Past

The 2d (massless) Sinh-Gordon model is amongst the simplest 2d quantum field theories that are expected to be integrable (= infinitely many symmetries), but without conformal symmetry. In this talk I will explain a rigorous construction of this model and its vertex correlations (= Laplace transforms) on the infinite cylinder using probability theory. A fundamental role is played by the Sinh-Gordon Hamiltonian and I will explain how the theory of Gaussian multiplicative chaos can be used to analyze this linear map. This talk will be based on joint work with Colin Guillarmou and Vincent Vargas.

We study the fixed points of the Berezin transform on the Fock-type spaces F^{2}_{m} with the weight e^{-|z|^{m}}, m > 0. It is known that the Berezin transform is well-defined on the polynomials in z and \bar{z}. In this talk from Ghazaleh Asghari from Reading University, we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many m \in (0,\infty). We also show that, in some particular cases, the fixed point polynomials are harmonic for all m.

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

Given a set $A$ and a scalar $\lambda$, how large must the sum of dilate $A+\lambda\cdot A=\{a+\lambda a'\mid a,a'\in A\}$ be in terms of $|A|$? In this talk, we will discuss two different settings of this problem, and how they relate to each other.

• For transcendental $\lambda\in \mathbb{C}$ and $A\subset \mathbb{C}$, how does $|A+\lambda\cdot A|$ grow with $|A|$?
• For a fixed large $\lambda\in \mathbb{Z}$ and even larger prime $p$, with $A\subset \mathbb{Z}/p\mathbb{Z}$, how does the density of $A+\lambda\cdot A$ depend on the density of $A$?

Joint with David Conlon.

Major depressive disorder (MDD) is a heterogeneous mental disorder. International guidelines present overall symptom severity as the key dimension for clinical characterisation. However, additional layers of heterogeneity may reside within severity levels related to how symptoms interact with one-another in a patient, called symptom dynamics. We investigate these individual differences by estimating the proportion of patients that display differences in their symptom dynamics while sharing the same diagnosis and overall symptom severity. We show that examining symptom dynamics provides information about the person-specific psychopathological expression of patients beyond severity levels by revealing how symptoms aggravate each other over time. These results suggest that symptom dynamics may serve as a promising new dimension for clinical characterisation. Areas of opportunity are outlined for the field of precision psychiatry in uncovering disorder evolution patterns (e.g., spontaneous recovery; critical worsening) and the identification of granular treatment effects by moving toward investigations that leverage symptom dynamics as their foundation. Future work aimed at investigating the cascading dynamics underlying depression onset and maintenance using the large-scale (N > 5.5 million) CIPA Study are outlined. 

In this talk, David Luo will use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $p$-adic $\text{GL}(2N, F)$ which we call \textit{middle supercuspidal representations}. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, David will show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$. Furthermore, David poses a conjecture which refines the local converse theorem for general supercuspidal representations of $\text{GL}(n, F)$.

The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.

 This is joint work with Renato Velozo Ruiz (Imperial College London).

For families of Calabi-Yau threefolds, we derive an explicit formula to count the number of points over $\mathbb{F}_{q}$ in terms of the periods of the holomorphic three-form, illustrated by the one-parameter mirror quintic and the 5-parameter Hulek-Verrill family. The formula holds for conifold singularities and naturally incorporates p-adic zeta values, the Yukawa coupling and modularity in the local zeta function. I will give a brief introduction on the physics motivation and how this framework links arithmetic, geometric and physics.

Random Fields with posses the Markov Property have played an important role in the development of Constructive Field Theory. They are related to their relativistic counterparts through Nelson Reconstruction. In this talk I will describe an attempt to understand the Markov Property of the $\Phi^4$ measure in 3 dimensions. We will also discuss the Properties of its Generator (i.e) the $\Phi^4_3$ Hamiltonian. This is based on Joint work with T. Gunaratnam.

A 2-connected, rational homotopy 7-sphere is classified up to diffeomorphism by three invariants: its (finite) 4th cohomology group, its q-invariant and its Eells-Kuiper invariant.  The q-invariant is a quadratic refinement of the linking form and determines the homeomorphism type, while the Eells-Kuiper invariant then pins down the diffeomorphism type.  In this talk, I will discuss the diffeomorphism classification of a certain family of non-negatively curved, 2-connected, rational homotopy 7-spheres, discovered by Sebastian Goette, Krishnan Shankar and myself, which contains, in particular, all $S^3$-bundles over $S^4$ and all exotic 7-spheres.

Transfer learning is a machine learning technique that leverages knowledge acquired in one domain to improve learning in another, related task. It is a foundational method underlying the success of large language models (LLMs) such as GPT and BERT, which were initially trained for specific tasks. In this talk, I will demonstrate how reinforcement learning (RL), particularly continuous time RL, can benefit from incorporating transfer learning techniques, especially with respect to convergence analysis. I will also show how this analysis naturally yields a simple corollary concerning the stability of score-based generative diffusion models.

Based on joint work with Zijiu Lyu of UC Berkeley.

How do you efficiently look for jobs?

How can you make the most of careers fairs?

What makes a CV or cover letter stand out?

Get practical advice and bring your questions!

In this talk I will give a gentle introduction to some aspects of the theory of hypergeometric functions as a natural language for addressing various integrals appearing in quantum field theory (QFT). In particular I will focus on the so-called intersection pairings as well as the differential equations satisfied by the integrals, and I will show how these aspects of the mathematical theory can find a natural interpretation in concrete QFT applications. I will mostly focus on Feynman integrals as paradigmatic example, where the language will shed new light on our most powerful method for computing Feynman integrals as well as their non-local symmetries. I will then give an outlook how these methods could allow us to also learn about integrals appearing in other places in field and string theory, such as Coulomb branch amplitudes, celestial holography and AdS (supergravity and string) amplitudes.

There will be no journal club this week to avoid conflicting with FPUK.

Over the last decade, adversarial attack algorithms have revealed instabilities in artificial intelligence (AI) tools. These algorithms raise issues regarding safety, reliability and interpretability; especially in high risk settings. Mathematics is at the heart of this landscape, with ideas from  numerical analysis, optimization, and high dimensional stochastic analysis playing key roles. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of successful attacks. I will present examples of attack algorithms for neural networks in image classification, for transformer models in optical character recognition and for large language models. I will also show how recent generative diffusion models can be used adversarially. From a more theoretical perspective, I will outline recent results on the overarching question of whether, under reasonable assumptions, it is inevitable that AI tools will be vulnerable to attack.

This talk will focus on various definitions of orientability for non-smooth spaces with Ricci curvature bounded from below. The stability of orientability and non-orientability will be discussed. As an application, we will prove the orientability of 4-manifolds with non-negative Ricci curvature and Euclidean volume growth. This work is based on a collaboration with E. Bruè and A. Pigati.

We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction.  The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition.  Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions.  We relate these dynamics to the solution on the real line.

This talk focuses on the logic side of the following result: the non-definability of free independence in the theory of tracial von Neumann algebras and C*-probability spaces. I will introduce continuous model theory, which is suitable for the study of metric structures. Definability in the continuous setting differs slightly from that in the discrete case. I will introduce its definition, give examples of definable sets, and prove an equivalent ultrapower condition of it. A. Berenstein and C. W. Henson exposited model theory for probability spaces in 2023, which was done with continuous model theory. It makes it natural for us to consider the definability of the notion of free independence in probability spaces. I will explain our result, which gives an example of a non-definable set.

This is work with William Boulanger and Emma Harvey, supervised by Jenny Pi and Jakub Curda.

Come take a break and get to know other Mathematrix members over some crafts! All supplies and sweet treats provided.

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus, using its representation as the value function of a stochastic control problem.

With the same techniques we also prove a bound for its oscillation, again independent of initial conditions, which is related to Harnack's inequality for the (rough) heat equation.

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

In this talk, Aaron Kettner, Institute of Mathematics, Czech Academy of Sciences, will show how to construct a C*-correspondence from a vector bundle together with a (partial) homeomorphism on the bundle's base space. The associated Cuntz-Pimsner algebras provide a class of examples that is both tractable and potentially quite large. Under reasonable assumptions, these algebras are classifiable in the sense of the Elliott program. If time permits, Aaron will sketch some K-theory calculations, which are work in progress.

This illustrated historical talk covers the period from around 1890, when graph theory was still mainly a collection of isolated results, to the 1990s, when it had become part of mainstream mathematics. Among many other topics, it includes material on graph and map colouring, factorisation, trees, graph structure, and graph algorithms. 

In this talk, I will introduce a new framework for working with moduli stacks in enumerative geometry, aimed at generalizing existing theories of enumerative invariants counting objects in linear categories, such as Donaldson–Thomas theory, to general, non-linear moduli stacks. This involves a combinatorial object called the component lattice, which is a globalization of the cocharacter lattice and the Weyl group of an algebraic group.

Several important results and constructions known in linear enumerative geometry can be extended to general stacks using this framework. For example, Donaldson–Thomas invariants can be defined for a general class of stacks, not only linear ones such as moduli stacks of sheaves. As another application, under certain assumptions, the cohomology of a stack, which is often infinite-dimensional, decomposes into finite-dimensional pieces carrying enumerative information, called BPS cohomology, generalizing a result of Davison–Meinhardt in the linear case.

This talk is based on joint works with Ben Davison, Daniel Halpern-Leistner, Andrés Ibáñez Núñez, Tasuki Kinjo, and Tudor Pădurariu.

Even in a totally discrete space $X$ you need to know how to move between distinct points. A path $P_{x,y}$ between two points $x,y \in X$ is a sequence of points in $X$ that starts with $x$ and ends with $y$. A path system is a collection of paths $P_{x,y}$, one per each pair of distinct points $x, y$ in $X$. We restrict ourselves to the undirected case where $P_{y,x}$ is $P_{x,y}$ in reverse.

Strictly metrical path systems are ubiquitous. They are defined as follows: There is some spanning, connected graph $(X, E)$ with positive edge weights $w(e)$ for all $e\in E$ and $P_{x,y}$ is the unique $w$-shortest $xy$ path. A metrical path system is defined likewise, but $w$-shortest paths need not be unique. Even more generally, a path system is called consistent  (no $w$ is involved here) if it satisfies the condition that when point $z$ is in $P_{x,y}$, then $P_{x,y}$ is $P_{x,z}$ concatenated with $P_{z,y}$. These three categories of path systems are quite different from each other and in our work we find quantitative ways to capture these differences.

Joint work with Daniel Cizma.

Recent foundational work by Ben-Bassat, Kelly, and Kremnitzer describes a model for derived analytic geometry as homotopical geometry relative to the infinity category of simplicial commutative complete bornological rings. In this talk, Rhiannon Savage will discuss a representability theorem for derived stacks in these contexts and will set out some new foundations for derived smooth geometry. Rhiannon will also briefly discuss the representability of the derived moduli stack of non-linear elliptic partial differential equations by an object we call a derived C∞-bornological affine scheme.

This tutorial will be a hands-on introduction to proving theorems in Lean, using its mathematical library Mathlib. It will not assume any previous knowledge about formal theorem provers. We will discover the Lean language, learn how to read a statement and a proof, and learn the essential "tactics" one can use to prove theorems in Lean.

Participants should come with a computer, and it would be best if they could install Lean before the tutorial by following the instructions at https://lean-lang.org/install/ . The installation should be easy and takes only a few minutes.

I will explain how the irreversibility of the renormalization group together with anomalies, including anomalies in the space of coupling constants, can be used to constrain the IR phases of defects in familiar quantum field theories. As an example, I will use these techniques to provide evidence for a conjectural "spin-flux duality" which describes how certain line operators are mapped across particle/vortex duality in 2+1d.

Einstein’s theory of general relativity reshaped our understanding of the universe. Instead of thinking of gravity as a force, Einstein showed it is the bending and warping of space and time caused by mass and energy. This radical idea not only explained how planets orbit stars, but also opened the door to astonishing predictions. In this seminar we will explore some of its most fascinating consequences from the expansion of the universe, to gravitational waves, and the existence of black holes.

The Ginzburg–Landau energy is often used to approximate the Dirichlet energy. As the perturbation parameter tends to zero, critical points of the Ginzburg–Landau energy converge, in an appropriate (bubbling) sense, to harmonic maps. In this talk I will first explain key analytical properties of this approximation procedure, then show that not every harmonic map can be approximated in this way. This is based on a rigidity theorem: under the energy threshold of 8pi, we classify all solutions of the associated nonlinear elliptic system from S2 to S2, thereby identifying exactly which harmonic maps can arise as Ginzburg–Landau limits in this regime.

A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction. No prior knowledge of stacks is needed for this talk!

TBC

I will present a collaborative project in which we formalized the construction of Brownian motion in Lean. Lean is an interactive theorem prover, with a large mathematical library called Mathlib. I will give an introduction to Lean and Mathlib, explain why one may want to formalize mathematics, and give a tour of the probability theory part of Mathlib. I will then describe the Brownian motion project, its organization, and some of the formalized results. For that project, we developed the theory of Gaussian measures and implemented a proof of Kolmogorov's extension theorem, as well as a modern version of the Kolmogorov-Chentsov continuity theorem based on Talagrand's chaining technique. Finally, I will discuss the next step of the project: formalizing stochastic integrals.

Symplectic implosion is an abelianisation construction in symplectic geometry. The implosion of the cotangent bundle of a group K plays a universal role in the implosion of manifolds with a K-action.  This universal implosion, which is usually a singular variety, can also be viewed as the non-reductive Geometric Invariant Theory quotient of the complexification G of K by its maximal unipotent subgroup. 

In this talk, we describe joint work with Johan Martens and Nick Proudfoot which uses point-counting techniques to calculate the intersection cohomology of the universal implosion.

Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in [Annales de l’Institut Henri Poincaré Probability and Statistics, 56(2):1465–1483, 2020]. We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.

I will review notions of categorical complexity, and the more recent work of Biran, Cornea and Zhang on fragmentation in triangulated persistence categories (TPCs), then go on to discuss applications of this to filtered topology. In particular, we will consider a suitable category of filtered topological spaces and detail some constructions and properties, before showing that an associated 'filtered stable homotopy category' is a TPC. I will then give some interesting results relating to this.

Prof Ian Griffiths (a mental health first aider in the department) will lead a discussion about how to protect your mental health when studying an intense graduate degree and outline the support and resources available within the Mathematical Institute.