Fri, 07 Feb 2025

12:00 - 13:00
Quillen Room

BGG resolutions and Grothendieck-Cousin Complexes

Xuanzuo Chen
(University of Oxford)
Abstract

Bernstein–Gelfand–Gelfand (BGG) resolutions and the Grothendieck–Cousin complex both play central roles in modern algebraic geometry and representation theory. The BGG approach provides elegant, combinatorial resolutions for important classes of modules especially those arising in Lie theory; while Grothendieck–Cousin complexes furnish a powerful framework for computing local cohomology via filtrations by support. In this talk, we will give an overview of these two constructions and illustrate how they arise from the same categorical consideration.

Fri, 07 Feb 2025

11:00 - 12:00
L4

Self-organized patterning in complex biological fluids

Dr Giulia Celora
(Mathematical Institute University of Oxford)
Abstract

Understanding how living systems dynamically self-organise across spatial and temporal scales is a fundamental problem in biology; from the study of embryo development to regulation of cellular physiology. In this talk, I will discuss how we can use mathematical modelling to uncover the role of microscale physical interactions in cellular self-organisation. I will illustrate this by presenting two seemingly unrelated problems: environmental-driven compartmentalisation of the intracellular space; and self-organisation during collective migration of multicellular communities. Our results reveal hidden connections between these two processes hinting at the general role that chemical regulation of physical interactions plays in controlling self-organisation across scales in living matter

Thu, 06 Feb 2025
17:00
L3

Asymptotic theories: from finite structures to infinite fields

Philip Dittmann
(University of Manchester)
Abstract

I will discuss several interesting examples of classes of structures for which there is a sensible first-order theory of "almost all" structures in the class, for certain notions of "almost all". These examples include the classical theory of almost all finite graphs due to Glebskij-Kogan-Liogon'kij-Talanov and Fagin (and many more examples from finite model theory), as well as more recent examples from the model theory of infinite fields: the theory of almost all algebraic extensions and the universal/existential theory of almost all completions of a global field (both joint work with Arno Fehm). Interestingly, such asymptotic theories are sometimes quite well-behaved even when the base theories are not.

Thu, 06 Feb 2025
17:00
L6

Parametrising complete intersections

Jakub Wiaterek
(University of Oxford )
Abstract

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

Thu, 06 Feb 2025
16:00
L4

Unramified Langlands: geometric and function-theoretic

Dennis Gaitsgory
(MPIM, Bonn)
Abstract

I will explain the content of Geometric Langlands (which is a theorem over the ground fields of characteristic 0 but still a conjecture in positive characteristic) and show how it implies a description of the space of automorphic functions in terms of Galois data. The talk will mostly follow a joint paper with Arinkin, Kazhdan, Raskin, Rozenblyum and Varshavsky from 2022.

Thu, 06 Feb 2025
16:00
L4

Unramified Langlands: geometric and function-theoretic

Dennis Gaitsgory
(MPI Bonn)
Abstract
I will explain the content of Geometric Langlands (which is a theorem over the ground fields of characteristic 0 but still a conjecture in positive characteristic) and show how it implies a description of the space of automorphic functions in terms of Galois data. The talk will mostly follow a joint paper with Arinkin, Kazhdan, Raskin, Rozenblyum and Varshavsky from 2022.
Thu, 06 Feb 2025
16:00
L5

First-best implementation in dynamic adverse selection models with news

Jacco Thijssen
(University of York)
Abstract

This paper shows that a simple sale contract with a collection of options implements the full-information first-best allocation in a variety of continuous-time dynamic adverse selection settings with news. Our model includes as special cases most models in the literature. The implementation result holds regardless of whether news is public (i.e., contractible) or privately observed by the buyer, and it does not require deep pockets on either side of the market. It is an implication of our implementation result that, irrespective of the assumptions on the game played, no agent waits for news to trade in such models. The options here do not play a hedging role and are, thus, not priced using a no-arbitrage argument. Rather, they are priced using a game-theoretic approach.

Thu, 06 Feb 2025

14:00 - 15:00
Lecture Room 3

Deflation Techniques for Finding Multiple Local Minima of a Nonlinear Least Squares Problem

Marcus Webb
(University of Manchester)
Abstract

Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators). 

 

In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems.  I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).

 

Main preprint: https://arxiv.org/abs/2409.14438

WoodburyLS preprint: https://arxiv.org/abs/2406.15120

Thu, 06 Feb 2025

12:00 - 13:00
L3

Modelling flying formations and vortex ring motions

Christiana Mavroyiakoumou
( Courant Institute of Mathematical Sciences)
Further Information

Christiana is an Assistant Professor at the Courant Institute of Mathematical Sciences (New York University) working in the Applied Math Lab, primarily with Leif Ristroph and Jun Zhang. Her interests are in using modeling, numerical simulations, and experiments to study fluid dynamical problems, with an emphasis on fluid-structure interactions.

Currently Christiana is working on understanding the role of flow interactions in flying bird formations and the hydrodynamics of swimming fish.

Abstract

We consider two problems in fluid dynamics: the collective locomotion of flying animals and the interaction of vortex rings with fluid interfaces. First, we present a model of formation flight, viewing the group as a material whose properties arise from the flow-mediated interactions among its members. This aerodynamic model explains how flapping flyers produce vortex wakes and how they are influenced by the wakes of others. Long in-line arrays show that the group behaves as a soft, excitable "crystal" with regularly ordered member "atoms" whose positioning is susceptible to deformations and dynamical instabilities. Second, we delve into the phenomenon of vortex ring reflections at water-air interfaces. Experimental observations reveal reflections analogous to total internal reflection of a light beam. We present a vortex-pair--vortex-sheet model to simulate this phenomenon, offering insights into the fundamental interactions of vortex rings with free surfaces.

Thu, 06 Feb 2025

12:00 - 12:30
Lecture room 5

A posteriori error estimation for randomized low-rank approximation

Yuji Nakatsukasa
((Oxford University))
Abstract

A number of algorithms are now available---including Halko-Martinsson-Tropp, interpolative decomposition, CUR, generalized Nystrom, and QR with column pivoting---for computing a low-rank approximation of matrices. Some methods come with extremely strong guarantees, while others may fail with nonnegligible probability. We present methods for efficiently estimating the error of the approximation for a specific instantiation of the methods. Such certificate allows us to execute "responsibly reckless" algorithms, wherein one tries a fast, but potentially unstable, algorithm, to obtain a potential solution; the quality of the solution is then assessed in a reliable fashion, and remedied if necessary. This is joint work with Gunnar Martinsson. 

Time permitting, I will ramble about other topics in Randomised NLA. 

Wed, 05 Feb 2025
16:00
Lecture Room 4

Twisting Higgs modules and applications to the p-adic Simpson correspondence II

Ahmed Abbes
(IHES)
Abstract

[This is the second in a series of two talks; the first talk will be in the Algebra Seminar of Tuesday Feb 4th https://www.maths.ox.ac.uk/node/70022]

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

Wed, 05 Feb 2025
16:00
L6

Semi-regular tilings and the d-chromatic number of the hyperbolic plane

Luke Waite
(University of Southampton)
Abstract

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

Tue, 04 Feb 2025
16:00
C3

Equivariant correspondences

Kenny de Commer
(VUB)
Abstract

Given two von Neumann algebras A,B with an action by a locally compact (quantum) group G, one can consider its associated equivariant correspondences, which are usual A-B-correspondences (in the sense of Connes) with a compatible unitary G-representation. We show how the category of such equivariant A-B-correspondences carries an analogue of the Fell topology, which is preserved under natural operations (such as crossed products or equivariant Morita equivalence). If time permits, we will discuss one particular interesting example of such a category of equivariant correspondences, which quantizes the representation category of SL(2,R). This is based on joint works with Joeri De Ro and Joel Dzokou Talla. 

Tue, 04 Feb 2025
15:30
L4

Global logarithmic deformation theory

Simon Felten
(Oxford)
Abstract

A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

Tue, 04 Feb 2025
15:00
L6

Random elements in finite groups

Michał Marcinkowski
Abstract

How to uniformly, or at least almost uniformly, choose an element from a finite group ? When G is too large to enumerate all its elements, direct (pseudo)random selection is impossible. However, if we have an explicit set of generators of G (e.g., as in the Rubik's cube group), several methods are available. This talk will focus on one such method based on the well-known product replacement algorithm. I will discuss how recent results on property (T) by Kaluba, Kielak, Nowak and Ozawa partially explain the surprisingly good performance of this algorithm.

Tue, 04 Feb 2025
14:00
C4

Mapping regularized network flows on networks with incomplete observations

Jelena Smiljanic
(Umea University)
Abstract

Real-world networks have a complex topology with many interconnected elements often organized into communities. Identifying these communities helps reveal the system’s organizational and functional structure. However, network data can be noisy, with incomplete link observations, making it difficult to detect significant community structures as missing data weakens the evidence for specific solutions. Recent research shows that flow-based community detection methods can highlight spurious communities in sparse networks with incomplete link observations. To address this issue, these methods require regularization. In this talk, I will show how a Bayesian approach can be used to regularize flows in networks, reducing overfitting in the flow-based community detection method known as the map equation.

Tue, 04 Feb 2025

14:00 - 15:00
L4

Normal covering numbers for groups and connections to additive combinatorics

Sean Eberhard
(University of Warwick)
Abstract

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of $\gamma(S_n)$ and $\gamma(A_n)$ as $n$ tends to infinity.

Tue, 04 Feb 2025
13:00
L5

Symmetries of Coupled Minimal Models

Connor Behan
(ITP Sao Paolo)
Abstract

When tensor products of N minimal models accumulate at central charge N, they also admit relevant operators arbitrarily close to marginality. This raises the tantalizing possibility that they can be use to reach purely Virasoro symmetric CFTs where the breaking of extended chiral symmetry can be seen in a controlled way. This talk will give an overview of the theories where this appears to be the case, according to a brute force check at low lying spins. We will also encounter an interesting non-example where the same type of analysis can be used to give a simpler proof of integrability.

Tue, 04 Feb 2025
10:00
L4

Twisting Higgs modules and applications to the p-adic Simpson correspondence I (special time!)

Ahmed Abbes
(IHES)
Abstract

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

Mon, 03 Feb 2025
16:30
L4

Shock Reflection and other 2D Riemann Problems in Gas Dynamics

Alexander Cliffe
(Università degli Studi di Padova)
Abstract

The Riemann problem is an IVP having simple piecewise constant initial data that is invariant under scaling. In 1D, the problem was originally considered by Riemann during the 19th century in the context of gas dynamics, and the general theory was more or less completed by Lax and Glimm in the mid-20th century. In 2D and MD, the situation is much more complicated, and very few analytic results are available. We discuss a shock reflection problem for the Euler equations for potential flow, with initial data that generates four interacting shockwaves. After reformulating the problem as a free boundary problem for a nonlinear PDE of mixed hyperbolic-elliptic type, the problem is solved via a sophisticated iteration procedure. The talk is based on joint work with G-Q Chen (Oxford) et. al. arXiv:2305.15224, to appear in JEMS (2025).

Mon, 03 Feb 2025
16:00
C3

The uniqueness theorem for Kasparov theory

Gabor Szabo
(KU Leuven)
Abstract

Kasparov's bivariant K-theory (or KK-theory) is an extremely powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated for a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem, which forms the technical basis behind many of the most important articles written over the past decade. In the recent landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds.

In this talk I want to present the affirmative answer to this question: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.

Mon, 03 Feb 2025
16:00
C6

Progress towards the Keating-Snaith conjecture for quadratic twists of elliptic curves

Nathan Creighton
(University of Oxford)
Abstract

The Keating-Snaith conjecture for quadratic twists of elliptic curves predicts the central values should have a log-normal distribution. I present recent progress towards establishing this in the range of large deviations of order of the variance. This extends Selberg’s Central Limit Theorem from ranges of order of the standard deviation to ranges of order of the variance in a variety of contexts, inspired by random walk theory. It is inspired by recent work on large deviations of the zeta function and central values of L-functions.
 

Mon, 03 Feb 2025
15:30
L3

Analyzing the Error in Score-Based Generative Models: A Stochastic Control Approach

Dr Giovanni Conforti
(University of Padova)
Abstract

Score-based generative models (SGMs), which include diffusion models and flow matching, have had a transformative impact on the field of generative modeling. In a nutshell, the key idea is that by taking the time-reversal of a forward ergodic diffusion process initiated at the data distribution, one can "generate data from noise." In practice, SGMs learn an approximation of the score function of the forward process and employ it to construct an Euler scheme for its time reversal.

In this talk, I will present the main ideas of a general strategy that combines insights from stochastic control and entropic optimal transport to bound the error in SGMs. That is, to bound the distance between the algorithm's output and the target distribution. A nice feature of this approach is its robustness: indeed, it can be used to analyse SGMs built upon noising dynamics that are different from the Ornstein-Uhlenbeck process . As an example, I will illustrate how to obtain error bounds for SGMs on the hypercube.

Based on joint works with A.Durmus, M.Gentiloni-Silveri, Nhi Pham Le Tuyet and Dario Shariatian
Mon, 03 Feb 2025
15:30
L5

Relative Thom conjectures

Matthew Hedden
(Michigan State University)
Abstract

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

Mon, 03 Feb 2025
14:15
L5

ALC G2-manifolds

Lorenzo Foscolo
(La Sapienza, Rome)
Abstract

ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.