Fri, 17 Oct 2025

14:00 - 15:00
L1

The Art of Maths Communication

Abstract

Join bestselling author Simon Singh and Oxford mathematician turned educator Junaid Mubeen for a session on maths communication! Learn how to present mathematics in a way that is both accessible and engaging, and how to apply these principles in a teaching context. Simon and Junaid will draw on their experiences in the Parallel Academy https://parallel.org.uk, an online initiative they set up in 2023, which has since grown to support thousands of keen and talented students to pursue maths beyond the curriculum. 

Fri, 17 Oct 2025
13:00
L6

Zero sets from the viewpoint of topological persistence

Vukašin Stojisavljević
(Oxford University)
Abstract

Studying the topology of zero sets of maps is a central topic in many areas of mathematics. Classical homological invariants, such as Betti numbers, are not always suitable for this purpose due to the fact that they do not distinguish between topological features of different sizes. Topological data analysis provides a way to study topology coarsely by ignoring small-scale features. This approach yields generalizations of a number of classical theorems, such as Bézout's theorem and Courant’s nodal domain theorem, to a wider class of maps. We will explain this circle of ideas and discuss potential directions for future research. The talk is partially based on joint works with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

Fri, 17 Oct 2025

12:00 - 13:00
N4.01

Mathematrix Welcome Pizza Lunch

(Mathematrix)
Abstract

Join us for an initial welcome pizza lunch to start the academic year to learn about what's happening in Mathematrix in 2025/26! Meet other students who are from underrepresented groups in mathematics and allies :) 

Please RSVP here to confirm your spot: https://form.jotform.com/252814345456864

Fri, 17 Oct 2025
12:00
L3

Multi-Entropy Measures for Topologically Ordered Phases in (2+1) Dimensions

Shinsei Ryu
(Princeton)
Abstract

 

Entanglement entropy has long served as a key diagnostic of topological order in (2+1) dimensions. In particular, the topological entanglement entropy captures a universal quantity (the total quantum dimension) of the underlying topological order. However, this information alone does not uniquely determine which topological order is realized, indicating the need for more refined probes. In this talk, I will present a family of quantities formulated as multi-entropy measures, including examples such as reflected entropy and the modular commutator. Unlike the conventional bipartite setting of topological entanglement entropy, these multi-entropy measures are defined for tripartite partitions of the Hilbert space and capture genuinely multipartite entanglement. I will discuss how these measures encode additional universal data characterizing topologically ordered ground states.

Thu, 16 Oct 2025
17:00
L3

Integration in finite terms and exponentially algebraic functions

Jonathan Kirby
(University of East Anglia)
Abstract

The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially-algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of their types in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions and give some new decidability results.

This is joint work with Rémi Jaoui, Lyon

Thu, 16 Oct 2025
16:00
L5

The Relative Entropy of Expectation and Price

Paul McCloud
(nomura)
Abstract

Understanding the relationship between expectation and price is central to applications of mathematical finance, including algorithmic trading, derivative pricing and hedging, and the modelling of margin and capital. In this presentation, the link is established via dynamic entropic risk optimisation, which is promoted for its convenient integration into standard pricing methodologies and for its ability to quantify and analyse model risk. As an example of the versatility of entropic pricing, discrete models with classical and quantum information are compared, with studies that demonstrate the effectiveness of quantum decorrelation for model fitting.

Thu, 16 Oct 2025
15:00
L6

Operator algebras meet (generalized) global symmetries

Andrea Antinucci
Abstract

Two different, almost orthogonal approaches to QFT are: (1) the study of von Neumann algebras of local observables in flat space, and (2) the study of extended and topological defects in general spacetime manifolds. While naively the two focus on different aspects, it has been recently pointed out that some of the axioms of approach (1) clash with certain expectations from approach (2). In this JC talk, I’ll give a brief introduction to both approaches and review the recent discussion in [2008.11748], [2503.20863], and [2509.03589], explaining (i) what the tensions are, (ii) a recent proposal to solve them, and (iii) why it can be useful.

Thu, 16 Oct 2025

14:00 - 15:00
Lecture Room 3

Piecewise rational finite element spaces of differential forms

Evan Gawlik
(Santa Clara University)
Abstract

The Whitney forms on a simplicial triangulation are piecewise affine differential forms that are dual to integration over chains.  The so-called blow-up Whitney forms are piecewise rational generalizations of the Whitney forms.  These differential forms, which are also called shadow forms, were first introduced by Brasselet, Goresky, and MacPherson in the 1990s.  The blow-up Whitney forms exhibit singular behavior on the boundary of the simplex, and they appear to be well-suited for constructing certain novel finite element spaces, like tangentially- and normally-continuous vector fields on triangulated surfaces.  This talk will discuss the blow-up Whitney forms, their properties, and their applicability to PDEs like the Bochner Laplace problem.  

Thu, 16 Oct 2025

12:00 - 13:00
L3

Think Global, Act Local: A Mathematician's Guide to Inducing Localised Patterns

Dan J. Hill
(University of Oxford)
Further Information

Dan is a recently appointed Hooke Fellow within OCIAM. His research focus is on pattern formation and the emergence of localised states in PDE models, with an emphasis on using polar coordinate systems to understand nonlinear behaviour in higher spatial dimensions. He received his MMath and PhD from the University of Surrey, with a thesis on the existence of localised spikes on the surface of a ferrofluid, and previously held postdoctoral positions at Saarland University, including an Alexander von Humboldt Postdoctoral Fellowship. www.danjhill.com

Abstract
The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples.
 
In this talk I will discuss how adding a compact region of spatial heterogeneity to a PDE model can not only induce the emergence of fully localised 2D patterns, but also allows us to rigorously prove and characterise their bifurcation. The idea is inspired by experimental and numerical studies of magnetic fluids and tornados, where our compact heterogeneity corresponds to a local spike in the magnetic field and temperature gradient, respectively. In particular, we obtain local bifurcation results for fully localised patterns both with and without radial or dihedral symmetry, and rigorously continue these solutions to large amplitude. Notably, the initial bifurcating solution (which can be stable at bifurcation) varies between a radially-symmetric spot and a 'dipole' solution as the width of the spatial heterogeneity increases. 
 
This work is in collaboration with David J.B. Lloyd and Matthew R. Turner (both University of Surrey).
 
 
Thu, 16 Oct 2025

12:00 - 12:30
Lecture Room 4

A C0-hybrid interior penalty method for the nematic Helmholtz-Korteweg equation

Tim van Beeck
(University of Göttingen)
Abstract

The nematic Helmholtz-Korteweg equation is a fourth-order scalar PDE modelling time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. Conforming discretizations typically require C1-conforming elements, for example the Argyris element, whose implementation is notoriously challenging - especially in three dimensions - and often demands a high polynomial degree. 
In this talk, we consider an alternative non-conforming C0-hybrid interior penalty method that is both stable and convergent for any polynomial degree greater than two. Classical C0-interior penalty methods employ an H1-conforming subspace and treat the non-conformity with respect to H2 with discontinuous Galerkin techniques. Building on this idea, we use hybridization techniques to improve the computational efficiency of the discretization. We provide a brief overview of the numerical analysis and show numerical examples, demonstrating the method's ability to capture anisotropic propagation of sound in two and three dimensions. 

Wed, 15 Oct 2025
16:00
L4

Pointwise bounds for 3-torsion (note: Wednesday)

Stephanie Chan
(UCL)
Abstract

For $\ell$ an odd prime number and $d$ a squarefree integer, a notable problem in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record due to Ellenberg—Venkatesh is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Peter Koymans.

Wed, 15 Oct 2025
15:00
L5

The Polynomial Conjecture for Monomial Representations of Exponential Lie Groups

Ali Baklouti
(University of SFAX Tunisia)
Abstract

Let \( G = \exp(\mathfrak{g}) \) be a connected, simply connected nilpotent Lie group with Lie algebra \( \mathfrak{g} \), and let \( H = \exp(\mathfrak{h}) \) be a closed subgroup with Lie algebra \( \mathfrak{h} \). Consider a unitary character \( \chi \) of \( H \), given by \(\chi(\exp X) = \chi_{f}(\exp X) = e^{i f(X)}, \  X \in \mathfrak{h}, \) for some \( f \in \mathfrak{g}^{\ast} \). Let \( \tau = \operatorname{Ind}_{H}^{G} \chi \) denote the monomial representation of \( G \) induced from \( \chi \).

The object of interest is the algebra \( D_{\tau}(G/H) \) of \( G \)-invariant differential operators acting on the homogeneous line bundle associated with the data \( (G, H, \chi) \). Under the assumption that \( \tau \) has finite multiplicities, it is known that \( D_{\tau}(G/H) \) is commutative.

In this talk, I will discuss the Polynomial Conjecture for the representation \( \tau \), which asserts that the algebra \( D_{\tau}(G/H) \) is isomorphic to  
\(\mathbb{C}[\Gamma_{\tau}]^{H}\),  the algebra of \( H \)-invariant polynomial functions on \( \Gamma_{\tau} \). Here, \( \Gamma_{\tau} = f + \mathfrak{h}^{\perp} \) denotes the affine subspace of \( \mathfrak{g}^{\ast} \).

I will present recent advances toward proving this conjecture, with a particular emphasis on Duflo's Polynomial Conjecture concerning the Poisson center of \( \Gamma_{\tau} \). Furthermore, I will discuss the case where \( \tau \) has discrete-type multiplicities in the exponential setting, shedding light on a counterexample to Duflo's conjecture.
 

Tue, 14 Oct 2025
16:00
C3

Homotopy groups of Cuntz classes in C*-algebras

Andrew Toms
(Leverhulme Visiting Professor, University of Oxford)
Abstract

The Cuntz semigroup of a C*-algebra A consists of equivalence classes of positive elements, where equivalence means roughly that two positive elements have the same rank relative to A.  It can be thought of as a generalization of the Murray von Neumann semigroup to positive elements and is an incredibly sensitive invariant. We present a calculation of the homotopy groups of these Cuntz classes as topological subspaces of A when A is classifiable in the sense of Elliott.  Remarkably, outside the case of compact classes, these spaces turn out to be contractible.  

Tue, 14 Oct 2025
15:30
L4

Vafa-Witten invariants from modular anomaly

Sergey Alexandrov
(Montpelier)
Abstract
I'll present a modular anomaly equation satisfied by generating functions of refined Vafa-Witten invariants 
for the gauge group $U(N)$ on complex surfaces with $b_1=0$ and $b_2^+=1$, 
which has been derived from S-duality of string theory.
I'll show how this equation can used to find explicit expressions for these generating functions
(and their modular completions) on $\mathbb{CP}^2$, Hirzebruch and del Pezzo surfaces.
The construction for $\mathbb{CP}^2$ suggests also a new form of blow-up identities.
Tue, 14 Oct 2025

14:00 - 15:00
L4

An exponential upper bound on induced Ramsey numbers

Marcelo Campos
(Instituto Nacional de Matemática Pura e Aplicada (IMPA))
Abstract
The induced Ramsey number $R_{ind}(H)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all red/blue colorings of its edges contain a monochromatic induced copy of $H$. In this talk I'll show there exists an absolute constant $C > 0$ such that, for every graph $H$ on $k$ vertices, these numbers satisfy $R_{ind}(H) ≤ 2^{Ck}$. This resolves a conjecture of Erdős from 1975.
 
This is joint work with Lucas Aragão, Gabriel Dahia, Rafael Filipe and João Marciano.
Tue, 14 Oct 2025
14:00
L6

The Laplace Transform on Lie Groups: A Representation-Theoretical Perspective

Ali Baklouti
(University of SFAX Tunisia)
Abstract

In this talk, I will present a representation-theoretical approach to constructing a non-commutative analogue of the classical Laplace transform on Lie groups. I will begin by discussing the motivations for such a generalization, emphasizing its connections with harmonic analysis, probability theory, and the study of evolution equations on non-commutative spaces. I will also outline some of the key challenges that arise when extending the Laplace transform to the setting of Lie groups, including the non-commutativity of the group operation and the complexity of its dual space.

The main part of the talk will focus on an explicit construction of the Laplace transform in the framework of connected, simply connected nilpotent Lie groups. This construction relies on Kirillov’s orbit method, which provides a powerful bridge between the geometry of coadjoint orbits and the representation theory of nilpotent groups.

As an application, I will describe an operator-theoretic analogue of the classical Müntz–Szász theorem, establishing a density result for a family of generalized polynomials in associated with the group setting. This result highlights the strength of the representation-theoretical approach and its potential for solving classical approximation problems in a non-commutative context.

Tue, 14 Oct 2025
13:00
L2

SymTFTs for continuous spacetime symmetries

Nicola Dondi
(ICTP)
Abstract

Symmetry Topological Field Theories (SymTFTs) are topological field theories that encode the symmetry structure of global symmetries in terms of a theory in one higher dimension. While SymTFTs for internal (global) symmetries have been highly successful in characterizing symmetry aspects in the last few years, a corresponding framework for spacetime symmetries remains unexplored. We propose an extension of the SymTFT framework to include spacetime symmetries. In particular, we propose a SymTFT for the conformal symmetry in various spacetime dimensions. We demonstrate that certain BF-type theories, closely related to topological gravity theories, possess the correct topological operator content and boundary conditions to realize the conformal algebra of conformal field theories living on boundaries. As an application, we show how effective theories with spontaneously broken conformal symmetry can be derived from the SymTFT, and we elucidate how conformal anomalies can be reproduced in the presence of even-dimensional boundaries.
 

Mon, 13 Oct 2025
16:45
L5

Varieties over free associative algebras

Zlil Sela
Abstract
In the 1960s and 1970s ring theorists (P. M. Cohn, G.Bergman and others) tried to study the structure of sets of solutions to systems of (polynomial) equations (varieties) over free associative algebras. They found significant pathologies that demonstrated the difficulty to achieve their goal.
 
In an ongoing joint work with A. Atkarskaya we modify techniques that were used to study varieties over free groups and semigroups to study the structure of varieties over associative algebras. Along the way we find new structures also in free groups and semigroups. 
Mon, 13 Oct 2025

16:30 - 17:30
L4

Local L^\infty estimates for optimal transport problems

Prof Lukas Koch 
(School of Mathematical and Physical Sciences University of Sussex)
Abstract

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

Mon, 13 Oct 2025
16:00
C3

Eigenvalues of non-backtracking matrices

Cedric Pilatte
(Mathematical Insitute, Oxford)
Abstract
Understanding the eigenvalues of the adjacency matrix of a (possibly weighted) graph is a problem arising in various fields of mathematics. Since a direct computation of the spectrum is often too difficult, a common strategy is to instead study the trace of a high power of the matrix, which corresponds to a high moment of the eigenvalues. The utility of this method comes from its combinatorial interpretation: the trace counts the weighted, closed walks of a given length within the graph.
 
However, a common obstacle arises when these walk-counts are dominated by trivial "backtracking" walks—walks that travel along an edge and immediately return. Such paths can mask the more meaningful structural properties of the graph, yielding only trivial bounds.
 
This talk will introduce a powerful tool for resolving this issue: the non-backtracking matrix. We will explore the fundamental relationship between its spectrum and that of the original matrix. This technique has been successfully applied in computer science and random graph theory, and it is a key ingredient in upcoming work on the 2-point logarithmic Chowla conjecture.