Tag - Algebraic topology

Craig Westerland: The Stable Homology of the Braid Group with Coefficients Arising from the Hyperelliptic Representation

2nd November, 2023

The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.

Dan Petersen: Moments of Families of Quadratic L-Functions Over Function Fields Via Homotopy Theory

2nd November, 2023

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. (This will be explained in Craig Westerland's lecture on Nov 2.) We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. (This will be explained in my lecture on Nov 3.) Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

Leovigildo Alonso Tarrio: Derivators in additive context

20th October, 2023

By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.

Oscar Randal-Williams: Homeomorphisms of Euclidean Space

6th October, 2023

The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.

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).

Tselil Schramm: Higher-dimensional Expansion of Random Geometric Complexes

25th July, 2023

A graph is said to be a (1-dimensional) expander if the second eigenvalue of its adjacency matrix is bounded away from 1, or almost-equivalently, if it has no sparse vertex cuts. There are several natural ways to generalize the notion of expansion to hypergraphs/simplicial complexes, but one such way is 2-dimensional spectral expansion, in which the local expansion of vertex links/neighborhoods (remarkably) witnesses global expansion. While 1-dimensional expansion is known to be achieved by, e.g., random regular graphs, very few examples of sparse 2-dimensional expanders are known, and at present all are algebraic. It is an open question whether sparse 2-dimensional expanders are natural and "abundant" or "rare." In this talk, we'll give some evidence towards abundance: we show that the set of triangles in a random geometric graph on a high-dimensional sphere yields an expanding simplicial complex of arbitrarily small polynomial degree.

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.

Corey Jones: K-theoretic classification of fusion category actions on locally semisimple algebras

14th May, 2023

An action of a tensor category C on an associative algebra A is a linear monoidal functor from C to the monoidal category of A-A bimodules. We consider the problem of classifying (unitary) actions of (unitary) fusion categories on inductive limits of semisimple associative algebras (called locally semisimple algebras). A theorem of Elliot classifies locally semisimple algebras by their ordered K0 groups. We extend this theorem to a K-theoretic classification of fusion category actions on locally semisimple algebras which have an inductive limit decomposition.