomposition theorem to various morphisms related to quiver varieties, thus prove a conjecture proposed by Lee-Li-Rupel-Zelevinsky in 2014.
LOCATION:C204A Wells Hall
DTSTART:20221018T163000Z
DTEND:20221018T173000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=30467
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BEGIN:VEVENT
UID:20221205T022328-30481@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Cluster structures on braid varieties
DESCRIPTION:Speaker\: Linhui Shen, Michigan State University\r\nLet G be a complex simple group. Let $\beta$ be a positive braid whose Demazure product is the longest Weyl group element. The braid variety X($\beta$) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bott-Samelson cells. We provide a concrete construction of the cluster structure on X($\beta$), using the weaves of Casals and Zaslow. We show that the coordinate ring of X($\beta$) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that X($\beta$) admits a natural Poisson structure and can be further quantized. If\r\ntime permits, I will explain several of its applications on representation theory and knot theory,\r\nincluding its connections with the Kazhdan-Lusztig R-polynomials and a geometric interpretation of the\r\nKhovanov-Rozansky homology (following the work of Lam-Speyer and Galashin-Lam). This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental (arXiv:2207.11607).
LOCATION:C304 Wells Hall
DTSTART:20221031T163000Z
DTEND:20221031T173000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=30481
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UID:20221205T022328-30473@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Determinantal inequalities for totally positive matrices
DESCRIPTION:Speaker\: Daniel Soskin, University of Notre Dame\r\nTotally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss two sourses of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with majorizing monotonicity of symmetrized Fischer's products which are known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.
LOCATION:C304 Wells Hall
DTSTART:20221107T173000Z
DTEND:20221107T183000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=30473
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BEGIN:VEVENT
UID:20221205T022328-30480@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Super Cluster Algebras from Surfaces
DESCRIPTION:Speaker\: Nicholas Ovenhouse, Yale University\r\nOne of the most well-known examples of a cluster structure comes from Penner's lambda-length coordinates on the decorated Teichmuller space of a surface. In 2019, Penner and Zeitlin defined a super-manifold generalizing the decorated Teichmuller space, which involves new anti-commuting variables. I wall talk about some recent work with Gregg Musiker and Sylvester Zhang, where we showed that the coordinates on the decorated super Teichmuller space have many of the nice properties associated to a cluster structure, such as a kind of Laurent phenomenon, positivity, and some interesting combinatorial interpretations of the Laurent expressions, involving double dimer covers of certain graphs.
LOCATION:C304 Wells Hall
DTSTART:20221114T173000Z
DTEND:20221114T183000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=30480
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BEGIN:VEVENT
UID:20221205T022328-31502@math.msu.edu
DTSTAMP:20221205T022328Z
SUMMARY:Decorated character varieties and their quantizations from factorization homology
DESCRIPTION:Speaker\: Gus Schrader, Northwestern University\r\n I will report on joint work with D. Jordan, I. Le and A. Shapiro in which we construct categorical invariants of decorated surfaces using the stratified factorization homology of Ayala, Francis and Tanaka, together with the representation theory of quantum groups. The categories we obtain can be regarded as `quantizations' of the categories of quasicoherent sheaves on the stacks of decorated local systems on surfaces, and satisfy strong functoriality and locality properties reminiscent of those of a TQFT. I will give an overview of their construction, and explain how to recover Fock-Goncharov-Shen's cluster quantizations of related moduli spaces within this framework.
LOCATION:C204A Wells Hall
DTSTART:20221129T210000Z
DTEND:20221129T220000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=31502
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