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Events happening on October 11, 2022

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Oct. 11, 2022 10am (MATH 350)	Number Theory Matthew de Courcy-Ireland (EPFL, Lausanne) X We discuss a family of cubic surfaces defined by x^2+y^2+z^2=xyz+k modulo prime numbers. The solutions form a graph, where each vertex (x,y,z) is joined to the other solution of the same quadratic in any of the three variables. These moves are related to a nonlinear action of the modular group PGL(2,Z) on the surface. We outline some ways these equations arise in geometry, and how we became interested in showing that the associated graphs cannot be embedded in the plane. We sketch a proof that 2, 3, and 7 are the only primes for which the graph is planar. For larger primes, we give an estimate of the Euler characteristic of a surface in which the graph can be embedded. Non-planarity of Markoff graphs mod p
Oct. 11, 2022 11am (MATH 350)	Number Theory Matthew Litman (University of California Davis) X Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran. Markoff-type K3 Surfaces: Local and Global Finite Orbits
Oct. 11, 2022 1:25pm (MATH 2…	Marcos Mazari-Armida (CU Boulder) X The classical model-theoretic notion of superstability was introduced by Shelah in the late sixties for first-order theories. In this talk we will show that superstability is a natural algebraic property by characterizing some classical classes of rings, such as noetherian rings and perfect rings, via superstability of certain classes of modules. Characterizing some classical rings via superstability
Oct. 11, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X Kazhdan and Lusztig defined partitions of an arbitrary Coxeter group W into left, right, and two-sided cells. Lusztig defined an integer a(w) for each element w of W, and proved that the a-function takes constant values on cells. In general, the problem of determining these cells, and the function a, is difficult. Elements of a-value 0 or 1 are well understood, and this talk will describe the structure of cells of elements of a-value 2 in the cases where the number of such elements is finite. This is joint work with Tianyuan Xu. The structure of Kazhdan—Lusztig cells of a-value 2

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