Thu, 19 Oct 2023
16:00
L5

Siegel modular forms and algebraic cycles

Aleksander Horawa
(Oxford University)
Abstract

(Joint work with Kartik Prasanna)

Siegel modular forms are higher-dimensional analogues of modular forms. While each rational elliptic curve corresponds to a single holomorphic modular form, each abelian surface is expected to correspond to a pair of Siegel modular forms: a holomorphic and a generic one. We propose a conjecture that explains the appearance of these two forms (in the cohomology of vector bundles on Siegel modular threefolds) in terms of certain higher algebraic cycles on the self-product of the abelian surface. We then prove three results:
(1) The conjecture is implied by Beilinson's conjecture on special values of L-functions. Amongst others, this uses a recent analytic result of Radzwill-Yang about non-vanishing of twists of L-functions for GL(4).
(2) The conjecture holds for abelian surfaces associated with elliptic curves over real quadratic fields.
(3) The conjecture implies a conjecture of Prasanna-Venkatesh for abelian surfaces associated with elliptic curves over imaginary quadratic fields.

Thu, 15 Jun 2023

14:00 - 15:00
Lecture Room 3

26 Years at Oxford

Nick Trefethen
(Oxford University)
Abstract

I will reflect on my time as Professor of Numerical Analysis.

Tue, 28 Feb 2023

14:00 - 15:00
L4

Some combinatorial applications of guided random processes

Peter Keevash
(Oxford University)
Abstract

Random greedy algorithms became ubiquitous in Combinatorics after Rödl's nibble (semi-random method), which was repeatedly refined for various applications, such as iterative graph colouring algorithms (Molloy-Reed) and lower bounds for the Ramsey number $R(3,t)$ via the triangle-free process (Bohman-Keevash / Fiz Pontiveros-Griffiths-Morris). More recently, when combined with absorption, they have played a key role in many existence and approximate counting results for combinatorial structures, following a paradigm established by my proofs of the Existence of Designs and Wilson's Conjecture on the number of Steiner Triple Systems. Here absorption (converting approximate solutions to exact solutions) is generally the most challenging task, which has spurred the development of many new ideas, including my Randomised Algebraic Construction method, the Kühn-Osthus Iterative Absorption method and Montgomery's Addition Structures (for attacking the Ryser-Brualdi-Stein Conjecture). The design and analysis of a suitable guiding mechanism for the random process can also come with major challenges, such as in the recent proof of Erdős' Conjecture on Steiner Triple Systems of high girth (Kwan-Sah-Sawhney-Simkin). This talk will survey some of this background and also mention some recent results on the Queens Problem (Bowtell-Keevash / Luria-Simkin / Simkin) and the Existence of Subspace Designs (Keevash-Sah-Sawhney). I may also mention recent solutions of the Talagrand / Kahn-Kalai Threshold Conjectures (Frankston-Kahn-Narayanan-Park / Park-Pham) and thresholds for Steiner Triple Systems / Latin Squares (Keevash / Jain-Pham), where the key to my proof is constructing a suitable spread measure via a guided random process.

Thu, 19 Jan 2023
16:00
L5

Néron models of Jacobians and Chai's conjecture

Otto Overkamp
(Oxford University)
Abstract

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

Thu, 02 Mar 2023

14:00 - 15:00
Lecture Room 3

Finite element computations for modelling skeletal joints

Jonathan Whiteley
(Oxford University)
Abstract

Skeletal joints are often modelled as two adjacent layers of poroviscoelastic cartilage that are permitted to slide past each other.  The talk will begin by outlining a mathematical model that may be used, focusing on two unusual features of the model: (i) the solid component of the poroviscoelastic body has a charged surface that ionises the fluid within the pores, generating a swelling pressure; and (ii) appropriate conditions are required at the interface between the two adjacent layers of cartilage.  The remainder of the talk will then address various theoretical and practical issues in computing a finite element solution of the governing equations.

 

Thu, 08 Dec 2022
15:00
L3

On the stability of minimal submanifolds in conformal spheres

Federico Trinca
(Oxford University)
Abstract

Minimal submanifolds are the critical points of the volume functional. If the second derivative of the volume is nonnegative, we say that such a minimal submanifold is stable.

After reviewing some basics of minimal submanifolds in a generic Riemannian manifold, I will give some motivations behind the Lawson--Simons conjecture, which claims that there are no stable minimal submanifolds in 1/4-pinched spheres. Finally, I will discuss my recent work with Giada Franz on the nonexistence of stable minimal submanifolds in conformal pinched spheres.

Tue, 01 Nov 2022

14:00 - 15:00
L5

Generating random regular graphs quickly

Oliver Riordan
(Oxford University)
Abstract

A random $d$-regular graph is just a $d$-regular simple graph on $[n]=\{1,2,\ldots,n\}$ chosen uniformly at random from all such graphs. This model, with $d=d(n)$, is one of the most natural random graph models, but is quite tricky to work with/reason about, since actually generating such a graph is not so easy. For $d$ constant, Bollobás's configuration model works well; for larger $d$ one can combine this with switching arguments pioneered by McKay and Wormald. I will discuss recent progress with Nick Wormald, pushing linear-time generation up to $d=o(\sqrt{n})$. One ingredient is reciprocal rejection sampling, a trick for 'accepting' a certain graph with a probability proportional to $1/N(G)$, where $N(G)$ is the number of certain configurations in $G$. The trick allows us to do this without calculating $N(G)$, which would take too long.

Tue, 18 Oct 2022

14:00 - 15:00
L5

Improved bounds for 1-independent percolation on $\mathbb{Z}^n$

Paul Balister & Michael Savery
(Oxford University)
Abstract

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollobás defined $p_{\textrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\textrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\textrm{max}}(\mathbb{Z}^n)\leq p_{\textrm{max}}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty}p_{\textrm{max}}(\mathbb{Z}^n)$.

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\textrm{max}}(\mathbb{Z}^2)$. We will also show that with high confidence we have $p_{\textrm{max}}(\mathbb{Z}^n)<p_{\textrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.

Tue, 18 Oct 2022
14:00
L6

The local Langlands correspondence and unitary representations of GL(n)

Adam Brown
(Oxford University)
Abstract

Harish-Chandra's Lefschetz principle suggests that representations of real and p-adic split reductive groups are closely related, even though the methods used to study these groups are quite different. The local Langlands correspondence (as formulated by Vogan) indicates that these representation theoretic relationships stem from geometric relationships between real and p-adic Langlands parameters. In this talk we will discuss how the geometric structure of real and p-adic Langlands parameters lead to functorial relationships between representations of real and p-adic groups. I will describe work in progress which applies this functoriality to the study of unitary representations and signatures of invariant hermitian forms for GL(n). The main result expresses signatures of invariant hermitian forms on graded affine Hecke algebra modules in terms of signature characters of Harish-Chandra modules, which are computable via the unitary algorithm for real reductive groups by Adams-van Leeuwen-Trapa-Vogan.

Mon, 10 Oct 2022
14:15
L5

Quantitative estimates for almost harmonic maps

Melanie Rupflin
(Oxford University)
Abstract

For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points $u_j$ can converge to a limiting object with different topology.

A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable.

In this talk we present a method that allows us to prove Lojasiewicz inequalities in the singular setting of almost harmonic maps that converge to a simple bubble tree and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces of positive genus into general analytic manifolds.

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