Past

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.

I will introduce various heights on number fields and outline how solving polynomial equations with heights conditions is related to Arakelov geometry and a continuous logic theory called GVF.

Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. To this end, a key step is the development of an effective discriminator to distinguish between time series distributions. In this talk, I will introduce the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance.  On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality. Joint work with Dr. Siran Li (Shanghai Jiao Tong Uni) and Hang Lou (UCL). Paper: [https://arxiv.org/pdf/2305.12511.pdf].

About 20 years ago, Dror Bar-Natan described an elegant generalisation
of Khovanov homology to tangles with any number of endpoints, by
considering certain quotients of two-dimensional relative cobordism
categories.  I claim that these categories are in general not
well-understood (not by me in any case).  However, if we restrict to
tangles with four endpoints, things simplify and Bar-Natan's category
turns out to be closely related to the wrapped Fukaya category of the
four-punctured sphere.  This relationship gives rise to a symplectic
interpretation of Khovanov homology that is useful both for doing
calculations and for proving theorems.  I will discuss joint work in
progress with Artem Kotelskiy and Liam Watson where we investigate what
happens when we fill in one of the punctures.
 

In this talk we give an introduction to the topic of Einstein metrics on spheres. In particular, we prove the existence of three non-round Einstein metrics with positive scalar curvature on $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$ This talk is based on joint work with Jan Nienhaus.

Optimization problems constrained on manifolds are prevalent across science and engineering. For example, they arise in (generalized) eigenvalue problems, principal component analysis, and low-rank matrix completion, to name a few problems. Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a (Riemannian) manifold.  Algorithms designed in this framework usually require some geometrical description of the manifold, i.e., tangent spaces, retractions, Riemannian gradients, and Riemannian Hessians of the cost function. However, in some cases, some of the aforementioned geometric components cannot be accessed due to intractability or lack of information.

In this talk, we present methods that allow for overcoming cases of intractability and lack of information. We demonstrate the case of intractability on canonical correlation analysis (CCA) and on Fisher linear discriminant analysis (FDA). Using Riemannian optimization to solve CCA or FDA with the standard geometric components is as expensive as solving them via a direct solver. We address this shortcoming using a technique called Riemannian preconditioning, which amounts to changing the Riemannian metric on the constraining manifold. We use randomized numerical linear algebra to form efficient preconditioners that balance the computational costs of the geometric components and the asymptotic convergence of the iterative methods. If time permits, we also show the case of lack of information, e.g., the constraining manifold can be accessed only via samples of it. We propose a novel approach that allows approximate Riemannian optimization using a manifold learning technique.

We will have a Q&A with a panel of academics and industry experts on applying to jobs both in and out of academia.

Tensor decomposition is an unsupervised learning methodology that has applications in a wide variety of domains, including chemometrics, criminology, and neuroscience. We focus on low-rank tensor decomposition using  canonical polyadic or CANDECOMP/PARAFAC format. A low-rank tensor decomposition is the minimizer according to some nonlinear program. The usual objective function is the sum of squares error (SSE) comparing the data tensor and the low-rank model tensor. This leads to a nicely-structured problem with subproblems that are linear least squares problems which can be solved efficiently in closed form. However, the SSE metric is not always ideal. Thus, we consider using other objective functions. For instance, KL divergence is an alternative metric is useful for count data and results in a nonnegative factorization. In the context of nonnegative matrix factorization, for instance, KL divergence was popularized by Lee and Seung (1999). We can also consider various objectives such as logistic odds for binary data, beta-divergence for nonnegative data, and so on. We show the benefits of alternative objective functions on real-world data sets. We consider the computational of generalized tensor decomposition based on other objective functions, summarize the work that has been done thus far, and illuminate open problems and challenges. This talk includes joint work with David Hong and Jed Duersch.

In topological data analysis, persistence barcodes record the
persistence of homological generators in a one-parameter filtration
built on the data at hand. In contrast, computing the pointwise Euler
characteristic (EC) of the filtration merely records the alternating sum
of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical
"signal/noise" dichotomy, EC tools are powerful descriptors, especially
when combined with new integral transforms mixing EC techniques with
Lebesgue integration. Our motivation is fourfold: their applicability to
multi-parameter filtrations and time-varying data, their remarkable
performance in supervised and unsupervised tasks at a low computational
cost, their satisfactory properties as integral transforms (e.g.,
regularity and invertibility properties) and the expectation results on
the EC in random settings. Along the way, we will give an insight into
the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and
the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].

Modern medicine has given us effective tools to treat some of the most significant and burdensome diseases. At the same time, it is becoming consistently more challenging and more expensive to develop new therapeutics. A key factor in this trend is that we simply don't understand the underlying biology of disease, and which interventions might meaningfully modulate clinical outcomes and in which patients. To achieve this goal, we are bringing together large amounts of high content data, taken both from humans and from human-derived cellular systems generated in our own lab. Those are then used to learn a meaningful representation of biological states via cutting edge machine learning methods, which enable us to make predictions about novel targets, coherent patient segments, and the clinical effect of molecules. Our ultimate goal is to develop a new approach to drug development that uses high-quality data and ML models to design novel, safe, and effective therapies that help more people, faster, and at a lower cost. 

Typically, the algebraic closure of a non-algebraically closed field F is an infinite extension of F. However, this doesn't always have to happen: for example consider $\mathbb{R}$ inside $\mathbb{C}$. Are there any other examples? Yes: for example you can consider the index two subfield of the algebraic numbers, defined by intersecting with $\mathbb{R}$. However this is still similar to the first example: the degree of the extension is two, and we extract a square root of $-1$ to obtain the algebraic closure. The Artin-Schreier Theorem tells us that amazingly this is always the case: if $F$ is a field for which the algebraic closure is a non trivial finite extension $L$, then this forces F to have characteristic 0, L is degree two over $F$, and $L = F(i)$ for some $i$ with $i^2 = -1$. I.e. all such extensions "look like" $\mathbb{C} / \mathbb{R}$. In this expository talk we will give an overview of the proof of this theorem, and try to get some feeling for why this result is true.

We propose a method to detect linear and nonlinear lead-lag relationships in stock returns.  Our approach uses pairwise Lévy-area and cross-correlation of returns to rank the assets from leaders to followers. We use the rankings to construct a portfolio that longs or shorts the followers based on the previous returns of the leaders, and the stocks are ranked every time the portfolio is rebalanced. The portfolio also takes an offsetting position on the SPY ETF so that the initial value of the portfolio is zero. Our data spans from 1963 to 2022 and we use an average of over 500 stocks to construct portfolios for each trading day. The annualized returns of our lead-lag portfolios are over  20%, and the returns outperform all lead-lag benchmarks in the literature. There is little overlap between the leaders and the followers we find and those that are reported in previous studies based on market capitalization, volume traded, and intra-industry relationships. Our findings support the slow information diffusion hypothesis; i.e., portfolios rebalanced once a day consistently outperform the bidiurnal, weekly, bi-weekly, tri-weekly, and monthly rebalanced portfolios.

(Joint work with Kartik Prasanna)

Siegel modular forms are higher-dimensional analogues of modular forms. While each rational elliptic curve corresponds to a single holomorphic modular form, each abelian surface is expected to correspond to a pair of Siegel modular forms: a holomorphic and a generic one. We propose a conjecture that explains the appearance of these two forms (in the cohomology of vector bundles on Siegel modular threefolds) in terms of certain higher algebraic cycles on the self-product of the abelian surface. We then prove three results:
(1) The conjecture is implied by Beilinson's conjecture on special values of L-functions. Amongst others, this uses a recent analytic result of Radzwill-Yang about non-vanishing of twists of L-functions for GL(4).
(2) The conjecture holds for abelian surfaces associated with elliptic curves over real quadratic fields.
(3) The conjecture implies a conjecture of Prasanna-Venkatesh for abelian surfaces associated with elliptic curves over imaginary quadratic fields.

Randomized least squares is a promising method but not yet widely used in practice. We show an example of its use for finding low-rank canonical polyadic (CP) tensor decompositions for large sparse tensors. This involves solving a sequence of overdetermined least problems with special (Khatri-Rao product) structure.

In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. We perform sketching through leverage score sampling, crucially relying on the fact that the problem structure enable efficient sampling from overestimates of the leverage scores with much less work. We discuss what it took to make the algorithm practical, including general-purpose improvements.

Numerical results on real-world large-scale tensors show the method is faster than competing methods without sacrificing accuracy.

*This is joint work with Brett Larsen, Stanford University.

Extrinsic flows are evolution equations whose speeds are determined by the extrinsic curvature of submanifolds in ambient spaces.  Some of the well-known ones are mean curvature flow, Gauss curvature flow, and Lagrangian mean curvature flow.

We focus on the special case in which the speed of a flow is given by powers of mean curvature for smooth convex hypersurfaces of graph type, i.e., ones that can be represented as the graph of a function.  Convergence and long-time existence of such flow will be discussed. Furthermore, C^2 estimates which are independent of height of the graph will be derived to see that the boundary of the domain of the graph is also a smooth solution for the same flow as a submanifold with codimension two in the classical sense.  Some of the main ideas, notably a priori estimates via the maximum principle, come from the work of Huisken and Ecker on mean curvature evolution of entire graphs in 1989.  This is a joint work with Ki-ahm Lee and Taehun Lee.

The classical model of evaporation of liquids hinges on Maxwell’s assumption that the air near the liquid’s surface is saturated. It allows one to find the evaporative flux without considering the interface separating liquid and air. Maxwell’s hypothesis is based on an implicit assumption that the vapour-emission capacity of the interface exceeds the throughput of air (i.e., its ability to pass the vapour on to infinity). If indeed so, the air adjacent to the liquid would get quickly saturated, justifying Maxwell’s hypothesis.

In the present paper, the so-called diffuse-interface model is used to account for the interfacial physics and, thus, derive a generalised version of Maxwell’s boundary condition for the near-interface vapour density. It is then applied to a spherical drop floating in air. It turns out that the vapour-emission capacity of the interface exceeds the throughput of air only if the drop’s radius is r_d ≳ 10μm, but for r_d ≈ 2μm, the two are comparable. For r_d ≲ 1μm, evaporation is interface-driven, and the resulting evaporation rate is noticeably smaller than that predicted by the classical model.

I will give an overview of new ideas showing up in arithmetic intersection theory based on some exciting talks that appeared at the very recent conference "Global invariants of arithmetic varieties". I will also outline connections to globally valued fields and some classical problems.

Algebraic fibring is the group-theoretic analogue of fibration over the circle for manifolds. Generalising the work of Agol on hyperbolic 3-manifolds, Kielak showed that many groups virtually fibre. In this talk we will discuss the geometry of groups which fibre, with some fun applications to Poincare duality groups - groups whose homology and cohomology invariants satisfy a Poincare-Lefschetz type duality, like those of manifolds - as well as to exotic subgroups of Gromov hyperbolic groups. No prior knowledge of these topics will be assumed.

Disclaimer: This talk will contain many manifolds.

A previously known function-theoretic characterisation of compactness for a weighted composition operator on BMOA is improved. Moreover, the same function-theoretic condition also characterises weak compactness and complete continuity. In order to close the circle of implications, the operator-theoretic property of fixing a copy of c0 comes in useful. 

We study the distribution of eigenvalues of kernel random matrices where each element is the empirical covariance between the feature map evaluations of a random fully-connected neural network. We show that, under mild assumptions on the non-linear activation function, namely Lipschitz continuity and measurability, the limiting spectral distribution can be written as successive free multiplicative convolutions between the Marchenko-Pastur law and a nonrandom measure specific to the neural network. The latter has no known analytical expression but can be simulated empirically, separately from the random matrices of interest.

Motivated by the desire to construct large independent sets in random graphs, Karp and Sipser modified the usual greedy construction to yield an algorithm that outputs an independent set with a large cardinal called the Karp-Sipser core. When run on the Erdős-Rényi $G(n,c/n)$ random graph, this algorithm is optimal as long as $c < e$. We will present the proof of a physics conjecture of Bauer and Golinelli (2002) stating that at criticality, the size of the Karp-Sipser core is of order $n^{3/5}$. Along the way we shall highlight the similarities and differences with the usual greedy algorithm and the $k$-core algorithm.
Based on a joint work with Nicolas Curien and Thomas Budzinski.

The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups.  This talk is based on joint works with García-Mejía, Pallier and Tessera.

A $k$-block in a graph is a set of $k$ vertices every two of which are joined by $k$ vertex disjoint paths. By a result of Weissauer, graphs with no $k$-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions for bounded tree width, a lot can be said about graphs with no $k$-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a $k$-block in a graphs. We will discuss this phenomenon and its consequences in the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

The resolutions of Slodowy slices S͂_e are symplectic varieties that contain the Springer fiber (G/B)_e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element ts for s a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions.