Past

The classical lower-tail problem for triangles in random graphs asks the following: given $\eta\in[0,1)$, what is the probability that $G(n,p)$ contains at most $\eta$ times the expected number of triangles?  When $p=o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and regularity or container methods in the latter case.

We prove for the first time asymptotic formulas for the logarithm of the lower tail probability when $p=c n^{-1/2}$ for $c$ constant.  Our results apply for all $c$ when $\eta \ge 1/2$ and for $c$  small enough when $\eta < 1/2$.  For the special case $\eta=0$ of triangle-freeness, our results prove that a phase transition occurs as $c$ varies (in the sense of a non-analyticity of the rate function), while for $\eta \ge 1/2$ we prove that no phase transition occurs.

Our method involves ingredients from algorithms and statistical physics including rapid mixing of Markov chains and the cluster expansion.  We complement our asymptotic formulas with efficient algorithms to approximately sample from $G(n,p)$ conditioned on the lower tail event.

Joint work with Will Perkins, Aditya Potukuchi and Michael Simkin.

Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families. 

We will be joined by Dr Brigitte Stenhouse from the Open University to discuss harassment, particularly in academic settings. 

We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.  Joint work with Yaoting Gui and Jürgen Jost.

The notion of Lip(gamma) Functions, for a parameter gamma > 0, introduced by Stein in the 1970s (building on earlier work of Whitney) is a notion of smoothness that is well-defined on arbitrary closed subsets (including, in particular, finite subsets) that is instrumental in the area of Rough Path Theory initiated by Lyons and central in recent works of Fefferman. Lip(gamma) functions provide a higher order notion of Lipschitz regularity that is well-defined on arbitrary closed subsets, and interacts well with the more classical notion of smoothness on open subsets. In this talk we will survey the historical development of Lip(gamma) functions and illustrate some fundamental properties that make them an attractive class of function to work with from a machine learning perspective. In particular, models learnt within the class of Lip(gamma) functions are well-suited for both inference on new unseen input data, and for allowing cost-effective inference via the use of sparse approximations found via interpolation-based reduction techniques. Parts of this talk will be based upon the works https://arxiv.org/abs/2404.06849 and https://arxiv.org/abs/2406.03232.

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. We construct a generalized fiber integration map associated to each Lie graph homology class and the main result is that this gives a faithful representation of graph homology. I will discuss how this leads to new possible interpretations of Lie graph homology classes as obstructions to, on one hand, smoothness of Poincaré duality fibrations, and, on the other hand, the existence of Poincaré duality algebra resolutions of the cochains of the total space as a dg module over the cochains of the base space.

I will discuss how some ideas from Geometric Langlands can be used to obtain new results in birational geometry and on the topology of algebraic varieties.

I will review recent progess on a topic of common interest to many
physicists https://www.arxiv.org/abs/2410.12043. Time permitting, I will
also comment about new sum rules for protected operators along RG flows:
https://arxiv.org/pdf/2409.09006. 

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions. How can you set out your ideas clearly, and what are the standard mathematical conventions? Please bring a pen or pencil!

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

I will describe a holomorphic-topological field theory in eleven-dimensions which captures a 1/16-BPS subsector of eleven-dimensional supergravity. Remarkably, asymptotic symmetries of the theory on flat space and on twisted versions of the AdS_4 x S^7 and AdS_7 x S^4 backgrounds recover three of the five infinite dimensional exceptional simple super-Lie algebras. I will discuss some applications of this fact, including character formulae for indices counting multigravitons and the contours of a program to holographically describe 1/16-BPS local operators in the 6d (2,0) SCFTs of type A_{N-1}. This talk is based on joint work, much in progress, with Fabian Hahner, Ingmar Saberi, and Brian Williams.

The Junior Algebra and Representation Theory Seminar will kick-off the start of the academic year with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Turing patterns have long been proposed as a mechanism for spatial organization in biology, but their relevance remains controversial due to the stringent fine-tuning often required. In this talk, I will present recent efforts to engineer synthetic Turing systems in bacterial colonies, highlighting both successes and limitations. While our three-node gene circuit generates patterns, challenges remain in extending these results to broader contexts. Additionally, I will discuss our exploration of machine learning methods to address the inverse problem of pattern formation, helping the design process down the road. This work addresses the ongoing task in translating theory into robust biological applications, offering insights into both current capabilities and future directions.

The important problem of backtesting financial models over long horizons inevitably leads to overlapping returns, giving rise to correlated samples. We propose a new method of dealing with this problem by decorrelation and show how this increases the discriminatory power of the resulting tests.


About the speaker
Nikolai Nowaczyk is a Risk Management & AI consultant who has advised multiple institutional clients in  projects around counterparty credit risk and xVA as well as data science and machine learning. 
Nikolai holds a PhD in mathematics from the University of Regensburg and has been an Academic Visitor at Imperial College London.
 

Registration for in-person attendance is required in advance.

Register here.

Von Neumann algebras which are not matrix algebras, yet still possess a unique trace, form a basic class called II$_1$ factors. The set of asymptotically commuting elements (or, the relative commutant of the algebra within its own ultrapower), dubbed the central sequence algebra, can take many different forms. In this talk, we discuss an elementary class of II$_1$ factors whose central sequence algebra is again a II$_1$ factor. We show that the class of infinitely generic II$_1$ factors possess this property, and ask some related questions about properties of other existentially closed II$_1$ factors. This is based on joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.

Given a singularity with a crepant resolution, a symmetry of the derived 
category of coherent sheaves on the resolution may often be constructed 
using the formalism of spherical functors. I will introduce this, and 
new work (arXiv:2409.19555) on general constructions of such symmetries 
for hypersurface singularities. This builds on previous results with 
Segal, and is inspired by work of Bodzenta-Bondal.

Let $K$ be an unramified extension of $\mathbb{Q}_p$ for a prime $p > 3$. The reduced part of the Emerton-Gee stack for $\mathrm{GL}_{2}$ can be viewed as parameterizing two-dimensional mod $p$ Galois representations of the absolute Galois group of $K$. In this talk, we will consider the extremely non-generic irreducible components of this reduced part and see precisely which ones are smooth or normal, and which have Gorenstein normalizations. We will see that the normalizations of the irreducible components admit smooth-local covers by resolution-rational schemes. We will also determine the singular loci on the components, and use these results to update expectations about the conjectural categorical $p$-adic Langlands correspondence. This is based on recent joint work with Ben Savoie.

Roe type algebras are operator algebras designed to catch the large-scale behaviour of metric spaces. This talk focuses on the following question: if two Roe type algebras associated to spaces (X,d_X) and (Y,d_Y) are isomorphic, how similar are X and Y? We provide positive results proved in the last 5 years, and, if time allows it, we show that sometimes answers to this question are subject to set theoretic considerations

Joint work with Paul Hacking (U Mass Amherst). We first explain how to 
prove homological mirror symmetry for a maximal normal crossing 
Calabi-Yau surface Y with split mixed Hodge structure. This includes the 
case when Y is a type III K3 surface, in which case this is used to 
prove a conjecture of Lekili-Ueda. We then explain how to build on this 
to prove an HMS statement for K3 surfaces. On the symplectic side, we 
have any K3 surface (X, ω) with ω integral Kaehler; on the algebraic 
side, we get a K3 surface Y with Picard rank 19. The talk will aim to be 
accessible to audience members with a wide range of mirror symmetric 
backgrounds.

Following the 2012 breakthrough in deep learning for classification and visions problems, the last decade has seen tremendous raise of interest in machine learning in a wider mathematical research community from foundational research through field specific analysis to applications. 

As data is at the core of any inverse problem, it was a natural direction for the field to investigate how machine learning could aid various aspects of inversion yielding numerous approaches from somewhat ad-hoc but very effective like learned unrolled methods to provably convergent learned regularisers with everything in between. In this talk I will review some on these developments through a lens of the research of our group.   

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

A number of moduli problems are, via Hodge theory, closely related to 
ball quotients. In this situation there is often a choice of possible 
compactifications such as the GIT compactification´and its Kirwan 
blow-up or the Baily-Borel compactification and the toroidal 
compactificatikon. The relationship between these compactifications is 
subtle and often geometrically interesting. In this talk I will discuss 
several cases, including cubic surfaces and threefolds and 
Deligne-Mostow varieties. This discussion links several areas such as 
birational geometry, moduli spaces of pointed curves, modular forms and 
derived geometry. This talk is based on joint work with S. 
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.

The celebrated Splitting Theorem by Cheeger-Gromoll states that a manifold with non-negative Ricci curvature which contains a line is isometric to a product, where one of the factors is the real line. A related result was later proved by Kasue. He showed that a manifold with non-negative Ricci curvature and two mean convex boundary components, one of which is compact, is also isometric to a product. In this talk, I will present a variant of Kasue’s result based on joint work with Andrea Mondino. We consider manifolds with non-negative Ricci curvature and disconnected mean convex boundary. We show that if one boundary component is parabolic and convex, then the manifold is a product, where one of the factors is an interval of the real line. The result is an application of recently developed tools in synthetic geometry and exploits the interplay between Ricci curvature and optimal transport.

We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications include in particular highly-oscillatory Kubo oscillators and spatial discretizations of the nonlinear Schrödinger equation with fast white noise dispersion. We construct a method of weak order two with computational cost and accuracy both independent of the stiffness of the oscillations. A geometric modification that conserves exactly quadratic invariants is also presented. If time allows, we will discuss ongoing work on uniformly accurate methods for such systems. This is a joint work with Gilles Vilmart.