Past

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.

Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage and these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

Bálint Koczor

Associate Professor in Quantum Information Theory

Mathematical Institute, University of Oxford

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In recent years, some progress has been made in the development of finite element complexes, particularly in the discretization of BGG complexes in two and three dimensions, including Hessian complexes, elasticity complexes, and divdiv complexes. In this talk, I will discuss distributional complexes in two and three dimensions. These complexes are simply constructed using geometric concepts such as vertices, edges, and faces, and they share the same cohomology as the complexes at the continuous level, which reflects that the discretization is structure preserving. The results can be regarded as a tensor generalization of the Whitney forms of the finite element exterior calculus. This talk is based on joint work with Snorre Christiansen (Oslo), Kaibo Hu (Edinburgh), and Qian Zhang (Michigan).

The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.

We will present some of the remarkable properties of eigenvalue trajectories for rank-one perturbations of random matrices, with an emphasis on two models of particular interest, namely weakly non-Hermitian and weakly non-unitary matrices. In both cases, precise estimates can be obtained for the critical timescale at which an outlier can be observed with high probability. We will outline the proofs of these results and highlight their significance in connection with quantum chaotic scattering. (Based on joint works with L. Erdös and J. Reker)

The notion of semi-uniform stability of a strongly continuous semi-group refers to the stability of classical solutions of a linear evolution equation, and this has analogues with the classical Katznelson-Tzafriri theorem. The co-generator of a strongly continuous semigroup is a bounded linear operator that comes from a particular discrete approximation to the semigroup. After reviewing some background on (quantified) stability theory for semigroups and the Katznelson-Tzafriri theorem, I will present some results relating the stability of a strongly continuous semigroup with that of its cogenerator. This talk is based on joint work with David Seifert.

The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss some recent applications to discrete notions of curvature on graphs. More details can be found in the recent preprint: https://arxiv.org/abs/2410.07756

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.

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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.