Past

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. Despite intense interest in the higher-order connectivity of these networks, their Betti numbers at higher dimensions have been largely unexplored.

In this talk, after a brief survey on random topology, we study the clique complexes of preferential attachment graphs, and we prove the asymptotics of the expected Betti numbers. If time allows, we will briefly discuss their homotopy connectedness as well. This is joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He, and it is based on the preprint https://arxiv.org/abs/2305.11259

A major reason for the failure of cancer treatment and disease progression is the heterogeneous composition of tumor cells at the genetic, epigenetic, and phenotypic levels. Despite extensive efforts to characterize the makeup of individual cells, there is still much to be learned about the interactions between heterogeneous cancer cells and between cancer cells and the microenvironment in the context of cancer invasion. Clinical studies and in vivo models have shown that cancer invasion predominantly occurs through collective invasion packs, which invade more aggressively and result in worse outcomes. In vitro experiments on non-small cell lung cancer spheroids have demonstrated that the invasion packs consist of leaders and followers who engage in mutualistic social interactions during collective invasion. Many fundamental questions remain unanswered: What is the division of labor within the heterogeneous invasion pack? How does the leader phenotype emerge? Are the phenotypes plastic? What's the role of the individual "cheaters"? How does the invasion pack interact with the stroma? Can the social interaction network be exploited to devise novel treatment strategies? I will discuss recent modeling efforts to address these questions and hope to convince you that identifying and perturbing the "weak links" within the social interaction network can disrupt collective invasion and potentially prevent the malignant progression of cancer. 

The Celestial Holography program encompasses recent efforts to understand the flat space hologram in terms of a CFT living on the celestial sphere. Here we have fun relating various extrapolate dictionaries in CCFT and examining tools we can apply when perturbing around a 4D CFT in the bulk.

Cluster algebras and their quantum counterparts were invented in the early 2000s in an attempt to construct elements of dual canonical bases. This turned out to be a harder goal than first realised. In this talk I will aim to give an introduction and overview of the theory and display the wide range of interesting maths which has gone into making steps in this area. I will try to assume as little possible prior knowledge and instead focus on interesting questions which remain open in this area.

I will present a generalisation of the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, to a version where one obtains an algebraic group action. In the end the conclusion will be similar, but with weaker assumptions. This is recent work with Tingxiang Zou.

We consider an offline learning problem for an agent who first estimates an unknown price impact kernel from a static dataset, and then designs strategies to liquidate a risky asset while creating transient price impact. We propose a novel approach for a nonparametric estimation of the propagator from a dataset containing correlated price trajectories, trading signals and metaorders. We quantify the accuracy of the estimated propagator using a metric which depends explicitly on the dataset. We show that a trader who tries to minimise her execution costs by using a greedy strategy purely based on the estimated propagator will encounter suboptimality due to spurious correlation between the trading strategy and the estimator. By adopting an offline reinforcement learning approach, we introduce a pessimistic loss functional taking the uncertainty of the estimated propagator into account, with an optimiser which eliminates the spurious correlation, and derive an asymptotically optimal bound on the execution costs even without precise information on the true propagator. Numerical experiments are included to demonstrate the effectiveness of the proposed propagator estimator and the pessimistic trading strategy.

Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x²+y²=z²)? This is one of the simplest questions in arithmetic Ramsey theory which is still open. I will present a recent partial result, showing that in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x²+y²=z² for some integer z which may be in a different cell. 

The proof consists, after some initial maneuvers inspired by ergodic theory, in controlling the behavior of completely multiplicative functions along certain quadratic polynomials. Considering separately aperiodic and "pretentious" functions, the last major ingredient is a concentration estimate for functions in the latter class when evaluated along sums of two squares.

The talk is based on joint work with Frantzikinakis and Klurman.

Alan Weinstein, introduced the concept of "fat bundle" as a tool to understand when the total space of a fiber bundle with totally geodesic fibers allows a metric with positive sectional curvature. 

In recent times, certain weaker notions than the condition of having a metric with positive sectional curvature have been studied due to the apparent scarcity of spaces that meet this condition. Positive k^th-intermediate Ricci curvature (Ric_k > 0) on a Riemannian manifold Mⁿ is a condition that bridges the gap between positive sectional curvature and positive Ricci curvature. Indeed, when k = 1, this condition corresponds to positive sectional curvature, and when k = n−1, it corresponds to positive Ricci curvature. 

In this talk, I will discuss an ongoing project with Miguel Domínguez Vázquez, David González-Álvaro, and Jason DeVito, which aims to create new examples of compact Riemannian manifolds with Ric₂ > 0. We achieve this by employing a certain generalisation of the "fat bundle" concept.

Optimization problems subject to PDE constraints constitute a mathematical tool that can be applied to a wide range of scientific processes, including fluid flow control, medical imaging, option pricing, biological and chemical processes, and electromagnetic inverse problems, to name a few. These problems involve minimizing some function arising from a particular physical objective, while at the same time obeying a system of PDEs which describe the process. It is necessary to obtain accurate solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems.

In this talk we consider iterative methods, in particular Krylov subspace methods, to solve such systems, accelerated by fast and robust preconditioning strategies. In particular, we will survey several new developments, including block preconditioners for fluid flow control problems, a circulant preconditioning framework for solving certain optimization problems constrained by fractional differential equations, and multiple saddle-point preconditioners for block tridiagonal linear systems. We will illustrate the benefit of using these new approaches through a range of numerical experiments.

This talk is based on work with Santolo Leveque (Scuola Normale Superiore, Pisa), Spyros Pougkakiotis (Yale University), Jacek Gondzio (University of Edinburgh), and Andreas Potschka (TU Clausthal).

A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. In this talk, we will present a reduced-order dynamical system derived from the lubrication model based on the nearest-neighbour droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights to the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics. This is joint work with Thomas Witelski.

The key role of physical and mechanical interactions in cancer emerges from a very large variety of data sources and methods - from genomics to bioimaging, from proteomics to clinical records. Thus, learning physics-driven relational information is crucial to characterize its progression at different scales.

In this talk I will discuss how mathematical and computational tools allow for learning  and better understanding of  the mechano-biology of cancer,  thanks to the integration of  patient-specific data and physics-based models. I will present a few applications developed in the last decade in which the development of  digital twins,  empowered by ad-hoc learning tools,  allows us to test new hypotheses,  to assess the model predictions against biological and clinical data, and to aid decision-making in a clinical setting.

Funding from MUR - PRIN 2020, Progetto di Eccellenza 2023-2027 and Regione Lombardia (NEWMED Grant, ID: 117599, POR FESR 2014-2020) is gratefully acknowledged.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.

The study of topologically ordered quantum phases has led to interesting connections with, for example, the study of subfactors. In this talk, I will introduce a new axiomatisation of such quantum models defined on d-dimensional square lattices in terms of nets of projections. These local topological order axioms are satisfied by known 2D models such as the toric code and Levin-Wen models built on a unitary fusion category. We show that these axioms lead to a definition of boundary algebras naturally living on a hyperplane. This boundary algebra encodes information about the excitations in the bulk theory, leading to a bulk-boundary correspondence. I will outline the main points, with an emphasis on interesting connections to operator algebras and fusion categories. Based on joint work with C. Jones, Penneys, and Wallick (arXiv:2307.12552).

It was proved by Selberg's in the 1940's that the typical values of the logarithm of the Riemann zeta function on the critical line is distributed like a complex Gaussian random variable. In this talk, I will present recent work with Emma Bailey that extends the Gaussian behavior for the real part to the large deviation regime. This gives a new proof of unconditional upper bounds of the $2k$-moments of zeta for $0\leq k\leq 2$, and lower bounds for $k>0$. I will also discuss the connections with random matrix theory and with the Moments Conjecture of Keating & Snaith. 

Given a group $G$, finding a geometric action of $G$ on a CAT(0) cube complex can be used to say some rather strong things about $G$. Such actions are not always easy to find, however, which makes it useful to have sufficient conditions, both for existence and for non-existence. This talk concerns the latter: we shall see a coarse geometric obstruction to a group admitting a cocompact cubulation. Based on joint work with Zach Munro.

In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.

I will talk about supersymmetric quantum field theories with 8 supercharges in dimensions 3-6. After a brief introduction I will mostly speak about the moduli space of vacua of such theories, and in particular their Higgs branches, which are so called symplectic singularities (or mild generalisations thereof). Powerful theorems from mathematics say that a singular Higgs branch is stratified into a disjoint union of smooth open subsets, so called symplectic leaves. This stratification matches exactly the pattern of partial Higgsings of the theory in question. After introducing the stratification and explaining its physical interpretation, I will show how brane systems and so called magnetic quivers can be used to compute it.

Random walks on random graphs are associated with diffusion in disordered media. In this talk, the graphs of interest are uniform spanning tree (UST) and loop-erased random walk (LERW). First I will demonstrate some asymptotic behavior of the simple random walk on the three-dimensional UST. Next I will discuss annealed transition probability of the simple random walk on high-dimensional LERWs.

 

This talk presents some recent progress for models coupling large-strain, geometrically nonlinear elasto-plasticity with the movement of dislocations. In particular, a new geometric language is introduced that yields a natural mathematical framework for dislocation evolutions. In this approach, the fundamental notion is that of 2-dimensional "slip trajectories" in space-time (realized as integral 2-currents), and the dislocations at a given time are recovered via slicing. This modelling approach allows one to prove the existence of solutions to an evolutionary system describing a crystal undergoing large-strain elasto-plastic deformations, where the plastic part of the deformation is driven directly by the movement of dislocations. This is joint work with T. Hudson (Warwick).

I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.

We prove convergence results of the simulation of the density solution to the McKean-Vlasov equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.

Harmonic maps from surfaces to other manifolds is a fundamental object of geometric analysis with many applications, for example to minimal surfaces. In particular, there are many available methods of constructing them such, such as using complex geometry, min-max methods or flow techniques. By contrast, much less is known for harmonic maps from higher dimensional manifolds. In the present talk I will explain the role of dimension in this problem and outline the recent joint work with D. Stern, where we provide a min-max construction for higher-dimensional harmonic maps. If time permits, an application to eigenvalue optimisation problems will be discussed. Based on joint work with D. Stern.

Job applications involve a lot of work and can be overwhelming. Join us for a workshop and Q+A session focused on breaking down academic applications: we’ll talk about approaching reference letter writers, writing research statements, and discussing what makes a great CV and covering letter.

In this talk, I will present joint work with Omer Bobrowski:  a series of statements regarding the behaviour of persistence diagrams arising from random point-clouds. I will present evidence that, viewed in the right way, persistence values obey a universal probability law, that depends on neither the underlying space nor the original distribution of the point-cloud.  I will present two versions of this universality: “weak” and “strong” along with progress which has been made in proving the statements.  Finally, I will also discuss some applications of this phenomena based on detecting structure in data.

Mathematical models are useful tools for guiding infectious disease outbreak control measures. Before a pathogen has even entered a host population, models can be used to determine the locations that are most at risk of outbreaks, allowing limited surveillance resources to be deployed effectively. Early in an outbreak, key questions for policy advisors include whether initial cases will lead on to a major epidemic or fade out as a minor outbreak. When a major epidemic is ongoing, models can be applied to track pathogen transmissibility and inform interventions. And towards the end of (or after) an outbreak, models can be used to estimate the probability that the outbreak is over and that no cases will be detected in future, with implications for when interventions can be lifted safely. In this talk, I will summarise the work done by my research group on modelling different stages of infectious disease outbreaks. This includes: i) Before an outbreak: Projections of the locations at-risk from vector-borne pathogens towards the end of the 21^st century under a changing climate; ii) Early in an outbreak: Methods for estimating the risk that introduced cases will lead to a major epidemic; and iii) During a major epidemic: A novel approach for inferring the time-dependent reproduction number during outbreaks when disease incidence time series are aggregated temporally (e.g. weekly case numbers are reported rather than daily case numbers). In addition to discussing this work, I will suggest areas for further research that will allow effective interventions to be planned during future infectious disease outbreaks.

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

A key ingredient in the proof of the model completeness of the real exponential field was a valuation inequality for polynomially bounded o-minimal structures. I shall briefly describe the argument, and then move on to the complex exponential field and Zilber's quasiminimality conjecture for this structure. Here, one can reduce the problem to that of establishing an analytic continuation property for (complex) germs definable in a certain o-minimal expansion of the real field and in order to study this question I propose notions of "complex Hardy fields" and "complex valuations".   Here, the value group is not necessarily ordered but, nevertheless, one can still prove a valuation inequality.

I will survey Cartan respectively diagonal subalgebras of nuclear C*-algebras. This setup corresponds to a presentation of the ambient C*-algebra as an amenable groupoid C*-algebra, which in turn means that there is an underlying structure akin to an amenable topological dynamical system.

The existence of such subalgebras is tightly connected to the UCT problem; the classification of Cartan pairs is largely uncharted territory. I will present new constructions of diagonals of the Jiang-Su algebra Z and of the Cuntz algebra O_2, and will then focus on distinguishing Cantor Cartan subalgebras of O_2.

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial, analytical and number theoretical.

Central clearing counterparty houses (CCPs) play a fundamental role in mitigating the counterparty risk for exchange traded options. CCPs cover for possible losses during the liquidation of a defaulting member's portfolio by collecting initial margins from their members. In this article we analyze the current state of the art in the industry for computing initial margins for options, whose core component is generally based on a VaR or Expected Shortfall risk measure. We derive an approximation formula for the VaR at short horizons in a model-free setting. This innovating formula has promising features and behaves in a much more satisfactory way than the classical Filtered Historical Simulation-based VaR in our numerical experiments. In addition, we consider the neural-SDE model for normalized call prices proposed by [Cohen et al., arXiv:2202.07148, 2022] and obtain a quasi-explicit formula for the VaR and a closed formula for the short term VaR in this model, due to its conditional affine structure.

The need to solve linear systems of equations is ubiquitous in scientific computing. Powerful methods for preconditioning such systems have been developed in cases where we can exploit knowledge of the origin of the linear system; a recent example from the solution of systems from PDEs is the Gen-EO domain decomposition method which works well, but requires a non-trival amount of knowledge of the underlying problem to implement.  

In this talk I will present a new spectral coarse space that can be constructed in a fully-algebraic way, in contrast to most existing spectral coarse spaces, and will give a theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient, especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

This is joint work with Hussam Al Daas, Pierre Jolivet and Jennifer Scott.

We formulate and solve a generalized inverse Navier–Stokes boundary value problem for velocity field reconstruction and simultaneous boundary segmentation of noisy Flow-MRI velocity images. We use a Bayesian framework that combines CFD, Gaussian processes, adjoint methods, and shape optimization in a unified and rigorous manner.
With this framework, we find the velocity field and flow boundaries (i.e. the digital twin) that are most likely to have produced a given noisy image. We also calculate the posterior covariances of the unknown parameters and thereby deduce the uncertainty in the reconstructed flow. First, we verify this method on synthetic noisy images of flows. Then we apply it to experimental phase contrast magnetic resonance (PC-MRI) images of an axisymmetric flow at low and high SNRs. We show that this method successfully reconstructs and segments the low SNR images, producing noiseless velocity fields that match the high SNR images, using 30 times less data.
This framework also provides additional flow information, such as the pressure field and wall shear stress, accurately and with known error bounds. We demonstrate this further on a 3-D in-vitro flow through a 3D-printed aorta and 3-D in-vivo flow through a carotid artery.

An old and challenging conjecture proposed by R.H. Fox in 1962 states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot are trapezoidal i.e. strictly increase, possibly plateau, then strictly decrease. We give a survey of the known results and use them to motivate the study of branched double covers. The second part of the talk focuses on the properties of the branched double covers of alternating knots.

Abstract: Shifted moments of the Riemann zeta function, introduced by Chandee, are natural generalizations of the moments of zeta. While the moments of zeta capture large values of zeta, the shifted moments also capture how the values of zeta are correlated along the half line. I will describe recent work giving sharp bounds for shifted moments assuming the Riemann hypothesis, improving previous work of Chandee and Ng, Shen, and Wong. I will also discuss some unconditional results about shifted moments with small exponents.

Let $SO(3,R)$ be the $3D$-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Confirming a conjecture by Breuillard and Green, we show that if $A$ is an open subset of $SO(3,R)$ with sufficiently small measure, then $\mu(A^2) > 3.99 \mu(A)$. This is joint work with Chieu-Minh Tran (NUS) and Ruixiang Zhang (Berkeley). 

The Ciarlet definition of a finite element has been used for many years to describe the requisite parts of a finite element. In that time, finite element theory and implementation have both developed and improved, which has left scope for a redefinition of the concept of a finite element. In this redefinition, we look to encapsulate some of the assumptions that have historically been required to complete Ciarlet’s definition, as well as incorporate more information, in particular relating to the symmetries of finite elements, using concepts from Group Theory. This talk will present the machinery of the proposed new definition, discuss its features and provide some examples of commonly used elements.

We show that every 2-coloring of the natural numbers and any finite coloring of the rationals contains monochromatic sets of the form $\{x, y, xy, x+y\}$. We also discuss generalizations and obstructions to extending this result to arbitrary finite coloring of the naturals. This is partially based on joint work with Marcin Sabok.

The two-colored Temperley-Lieb algebra is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. In this talk, I will give conditions for the existence and rotatability of the two-colored Jones-Wenzl projector in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe’s category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.

 

The PDE formulation of Mean Field Games (MFG) is described by nonlinear systems in which a Hamilton—Jacobi—Bellman (HJB) equation and a Kolmogorov—Fokker—Planck (KFP) equation are coupled. The advective term of the KFP equation involves a partial derivative of the Hamiltonian that is often assumed to be continuous. However, in many cases of practical interest, the underlying optimal control problem of the MFG may give rise to bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we present results on the analysis and numerical approximation of second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians.
In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the partial derivative of the Hamiltonian in terms of subdifferentials of convex functions.

We present theorems that guarantee the existence of unique weak solutions to MFG PDIs under a monotonicity condition similar to one that has been considered previously by Lasry & Lions. Moreover, we introduce a monotone finite element discretization of the weak formulation of MFG PDIs and prove the strong convergence of the approximations to the value function in the H¹-norm and the strong convergence of the approximations to the density function in L^q-norms. We conclude the talk with some numerical experiments involving non-smooth solutions. 

This is joint work with my supervisor Iain Smears. 

We review the construction of non-invertible duality defects in various dimensions. We explain how they can be preserved along RG flows and how their realization on gapped phases contains their 't Hooft anomalies. We finally give a presentation of the anomalies from the Symmetry TFT. Time permitting I will discuss some possible future applications.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Akshay Hegde, supervisor Dmitry Beylaev

Julius Villar, supervisor Dmitry Beylaev

Csaba Toth, supervisor Harald Oberhauser 

We study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation, in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We show that these systems converge to the same graph-limit equation, in which the interaction is prescribed by a constant-in-time graphon.