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Abstract: Schubert calculus connects problems in algebraic geometry to combinatorics, classically resolving the question of counting points in the intersection of certain subvarieties of the Grassmannian with Young tableaux. Subsequent research has been dedicated to carrying out a similar program in more intricate settings. A recent breakthrough in the Schubert calculus program concerning the homology of the affine Grassmannian and quantum cohomology of flags was made by identifying k-Schur functions with a new class of symmetric functions called Catalan functions. In this talk, we will discuss a K-theoretic refinement of this theory and how it sheds light on K-k-Schur functions, the Schubert representatives for the K homology of the affine Grassmannian.
Subject codes: 05E05, 14N15
Link to paper: https://arxiv.org/abs/2010.01759
Speaker's Contact Information:
Email: ghs9ae@virginia.edu
Website: https://ghseeli.github.io/
Abstract: Let X be an elliptic curve over C with one point removed, and consider the unordered configuration spaces Conf^n(X)={(x_1,...,x_n): x_i\neq x_j for i\neq j} / S_n. We present a rational function in two variables from whose coefficients we can read off the i-th Betti numbers of Conf^n(X) for all i and n. The key of the proof is a property called "purity", which was known to Kim for (ordered or unordered) configuration spaces of the complex plane with r \geq 0 points removed. We show that the unordered configuration spaces of X also have purity (but with different weights). This is a joint work with G. Cheong.
Link to paper: https://arxiv.org/abs/2009.07976
Speaker's Contact Information:
Email: huangyf@umich.edu
Abstract: The aim of this thesis is to review and improve upon an unpublished thesis by Green, whose goal was to construct Chern classes of coherent analytic sheaves in de Rham cohomology that respect the Hodge filtration. The second part of this thesis is dedicated to the construction of a categorical enrichment of the bounded derived category of complexes of coherent sheaves on an arbitrary complex manifold: the objects are ‘simplicial’ vector bundles endowed with a certain type of simplicial connection. This construction uses the theory of twisting cochains, developed in this setting by O’Brian, Toledo, and Tong. The first part is dedicated to defining a categorical lift of the Chern character in de Rham cohomology that respects the Hodge filtration, and for this we use the categorical model mentioned above. This construction can be undertaken by adapting classical Chern-Weil theory to the simplicial setting, using Dupont’s theory of fibre integration.
Link to papers: https://arxiv.org/abs/2003.10023
https://arxiv.org/abs/2003.10591
Speaker's Contact Information:
Email: timhosgood@posteo.net
Website: https://thosgood.com/
Abstract: The degree of irrationality of a complex projective n-dimensional variety X is the minimal degree of a dominant rational map from X to n-dimensional projective space. It is a birational invariant that measures how far X is from being rational. Accordingly, one expects the computation of this invariant to be a difficult problem in general. Alzati and Pirola showed in 1993 that the degree of irrationality of any abelian g-fold is at least g+1 using inequalities on holomorphic length. Tokunaga and Yoshihara later proved that this bound is sharp for abelian surfaces and Yoshihara asked for examples of abelian surfaces with degree of irrationality at least 4. Recently, Chen and Chen Stapleton showed that the degree of irrationality of any abelian surface is at most 4. In this work, I provide the first examples of abelian surfaces with degree of irrationality 4. In fact, I show that most abelian surfaces have degree of irrationality 4. For instance, a very general (1,d)-polarized abelian surface has degree of irrationality 4 if d does not divide 6. The proof is very short and uses nothing beyond Mumford's theorem on rational equivalences of zero-cycles on surfaces with p_g<0.
Link to paper: https://arxiv.org/abs/1911.00296
Speaker's Contact Information:
Email: olivier6martin@gmail.com
Abstract: The Brauer group of a variety can detect both algebraic and arithmetic properties of the underlying object. In particular, the Brauer-Manin obstruction that lies in the Brauer group can obstruct the existence of rational points. In this talk, I will discuss an algorithm to compute the prime torsion of the Brauer group of an elliptic curve E explicitly over various ground fields k. This algorithm gives generators and relations of the torsion subgroup as tensor products of symbol algebras over the function field of the elliptic curve. As a consequence of the algorithm, I will give an upper bound on the symbol length of the prime torsion of Br(E)/Br(k).
Link to paper: https://arxiv.org/abs/1909.05317
Subject Codes: 16K50, 14H52, 14F22
Speaker's Contact Information:
Email: cu9da@virginia.edu
Abstract: We discuss a construction of nearly overconvergent modular forms for the purpose of interpolating differential operators which act on them. Motivation for the construction comes from the theory of p-adic L-functions.
Link to paper: https://arxiv.org/abs/2006.02543
Speaker's Contact Information:
Email: jaycock@uoregon.edu
Website: pages.uoregon.edu/jaycock