Tag - K-theory

Matthew Morrow: Algebraic K-theory and p-adic arithmetic geometry

22nd April, 2024

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information: depending on the ring, its K-groups can be related to zeta functions, corbordisms, algebraic cycles and the Hodge conjecture, elliptic operators, Grothendieck's theory of motives, and so on.

Our understanding of algebraic K-groups, at least as far as they appear in algebraic and arithmetic geometry, has rapidly improved in the past few years. This talk will present some of the fundamentals of the subject and explain why K-groups are related to the ongoing special year in p-adic arithmetic geometry. The intended audience is non-specialists.

Maxime Ramzi: Categorifying spectra and the theorem of the heart

28th March, 2024

The goal of this talk will be to present the results from my recent joint work with Vova Sosnilo and Christoph Winges, where we prove that every spectrum is the (non-connective) K-theory spectrum of a stable category. Our main application of this is the disproof of a conjecture by Antieau-Gepner-Heller about a non-connective version of the theorem of the heart in the non-noetherian setting; but I will also try to mention other perspectives on this result.

Maxime Ramzi: On Endomorphisms of THH

27th March, 2024

Topological Hochschild homology is an important invariant, closely related to algebraic K-theory, and can be seen as a non-commutative analogue of de Rham chains.

In this talk, I will describe various computations of the ring/monoid of endomorphisms of THH in different variants, with an approach based on a generalized version of the Dundas-McCarthy theorem.

Jay Shah: Real topological Hochschild homology, C2-stable trace theories, and Poincaré cyclic graphs

29th February, 2024

To study topological Hochschild homology as an invariant of stable ∞-categories and endow it with its universal property in this context, Nikolaus introduced the formalism of stable cyclic graphs and trace theories (after Kaledin). On the other hand, Poincaré ∞-categories are a C2-refinement of stable ∞-categories that provide an adequate formalism for studying real and hermitian algebraic K-theory, which should be then well-approximated by the real cyclotomic trace. In this talk, we explain how to systematically provide Poincaré refinements of all the components of Nikolaus's approach to stable trace theories.

Tess Bouis: Motivic Cohomology of Mixed Characteristic Schemes

14th February, 2024

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.

Shachar Carmeli: Cyclotomic Redshift

18th January, 2024

I will discuss a joint work with Ben-Moshe, Schlank, and Yanovski, proving the compatibility of T(n+1)-local algebraic K-theory with the formation of homotopy limits with respect to p-local π-finite group actions on T(n)-local categories. This is a generalization of the results of Thomason for height 0 and Clausen, Mathew, Naumann, and Noel for actions of discrete p-groups in arbitrary chromatic height. I will then discuss the compatibility of K-theory with the chromatic cyclotomic extensions, chromatic Fourier transform, and higher Kummer theory from previous works with Barthel, Schlank, and Yanovski, phenomena we refer to as "cyclotomic redshift''. Finally, I will explain how cyclotomic redshift gives hyperdescent for K-theory along the cyclotomic tower after K(n+1)-localization.

Weihong Xu: Quantum K-theory of partial flag varieties

4th December, 2023

The quantum K-theory ring of a smooth projective variety is a deformation of its K-theory ring of vector bundles. Inspired by supersymmetric gauge theory and generalizing a result of Gu, Mihalcea, Sharpe, and Zou for Grassmannians, we give conjectural relations in the quantum K-theory rings of partial flag varieties that deform Whitney relations. If these relations hold, then they form a complete set of relations. We prove these relations for Fl(1,n-1;n) relying on the proof of a quantum K divisor axiom conjectured by Buch and Mihalcea for partial flag varieties. I will discuss the geometry of the quantum K divisor axiom and a few other implications of it.

George Elliott: K-theory and C*-algebras

8th September, 2023

This is a 35-lecture course, with each lecture being an hour, given by George Elliott. Note that the 32nd lecture was not recorded. The first 31 lectures are still of great interest, but this needs to be known.

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras, both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras - what Bratteli called AF algebras - is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Alina Vdovina: Higher structures in Algebra, Geometry and C*-algebras

23rd June, 2023

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.