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Events happening on December 2, 2014

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Dec. 02, 2014 11am (Math 220)	Noncomm Geometry Alex Nita Fredholm Modules (continued)
Dec. 02, 2014 11am (MATH 350)	Number Theory Doug Ulmer (Georgia Tech) X For every genus g>0 and most primes p, we write down a curve over Fp(t) with Jacobian of large rank and with explicit divisors to fill out a finite index subgroup of the Mordell-Weil group. The height pairings among the explicit points have a very nice group-theoretic description, and we have an analytic class number formula of the type \|sha\|=square of index of the explicit points in the full Mordell-Weil group. Explicit high ranks in higher genus
Dec. 02, 2014 2pm (MATH 350)	Lie Theory Katherine Stange (University of Colorado Boulder) X This is a number theory talk of interest to those who care about matrix groups. A Schmidt arrangement is the orbit of the real line under the Bianchi group PSL(2,OK) (realised as Mobius transformations) where OK is the ring of integers in an imaginary quadratic field K. The arrangement takes the form of a dense collection of intricately nested circles in the complex plane. I'll explain how the number theory of K influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest. In other words, for those who walk the third floor, I will be explaining the circles on my door. The talk will require some basic algebraic number theory (e.g. the ideal class group), and a passing familiarity with Mobius transformations. The Apollonian structure of quadratic imaginary fields
Dec. 02, 2014 2pm (MATH 350)	Number Theory Katherine Stange (University of Colorado Boulder) X This is a number theory talk of interest to those who care about matrix groups. A Schmidt arrangement is the orbit of the real line under the Bianchi group PSL(2,OK) (realised as Mobius transformations) where OK is the ring of integers in an imaginary quadratic field K. The arrangement takes the form of a dense collection of intricately nested circles in the complex plane. I'll explain how the number theory of K influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest. In other words, for those who walk the third floor, I will be explaining the circles on my door. The talk will require some basic algebraic number theory (e.g. the ideal class group), and a passing familiarity with Mobius transformations. (SPECIAL TIME) The Apollonian structure of quadratic imaginary fields / cross listed with Lie Theory Seminar
Dec. 02, 2014 3pm (Math 350)	Algebraic Geometry Martin Ulirsch (Brown University) X This talk will begin with a quick introduction to the topology of Berkovich analytic spaces and the tropical geometry of toric varieties. In particular, we are going to talk about skeletons of toric varieties and their relationship to tropicalization. Moreover, I am going to explain how a generalization of these notions to the setting of toroidal embeddings, open embeddings that are étale locally isomorphic to the torus embedding of a toric variety, can be used to study the tropical geometry of moduli spaces. Part I: Tropicalization of toric varieties and moduli spaces Sponsored by the Meyer Fund
Dec. 02, 2014 4pm (Math 350)	Algebraic Geometry Martin Ulirsch (Brown University) X Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety X, or more generally of a logarithmic scheme X, can be formalized as a "Berkovich analytification" of a scheme over the field with one element that is canonically associated to X. The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy F₁-geometric objects to the more familiar realm of algebraic stacks. Artin fans are étale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans. I am going to explain how to use the ideas surrounding the notion Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions. Part II: Artin fans in tropical geometry Sponsored by the Meyer Fund

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