Past

We will be joined by Tim LaRock, president of the UCU, to talk about everything money and work related, and how these issues intersect with being a minority. Lunch will be provided. 

Speaker: Cedric Pilatte 
Title: Convolution of integer sets: a galaxy of (mostly) open problems

Title: Manifold-Free Riemannian Optimization

Abstract: Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

Unraveling the intricacies of intracellular signalling through predictive mathematical models holds great promise for advancing precision medicine and enhancing our foundational comprehension of biology. However, navigating the labyrinth of biological mechanisms governing signalling demands a delicate balance between a faithful description of the underlying biology and the practical utility of parsimonious models.
In this talk, I will present methods that enable training of large ordinary differential equation models of intracellular signalling and showcase application of such models to predict sensitivity to anti-cancer drugs. Through illustrative examples, I will demonstrate the application of these models in predicting sensitivity to anti-cancer drugs. A critical reflection on the construction of such models will be offered, exploring the perpetual question of complexity and how intricate these models should be.
Moreover, the talk will explore novel approaches that meld machine learning techniques with mathematical modelling. These approaches aim to harness the benefits of simplistic and unbiased phenomenological models while retaining the interpretability and biological fidelity inherent in mechanistic models.
 

This will be an informal discussion seminar based on https://arxiv.org/abs/2211.07592:

The constrained Hamiltonian analysis of geometric actions is worked out before applying the construction to the extended Bondi-Metzner-Sachs group in four dimensions. For any Hamiltonian associated with an extended BMS4 generator, this action provides a field theory in two plus one spacetime dimensions whose Poisson bracket algebra of Noether charges realizes the extended BMS4 Lie algebra. The Poisson structure of the model includes the classical version of the operator product expansions that have appeared in the context of celestial holography. Furthermore, the model reproduces the evolution equations of non-radiative asymptotically flat spacetimes at null infinity.

Let K be a non-archimedean field of mixed characteristic (0,p), and let L be a finite extension of
the p-adic numbers contained in K. The speaker is interested in the continuous representations of a
given L-analytic group G in locally convex (usually infinite dimensional) topological vector spaces over K.
This is, up to technicalities, equivalent to studying certain topological modules over the locally
analytic distribution algebra D(G,K) of G. But doing algebra with topological objects is hard!
In this talk, we present an excellent remedy, found by Schneider and Teitelbaum in the early 2000s.

The Hardy–Littlewood circle method is a well-known analytic technique that has successfully solved several difficult counting problems in number theory. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. I will explain how one can implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, leading thus to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.

We build universal approximators of continuous maps between arbitrary Polish metric spaces X and Y using universal approximators between Euclidean spaces as building blocks. Earlier results assume that the output space Y is a topological vector space. We overcome this limitation by "randomization": our approximators output discrete probability measures over Y. When X and Y are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for Hölder-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of X and Y. For barycentric Y, including Banach spaces, R-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to Y-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning. 

As an application, we show that the solution operator to an RDE can be approximated within our framework.

Based on the following articles: 

•         An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning (2023) - Chong Liu, Matti Lassas, Maarten V. de Hoop, and Ivan Dokmanić (ArXiV 2304.12231)

•         Designing universal causal deep learning models: The geometric (Hyper)transformer (2023) B. Acciaio, A. Kratsios, and G. Pammer, Math. Fin. https://onlinelibrary.wiley.com/doi/full/10.1111/mafi.12389

•         Universal Approximation Under Constraints is Possible with Transformers (2022) - ICLR Spotlight - A. Kratsios, B. Zamanlooy, T. Liu, and I. Dokmanić.

Causal optimal transport and the related adapted Wasserstein distance have recently been popularized as a more appropriate alternative to the classical Wasserstein distance in the context of stochastic analysis and mathematical finance. In this talk, we establish some interesting consequences of causality for transports on the space of continuous functions between the laws of stochastic differential equations.
 

We first characterize bicausal transport plans and maps between the laws of stochastic differential equations. As an application, we are able to provide necessary and sufficient conditions for bicausal transport plans to be induced by bi-causal maps. Analogous to the classical case, we show that bicausal Monge transports are dense in the set of bicausal couplings between laws of SDEs with unique strong solutions and regular coefficients.

 This is a joint work with Rama Cont.

This talk explores recent advancements in stress and flux-based finite element methods. It focuses on addressing the limitations of traditional finite elements, in order to describe complex material behavior and engineer new metamaterials.

Stress and flux-based finite element methods are particularly useful in error estimation, laying the groundwork for adaptive refinement strategies. This concept builds upon the hypercircle theorem [1], which states that in a specific energy space, both the exact solution and any admissible stress field lie on a hypercircle. However, the construction of finite element spaces that satisfy admissible states for complex material behavior is not straightforward. It often requires a relaxation of specific properties, especially when dealing with non-symmetric stress tensors [2] or hyperelastic materials.

Alternatively, methods that directly approximate stresses can be employed, offering high accuracy of the stress fields and adherence to physical conservation laws. However, when approximating eigenvalues, this significant benefit for the solution's accuracy implies that the solution operator cannot be compact. To address this, the solution operator must be confined to a subset of the solution that excludes the stresses. Yet, due to compatibility conditions, the trial space for the other solution components typically does not yield the desired accuracy. The second part of this talk will therefore explore the Least-Squares method as a remedy to these challenges [3].

To conclude this talk, we will emphasize the integration of those methods within global solution strategies, with a particular focus on the challenges regarding model order reduction methods [4].

[1] W. Prager, J. Synge. Approximations in elasticity based on the concept of function space.

Quarterly of Applied Mathematics 5(3), 1947.

[2] FB, K. Bernhard, M. Moldenhauer, G. Starke. Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity, Numerical Methods for Partial Differential Equations 37(4), 2021.

[3] FB, D. Boffi. First order least-squares formulations for eigenvalue problems, IMA Journal of Numerical Analysis 42(2), 2023.

[4] FB, D. Boffi, A. Halim. A reduced order model for the finite element approximation of eigenvalue problems,Computer Methods in Applied Mechanics and Engineering 404, 2023.

We study “bottom-up” models for the collective motion of large groups of animals. Similar models can be encoded into (micro)robotic matter, capable of sensing light and processing information. Agents are endowed only with visual sensing and information processing. We study a model in which moving agents reorientate to maximise the path-entropy of their visual environment over paths into the future. There are general arguments that principles like this that are based on retaining freedom in the future may confer fitness in an uncertain world. Alternative “top-down” models are more common in the literature. These typically encode coalignment and/or cohesion directly and are often motivated by models drawn from physics, e.g. describing spin systems. However, such models can usually give little insight into how co-alignment and cohesion emerge because these properties are encoded in the model at the outset, in a top-down manner. We discuss how our model leads to dynamics with striking similarities with animal systems, including the emergence of coalignment, cohesion, a characteristic density scaling anddifferent behavioural phenotypes. The dynamics also supports a very unusual order-disorder transition in which the order (coalignment) initially increases upon the addition of sensory or behavioural noise, before decreasing as the noise becomes larger.

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

I will present novel freeness results in ultraproducts of tracial von Neumann algebras. As a particular case, I will show that if a and b are the generators of the free group F_2, then the relative commutants of a and b in the ultraproduct of the free group factor are free with respect to the ultraproduct trace. The proof is based on a surprising application of Lp-boundedness results of Fourier multipliers in free group factors for p > 2. I will describe applications of these results to absorption and model theory of II_1 factors. This is joint work with Adrian Ioana.

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected: for instance, in different types of combinatorial statistics on perfect matchings that encode moments of noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.

Random temporal graphs are a version of the classical Erdős-Rényi random graph $G(n,p)$ where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We are interested in the asymptotics for the distances in such graphs, mostly in the regime of interest where the average degree $np$ is of order $\log n$ ('near' the phase transition).

More specifically, we will discuss the asymptotic lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, as well as the maxima between any two vertices; this also covers the (temporal) diameter. In the regime $np \gg \log n$, longest increasing paths were studied by Angel, Ferber, Sudakov and Tassion.

The talk contains joint work with Nicolas Broutin and Gábor Lugosi.

The classification of measure preserving actions up to orbit equivalence has attracted a lot of interest in the last 25 years. The goal of this talk is to survey the major discoveries in the field, including Popa's cocycle and orbit equivalence superrigidity theorem and discuss some recent superrigidity results for dense subgroups of Lie groups acting by translation.

Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

In this talk we will review the problem of sharpness in percolation, tracing its origins back to the seminal works of Menshikov, Grimmett-Marstrand and Aizenman-Barsky, which successfully settled the question in the context of Bernoulli independent percolation. Then we will present some recent advancements on the field, which have opened up the possibility of investigating dependent percolation models. Special emphasis will be given to the Interpolation technique, which has proved itself very effective. In particular, it has been used to establish the sharpness for Interlacements Percolation, a model introduced by Sznitman with notoriously intricate dependencies.

This talk is based on a joint work with Duminil-Copin, Goswami, Rodriguez and Severo

Let G be a reductive group defined over a local field of characteristic 0 (real or p-adic). By Harish-Chandra’s regularity theorem, the character Θ_π of an irreducible representation π of G is given by a locally integrable function f_π on G. It turns out that f_π has even better integrability properties, namely, it is locally L^{1+r}-integrable for some r>0. This gives rise to a new singularity invariant of representations \e_π by considering the largest such r.

We explore \e_π, show it is bounded below only in terms of the group G, and calculate it in the case of a p-adic GL(n). To do so, we relate \e_π to the integrability of Fourier transforms of nilpotent orbital integrals appearing in the local character expansion of Θ_π. As a main technical tool, we use explicit resolutions of singularities of certain hyperplane arrangements. We obtain bounds on the multiplicities of K-types in irreducible representations of G for a p-adic G and a compact open subgroup K.

Based on a joint work with Itay Glazer and Julia Gordon.

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows high-quality nonparametric learning via methods such as Gaussian process regression. Nevertheless, its learning phase requires unrealistic massive computations for large datasets. In this talk, we present a quadrature-based approach for scaling up Gaussian process regression via a low-rank approximation of the kernel matrix. The low-rank structure is utilized to achieve effective hyperparameter learning, training, and prediction. Our Gauss-Legendre features method is inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high-quality kernel approximation using a number of features that is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when using random Fourier features. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.

The Bethe-Gauge Correspondence (BGC) of Nekrasov and Shatashvili, linking quantum integrable spin chains to two-dimensional supersymmetric gauge theories with N=2 supersymmetry, stands out as a significant instance of the deep connection between supersymmetric gauge theories and integrable models. In this talk, I will delve into this correspondence and its origins for superspin chains. To achieve this, I will first elucidate the Bethe Side and its corresponding Gauge Side of the BGC. Subsequently, it becomes evident that the BGC can be naturally realized within String Theory. I will initially outline the brane configuration for the realization of the Gauge Side. Through the use of string dualities, this brane configuration will be mapped to another, embodying the Bethe Side of the correspondence. The 4D Chern-Simons theory plays a crucial role in this latter duality frame, elucidating the integrability of the Bethe Side. Lastly, I will elaborate on computing the main object of interest for integrable superspin chains—the R-matrix—from the BGC. While this provides a rather comprehensive picture of the correspondence, some important questions remain for further clarification. I will summarize some of the most interesting ones at the end of the talk.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method can yield short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

In a recent preprint, together with Bailleul and Chevyrev we introduced a class of random fields which try to model the basic properties of quantum fields. I will try to explain the basic ideas and some of the many open problems.

To read the preprint, please click here.

Busenberg and Travis suggested in 1983 a population system that exhibits complete segregation of the species. This system can be rigorously derived from interacting particle systems in a mean-field-type limit. It consists of parabolic cross-diffusion equations with an indefinite diffusion matrix. It is known that this system can be formulated in terms of so-called entropy variables such that the transformed equations possess a positive semidefinite diffusion matrix. We consider in this talk the case of incomplete diffusion, which means that the diffusion matrix has zero eigenvalues, and the problem is not parabolic in the sense of Petrovskii. 

We show that the cross-diffusion equations can be written as a normal form of symmetric hyperbolic-parabolic type beyond the Kawashima-Shizuta theory. Using results for symmetric hyperbolic systems, we prove the existence of a unique local classical solution. As solutions may become discontinuous in finite time, only global solutions with very low regularity can be expected. We prove the existence of global dissipative measure-valued solutions satisfying a weak-strong uniqueness property. The proof is based on entropy methods and a finite-volume approximation with a mesh-dependent artificial diffusion. 

Many congruences between modular forms (or at least their q-expansions) can be explained by the theory of $p$-adic families of modular forms. In this talk, I will discuss properties of eigenvarieties, a geometric interpretation of the idea of $p$-adic families. In particular, focusing initially on the well-understood case of (elliptic) modular forms, before delving into the considerably murkier world of Bianchi modular forms. In this second case, this work gives numerical verification of a couple of conjectures, including BSD by work of Loeffler and Zerbes.

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting  the temporal dynamics of the time series distribution. We  estimate the drift function from data samples by nonparametric, e.g. kernel regression methods,  and the simulation of the SB diffusion  yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments.  First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion,  and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. 

Let $(M, g)$ be a Riemannian manifold equipped with a calibration $k$-form $\alpha$. In earlier work with Cheng and Madnick (AJM 2021), we studied the analytic properties of a special class of $k$-harmonic maps into $M$ satisfying a first order nonlinear PDE, whose images (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith immersions, as they were originally introduced in an unpublished preprint of Aaron Smith. They have nice properties related to conformal geometry, and are higher-dimensional analogues of the $J$-holomorphic map equation. In new joint work (arXiv:2311.14074) with my PhD student Anton Iliashenko, we have obtained analogous results for maps out of $M$. Slightly more precisely, we define a special class of $k$-harmonic maps out of $M$, satisfying a first order nonlinear PDE, whose fibres (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith submersions. I will give an introduction to both of these sets of equations, and discuss many future questions.

Mathematical methods are developed to characterize the asymptotics of recurrent neural networks (RNN) as the number of hidden units, data samples in the sequence, hidden state updates, and training steps simultaneously grow to infinity. In the case of an RNN with a simplified weight matrix, we prove the convergence of the RNN to the solution of an infinite-dimensional ODE coupled with the fixed point of a random algebraic equation. 
The analysis requires addressing several challenges which are unique to RNNs. In typical mean-field applications (e.g., feedforward neural networks), discrete updates are of magnitude O(1/N ) and the number of updates is O(N). Therefore, the system can be represented as an Euler approximation of an appropriate ODE/PDE, which it will converge to as N → ∞. However, the RNN hidden layer updates are O(1). Therefore, RNNs cannot be represented as a discretization of an ODE/PDE and standard mean-field techniques cannot be applied. Instead, we develop a fixed point analysis for the evolution of the RNN memory state, with convergence estimates in terms of the number of update steps and the number of hidden units. The RNN hidden layer is studied as a function in a Sobolev space, whose evolution is governed by the data sequence (a Markov chain), the parameter updates, and its dependence on the RNN hidden layer at the previous time step. Due to the strong correlation between updates, a Poisson equation must be used to bound the fluctuations of the RNN around its limit equation. These mathematical methods allow us to prove a neural tangent kernel (NTK) limit for RNNs trained on data sequences as the number of data samples and size of the neural network grow to infinity.

Say we want to view the function-Rips multifiltration as an estimator. Then, what is the target? And what kind of consistency, bias, or convergence rate, should we expect? In this talk I will present on-going joint work with Ethan André (Ecole Normale Supérieure) that aims at laying the algebro-topological ground to start answering these questions.

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for cells migrating along one-dimensional lines, exhibiting diverse migration modes. When arriving at a symmetric Y-junction and extending protrusions along the different paths that emanate from the junction. The model predicts the spontaneous emergence of deterministic oscillations between competing protrusions, whereby the cellular polarization and growth alternates between the competing protrusions. These predicted oscillations are found experimentally for two different cell types, noncancerous endothelial and cancerous glioma cells, migrating on patterned network of thin adhesive lanes with junctions. Finally we present an analysis of the migration modes of multicellular "trains" along one-dimensional tracks.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary term 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.

We introduce a methodology based on Multireference Alignment (MRA) for lead-lag detection in multivariate time series, and demonstrate its applicability in developing trading strategies. Specifically designed for low signal-to-noise ratio (SNR) scenarios, our approach estimates denoised latent signals from a set of time series. We also investigate the impact of clustering the time series on the recovery of latent signals. We demonstrate that our lead-lag detection module outperforms commonly employed cross-correlation-based methods. Furthermore, we devise a cross-sectional trading strategy that capitalizes on the lead-lag relationships uncovered by our approach and attains significant economic benefits. Promising backtesting results on daily equity returns illustrate the potential of our method in quantitative finance and suggest avenues for future research.

In ring theory, Morita equivalence is an invariant for many properties, generalising the isomorphism of commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years, there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphisms, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.

We present a general framework for preconditioning Hermitian positive definite linear systems based on the Bregman log determinant divergence. This divergence provides a measure of discrepancy between a preconditioner and a target matrix, giving rise to

the study of preconditioners given as the sum of a Hermitian positive definite matrix plus a low-rank correction. We describe under which conditions the preconditioner minimises the $\ell^2$ condition number of the preconditioned matrix, and obtain the low-rank 

correction via a truncated singular value decomposition (TSVD). Numerical results from variational data assimilation (4D-VAR) support our theoretical results.

We also apply the framework to approximate factorisation preconditioners with a low-rank correction (e.g. incomplete Cholesky plus low-rank). In such cases, the approximate factorisation error is typically indefinite, and the low-rank correction described by the Bregman divergence is generally different from one obtained as a TSVD. We compare these two truncations in terms of convergence of the preconditioned conjugate gradient method (PCG), and show numerous examples where PCG converges to a small tolerance using the proposed preconditioner, whereas PCG using a TSVD-based preconditioner fails. We also consider matrices arising from interior point methods for linear programming that do not admit such an incomplete factorisation by default, and present a robust incomplete Cholesky preconditioner based on the proposed methodology.

The talk is based on papers with Martin S. Andersen (DTU).

During the last fifteen years, there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the μ(I)-rheology (where μ is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This talk details theoretical and numerical frameworks (based on OpenFOAM®) which unify these disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible μ(I)-rheology, which is coupled to a theory for gravity/shear-driven segregation (Gray & Ancey, J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. The model is used to simulate the complex particle-size segregation patterns that form in a partially filled triangular rotating drum. There are many other applications of the theory to industrial granular flows, which are the second most common material used after fluids. The same processes also occur in geophysical flows, such as snow avalanches, debris flows and dense pyroclastic flows. Depth-averaged models, that go beyond the μ(I)-rheology, will also be derived to capture spontaneous self-channelization and levee formation, as well as complex segregation-induced flow fingering effects, which enhance the run-out distance of these hazardous flows.

The Ivanov conjecture is equivalent to the statement that every covering map of surfaces has the so-called Putman-Wieland property. I will discuss my recent work with Vlad Marković, where we prove it for asymptotically all coverings as the degree grows. I will give some overview of our main tool: spectral geometry, which is related to objects like the heat kernel of a hyperbolic surface, or Cheeger connectivity constant.

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this talk, we will present a new approach by applying a natural scaling on the Boltzmann equation. The proof is based on an improved well-posedness theory for the Boltzmann equation without angular cutoff in the regime with an optimal range of parameters so that the grazing limit can be justified directly that includes the Coulomb potential. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two parts. The first one converges to the Landau operator when the parameter of deviation angle tends to its singular value and the second one vanishes in the limit. Hence, the scaling and limiting process exactly capture the grazing collisions. The talk is based on a recent joint work with Yu-Long Zhou.

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.

A rooted tree $T$ has degree sequence $(d_1,\ldots,d_n)$ if $T$ has vertex set $[n]$ and vertex $i$ has $d_i$ children for each $i$ in $[n]$. 

I will describe a line-breaking construction of random rooted trees with given degree sequences, as well as a way of coupling random trees with different degree sequences that also couples their heights to one another. 

The construction and the coupling have several consequences, and I'll try to explain some of these in the talk.

First, let $T$ be a branching process tree with critical—mean one—offspring distribution, and let $T_n$ have the law of $T$ conditioned to have size $n$. Then the following both hold.
1) $\operatorname{height}(T_n)/\log(n)$ tends to infinity in probability. 
2) If the offspring distribution has infinite variance then $\operatorname{height}(T_n)/n^{1/2}$ tends to $0$ in probability. This result settles a conjecture of Svante Janson.

The next two statements relate to random rooted trees with given degree sequences. 
1) For any $\varepsilon > 0$ there is $C > 0$ such that the following holds. If $T$ is a random tree with degree sequence $(d_1,\ldots,d_n)$ and at least $\varepsilon n$ leaves, then $\mathbb{E}(\operatorname{height}(T)) < C \sqrt{n}$. 
2) Consider any random tree $T$ with a fixed degree sequence such that $T$ has no vertices with exactly one child. Then $\operatorname{height}(T)$ is stochastically less than $\operatorname{height}(B)$, where $B$ is a random binary tree of the same size as $T$ (or size one greater, if $T$ has even size). 

In this talk, I will discuss correlation functions in 6d (2, 0) theories of two 1/2-BPS operators inserted away from a 1/2-BPS surface defect. In the large central charge limit the leading connected contribution corresponds to sums of tree-level Witten diagram in AdS7×S4 in the presence of an AdS3 defect. I will show that these correlators can be uniquely determined by imposing only superconformal symmetry and consistency conditions, eschewing the details of the complicated effective Lagrangian. I will present the explicit result of all such two-point functions, which exhibits remarkable hidden simplicity.

A representation on the space of paths is a map which is compatible with the concatenation operation of paths, such as the path signature and Cartan development (or equivalently, parallel transport), and has been used to define characteristic functions for the law of stochastic processes. In this talk, we consider representations of surfaces which are compatible with the two distinct algebraic operations on surfaces: horizontal and vertical concatenation. To build these representations, we use the notion of higher parallel transport, which was first introduced to develop higher gauge theories. We will not assume any background in geometry or category theory. This is a continuation of the previous talk based on a recent preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.

In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems.