Abstract: I am Jonathan, and I am interested in connections between logic and higher category theory. In this talk, I am explaining how to reason about fibrations of (∞,1)-categories in a synthetic way. Technically speaking, the logical system used is a version of homotopy type theory (HoTT) which has been extended in 2017 by Riehl and Shulman to provide a convenient setting for reasoning synthetically about Segal and complete Segal aka Rezk spaces. We motivate the basic definitions and state a few familiar theorems that one can prove in this setup, such as a version of the 2-Yoneda Lemma. This is based on joint work with Ulrik Buchholtz and my recent PhD dissertation supervised by Thomas Streicher at TU Darmstadt, Germany.
Link to paper: https://arxiv.org/abs/2105.01724
Subject codes: 03B38, 18N45, 18N60, 18D30, 55U35
Speaker's Contact Information:
Email: weinberger@mathematik.tu-darmstadt.de
Abstract: In order to understand problems in dynamics which are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the circle to the setting of rotations on adelic tori. One such result is the classical three gap theorem, which is also referred to as the three distance theorem and as the Steinhaus problem. It states that, for any real number, a, and positive integer, N, the collection of points na mod 1, where n runs from 1 to N, partitions the circle into component arcs having one of at most three distinct lengths. Since the 1950s, when this theorem was first proved independently by multiple authors, it has been reproved numerous times and generalized in many ways. One of the more recent proofs has been given by Marklof and Strömbergsson using a lattice based approach to gaps problems in Diophantine approximation. In this talk, we use an adaptation of this approach to the adeles to prove a natural generalization of the classical three gap theorem for rotations on adelic tori. This is joint work with Alan Haynes.
Link to paper: https://arxiv.org/pdf/2107.05147.pdf
Subject codes: 11J71, 11S82, 37P05, 37A45
Speaker's Contact Information:
Email: atdas@cougarnet.uh.edu
Website: https://www.math.uh.edu/~atdas/
Abstract: Algebraic K-theory is an important invariant encoding at the same time arithmetic, topological, and geometric information. Recently, there has been a renewed interest in refinements of K-theory that take into account additional 'symmetries' of the input category, for example through the work of Merling, May, and others on G-equivariant algebraic K-theory (for a fixed finite group G) or through Schwede's construction of global algebraic K-theory. While these two concepts are somewhat similar in spirit, they are ultimately quite different, and neither of them specializes to the other. In this talk I will introduce G-global algebraic K-theory as a synthesis of the above two approaches and go into some of the theory behind it. I will then explain how one can use this theory to generalize Thomason's classical result that K-theory exhibits symmetric monoidal categories as a model of connective stable homotopy theory to G-equivariant, global, and G-global contexts.
Subject Codes: 55P91, 19D23, 55P48, 18G55
Link to paper: https://arxiv.org/abs/2012.12676, https://arxiv.org/abs/2009.07004
Speaker's Contact Information:
Email: lenz@math.uni-bonn.de
Abstract: The notion of beta-rings is built to capture operations on a ring indexed by finite sets with permutation actions. Examples of such operations are taking a power of a set with an action by a finite group G, possibly imposing relations like symmetry. A similar algebraic structure is that of a lambda-ring. However, where the theory of lambda-rings is well behaved and studied extensively classically, the theory of beta-rings is still quite mysterious. I will present a new approach of arriving at beta-ring structures coming from equivariant algebra. There is a natural way of constructing from equivariant multiplications on a ring, so called power operations, the structure of a beta-ring. This is useful, since these power operations are well-behaved and are intimately tied to multiplications on structured ring spectra. For example, such power operations are equivalent to structured multiplications on Moore-spectra. Moreover, many classical examples of beta-rings are recovered this way.
Link to paper: https://arxiv.org/abs/2007.14304
Speaker's Contact Information:
Email: stahlhau@math.uni-bonn.de
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: In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk I will discuss some aspects of a recent joint work with Melanie Rupflin and Peter M. Topping in which we examined this problem for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface. On one hand, we were able to construct explicit targets so that the flow exhibited non-uniqueness. On the other hand, when the target is real analytic, we were able to prove a Lojasiewicz-Simon inequality and show convergence to a unique limit in the absence of singular behaviour.
Link to paper: https://doi.org/10.1515/acv-2019-0086
Speaker's Contact Information:
Email: kohout@maths.ox.ac.uk
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: Let C2 denote the cyclic group of order two. Given a manifold with a C2-action, we can consider its equivariant Bredon RO(C2)-graded cohomology. In this talk, we explain how a version of the Thom isomorphism theorem in RO(C2)-graded cohomology in constant Z/2 coefficients can be used to develop a theory of fundamental classes for equivariant submanifolds. We then show these classes can be used to understand the cohomology of C2-surfaces, including the ring structure.
Link to paper: https://arxiv.org/abs/1907.07284
Subject Codes: 55N91, 55P91
Contact Information:
Email: chazel@math.ucla.edu
Website: math.ucla.edu/~chazel