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

Website: https://www.math.uni-bonn.de/people/lenz/

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

Website: http://www.math.uni-bonn.de/people/stahlhau/

Abstract: Using Kuperberg's B2/C2 webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for the Lie algebra of type C2 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when quantum two is invertible, the Karoubi envelope of the C2 web category is equivalent to the category of tilting modules for the divided powers quantum group.

Link to paper: https://arxiv.org/abs/2009.13786

Speaker's Contact Information:

Email: ebodish@uoregon.edu

Website: https://sites.google.com/view/elijahbodish/home

Abstract: The snake graph construction from Musiker-Schiffler-Williams provides an expansion formula for cluster variables in cluster algebras from surfaces. We generalize this construction to generalized cluster algebras from orbifolds. This is joint work with Elizabeth Kelley.

Link to Paper: https://arxiv.org/abs/2003.13872

Speaker's Contact Information:

Email: banai003@umn.edu

Abstract: For a monoid M, we will look at the topos of sets with an action of M. It turns out that many "geometric" properties of this topos correspond to algebraic properties of the monoid M. The topos-theoretic properties that we will look at are for example the properties of being atomic, strongly compact, local, totally connected or cohesive.

In this way, toposes of sets with a monoid action become a new family of toposes for which these properties are well-understood. Another such family is the toposes of sheaves on topological spaces, and we can compare what happens in the two different situations.

This is joint work with Morgan Rogers (Università degli Studi dell'Insubria).

Subject codes: 18B25, 20M30

Link to paper: https://arxiv.org/abs/2004.10513

Speaker's Contact Information:

Email: jens.hemelaer@uantwerpen.be

Website: https://www.uantwerpen.be/en/staff/jens-hemelaer/

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

Website: http://uva.theopenscholar.com/charlotte-ure