Fri, 28 Apr 2023

12:00 - 13:00
N3.12

The “Galois to Automorphic” Direction of Categorical Geometric Langlands

Ken Lee
(University of Oxford)
Abstract

In this talk, I give a statement of the “Galois to automorphic” direction of categorical geometric Langlands. I will describe the Galois and automorphic side, the Hecke action on both sides, and the definition of Hecke eigensheaves. On the way, I hope to give motivation for the various objects at play : the stack of $G^L$ local systems on the fixed curve $X$, the stack of $G$ bundles on $X$, $D$-modules, arc groups, loop groups, the affine Grassmannian, and geometric Satake.

Fri, 09 Jun 2023

12:30 - 13:30
C1

The Harish-Chandra local character expansion and canonical dimensions for p-adic reductive groups

Mick Gielen
(University of Oxford)
Abstract

A complex irreducible admissible representation of a reductive p-adic group is typically infinite-dimensional. To quantify the "size" of such representations, we introduce the concept of canonical dimension. To do so we have to discuss the Moy-Prasad filtrations. These are natural filtrations of the parahoric subgroups. Next, we relate the canonical dimension to the Harish-Chandra local character expansion, which expresses the distribution character of an irreducible representation in terms of nilpotent orbital integrals. Using this, we consider the wavefront set of a representation. This is an invariant the naturally arises from the local character expansion. We conclude by explaining why the canonical dimension might be considered a weaker but more computable alternative to the wavefront set.

Tue, 16 May 2023

16:00 - 17:00
L5

Some extensions of the Katznelson-Tzafriri theorem

Charles Batty
(University of Oxford)
Abstract

In 1986, Katznelson and Tzafriri proved that, if $T$ is a power-bounded operator on a Banach space $X$, and the spectrum of $T$ meets the unit circle only at 1, then $\|T^n(I-T)\| \to 0$ as $n\to\infty$. Actually, they went further and proved that $\|T^nf(T)\| \to 0$ if $T$ and $f$ satisfy certain conditions. Soon afterward, analogous results were obtained for bounded $C_0$-semigroups $(T(t))_{t\ge0}$. Further extensions and variants were proved later. I will speak about several extensions to the Katznelson-Tzafriri theorem(s), including in particular a recent result(s) obtained by David Seifert and myself.

Tue, 25 Apr 2023

16:00 - 17:00
C1

Anomalous symmetries and invariants of operator algebras

Sergio Giron Pacheco
(University of Oxford)
Abstract

An anomalous symmetry of an operator algebra A is a mapping from a group G into the automorphism group of A which is multiplicative up to inner automorphisms. To any anomalous symmetry, there is an associated cohomology invariant in H^3(G,T). In the case that A is the Hyperfinite II_1 factor R and G is amenable, the associated cohomology invariant is shown to be a complete invariant for anomalous actions on R by the work of Connes, Jones, and Ocneanu.

In this talk, I will introduce anomalous actions from the basics discussing examples and the history of their study in the literature. I will then discuss two obstructions to possible cohomology invariants of anomalous actions on simple C*-algebras which arise from considering K-theoretic invariants of the algebras. One of the obstructions will be of algebraic flavour and the other will be of topological flavour. Finally, I will discuss the classification question for certain classes of anomalous actions.

Tue, 16 May 2023

14:00 - 15:00
L6

Profinite completion of free profinite groups

Tamar Bar-On
(University of Oxford)
Abstract

The pro-C completion of a free profinite group on an infinite set of generators is a profinite group of a greater rank. However, it is still not known whether it is a free profinite group too.  We will discuss this question, present a positive answer for some special varieties, and show partial results regarding the general case. In addition, we present the infinite tower of profinite completions, which leads to a generalisation for completions of higher orders. 

Tue, 16 May 2023
14:30
L3

On the Initialisation of wide Neural Networks: the Edge of Chaos

Thiziri Nait Saada
(University of Oxford)
Abstract

 Wide Neural Networks are well known for their Gaussian Process behaviour. Based upon this fact, an initialisation scheme for the weights and biases of a network preserving some geometrical properties of the input data is presented — The edge-of-chaos. This talk will introduce such a scheme before briefly mentioning a recent contribution related to the edge-of-chaos dynamics of wide randomly initialized low-rank feedforward networks. Formulae for the optimal weight and bias variances are extended from the full-rank to low-rank setting and are shown to follow from multiplicative scaling. The principle second order effect, the variance of the input-output Jacobian, is derived and shown to increase as the rank to width ratio decreases. These results inform practitioners how to randomly initialize feedforward networks with a reduced number of learnable parameters while in the same ambient dimension, allowing reductions in the computational cost and memory constraints of the associated network.

Tue, 16 May 2023
14:00
L3

Discrete Tensor-Product BGG Sequences: Splines and Finite Elements

Duygu Sap
(University of Oxford)
Abstract

Placeholder entry; date+time TBC. 

Abstract for talk: In this talk, we present a systematic discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over cubical meshes of arbitrary dimension via the use of tensor-product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div-div complexes as examples for our construction.

Tue, 30 May 2023
14:30
L3

High-Order Finite Element Schemes for Multicomponent Flow Problems

Aaron Baier-Reinio
(University of Oxford)
Abstract

The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

Tue, 09 May 2023
15:30
C4

Multivalued Dir-Minimizing Functions

Dr Immanuel Ben Porat
(University of Oxford)
Further Information

The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.

Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs, and some basic geometric measure theory are recommended.

Sessions led by  Dr Immanuel Ben Porat will take place on

09 May 2023 15:30 - 17:30 C4

16 May 2023 15:30 - 17:30 C4

23 May 2023 15:30 - 17:30 C4

30 May 2023 15:30 - 17:30 C4

Should you be interested in taking part in the course, please send an email to @email.

Abstract

COURSE_PROPOSAL (12).pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

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