This week

Models of random matrices can be viewed as zero-dimensional analogs of usual field theory. Despite decades of exploration, matrix models remain at the forefront of intensive research, motivated by a rich web of connections to string theory, quantum gravity, integrability, Yang-Mills theory, combinatorics, geometry and representation theory. These lectures will present a pedagogical introduction to the subject.

​Lecture 1. Motivation and basic definitions. Hermitian matrix models: Feynman rules, ribbon graphs, large N genus expansion.
Lecture 2. Reduction to eigenvalues. Large N limit, Coulomb gas approach, saddle point equations.
Lecture 3. Continuum limit of saddle point equations. Eigenvalue density and spectral curve. Examples.
Lecture 4. Orthogonal polynomials. Relation to 2d gravity and phase transitions (sketch). Outlook: loop equations, topological recursion, integrability.

I present results on the quantum dynamics of a minimal model with spatial structure and local interactions. The model features a time-independent Hamiltonian, in contrast to the widely studied quantum circuits, and is analytically tractable in the limit of large local Hilbert space dimension and weak intersite coupling. In this regime, I show that the energy dynamics are governed by a classical master equation exhibiting diffusive behavior. Furthermore, I demonstrate that the spectral form factor can be expressed exactly in terms of the solution to this master equation, demonstrating how the linear ramp emerges at long times, while locality gives rise to an additional enhancement at short times.

In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of noncommutative probability theory - examples will include the stochastic quantization of fermionic quantum field theories and also the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.

In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of noncommutative probability theory - examples will include the stochastic quantization of fermionic quantum field theories and also the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.

In this talk, I will first introduce the adapted Wasserstein (AW) distance — an extension of the classical Wasserstein distance to stochastic processes. It captures the filtration generated by the underlying processes and plays a fundamental role in the study of stochastic analysis and optimal control problems. We then turn to applications in distributionally robust optimization (DRO) problems in a dynamic context. This framework addresses decision-making under model uncertainty by optimizing against the worst-case scenario, where the potential model lies in an adapted Wasserstein ball around a given reference model. I will discuss tractable reformulations of the worst-case performance via duality and sensitivity approaches. Both discrete- and continuous-time results will be presented.

In this talk, I will first introduce the adapted Wasserstein (AW) distance — an extension of the classical Wasserstein distance to stochastic processes. It captures the filtration generated by the underlying processes and plays a fundamental role in the study of stochastic analysis and optimal control problems. We then turn to applications in distributionally robust optimization (DRO) problems in a dynamic context. This framework addresses decision-making under model uncertainty by optimizing against the worst-case scenario, where the potential model lies in an adapted Wasserstein ball around a given reference model. I will discuss tractable reformulations of the worst-case performance via duality and sensitivity approaches. Both discrete- and continuous-time results will be presented.

AI tools such as ChatGPT and Gemini are permeating almost all aspects of our life including how we research and teach mathematics. In this talk I will describe how the use of AI is transforming how we teach mathematics and some ways in which we can incorporate these tools into our courses. Along the way I will highlight things which AI is good and bad at and how we can make the most of this.

Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic manifolds and discuss how "almost cycles" and "almost boundaries" relate to the "size" of homology.

Swimming on the microscale has long been the subject of intense research efforts, from experimental studies of bacteria, sperm, and algae through to varied theoretical questions of low-Reynolds-number fluid mechanics. The biological and biophysical settings that drive this ongoing research are often confoundingly complex, a fact that has driven the development and use of simple models of microswimmers. In this talk, we will showcase how we can often exploit separated scales present in these problems and models to reveal surprisingly simple emergent dynamics, including predictions of globally attracting, long-term behaviours. In doing so, we'll also uncover a surprising cautionary tale that calls into question much of the intuition gained from commonplace models of microswimming. In particular, we'll see that a wave-of-the-hands, which I have been guilty of before, can drastically and qualitatively change the dynamics that simple models predict, and we'll see how such missteps can be addressed through systematic multiscale methods.

Despite recent progress, a complete formulation of a holographic correspondence for flat spacetimes remains elusive.
Any viable formulation of flat space holography should be based on a correspondence between bulk and boundary states built upon the equivalence of unitary irreducible representations (UIRs) of the asymptotic symmetry group of flat spacetimes, the BMS group. In this talk, I will present explicit wavefunctions for the UIRs of the BMS group (the so-called BMS particles). These are functions on supermomentum space that generalize the familiar notion of Poincaré particles by incorporating additional soft degrees of freedom. I will discuss their connections to infrared physics and outline prospects for defining an S-matrix for BMS states that is free from infrared divergences. This talk is based on joint work with X. Bekaert and Y. Herfray.

There is a broad literature for the statistical analysis of a network as a snapshot, while the formulation of statistical frameworks for modelling populations of network data is still a developing area. Network populations are data sets where now each observation in the data comprises a network rather than a scalar quantity. In this talk I will present two modelling frameworks for making inferences for network populations. The first study provides a model-based approach for clustering network observations in a population using the Bayesian machinery. The second study provides a framework that allows inferences that explicitly capture information on cycles in a network population. Both methods will be illustrated using two real data applications in neuroscience and ecology, respectively.

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