Past

In long-range percolation on $\mathbb{Z}^d$, each potential edge $\{x,y\}$ is included independently at random with probability roughly $\beta\|x-y\|-d-\alpha$, where $\alpha > 0$ controls how long-range the model is and $\beta > 0$ is an intensity parameter. The smaller $\alpha$ is, the easier it is for very long edges to appear. We are normally interested in fixing $\alpha$ and studying the phase transition that occurs as $\beta$ is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when $\alpha$ is small: It is a theorem of Noam Berger that if $\alpha < d$ then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of $\beta$. (Proving the analogous result for nearest neighbour percolation is a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality.
    As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. This inequality can also be used to give streamlined new proofs of various classical results on e.g. Erdős-Rényi random graphs, which I will hopefully have time to talk a little bit about also.

Let $G$ be a connected reductive algebraic group, and let $G(\mathbb{F}_q)$ be its group of $\mathbb{F}_q$-rational points. Denote by $\mathrm{Irr}(G(\mathbb{F}_q))$ the set of (equivalence classes) of irreducible finite-dimensional representations. Deligne and Lusztig defined a finite subset $$\mathrm{Unip}(G(\mathbb{F}_q)) \subset \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$$ 
of unipotent representations. These representations play a distinguished role in the representation theory of $G(\mathbb{F}_q)$. In particular, the classification of $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ reduces to the classification of $\mathrm{Unip}(G(\mathbb{F}_q))$. 

Now replace $\mathbb{F}_q$ with a local field $k$ and replace $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ with $\mathrm{Irr}_{\mathrm{u}}(G(k))$ (irreducible unitary representations). Vogan has predicted the existence of a finite subset 
$$\mathrm{Unip}(G(k)) \subset \mathrm{Irr}_{\mathrm{u}}(G(k))$$ 
which completes the following analogy
$$\mathrm{Unip}(G(k)) \text{ is to } \mathrm{Irr}_{\mathrm{u}}(G(k)) \text{ as } \mathrm{Unip}(G(\mathbb{F}_q)) \text{ is to } \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q)).$$
In this talk I will propose a definition of $\mathrm{Unip}(G(k))$ when $k = \mathbb{C}$. The definition is geometric and case-free. The representations considered include all of Arthur's, but also many others. After sketching the definition and cataloging its properties, I will explain a classification of $\mathrm{Unip}(G(\mathbb{C}))$, generalizing the well-known result of Barbasch-Vogan for Arthur's representations. Time permitting, I will discuss some speculations about the case of $k=\mathbb{R}$.

This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.

(joint work with T. Kania, Academy of Sciences of the Czech Republic, Prague)
In this talk we discuss our recent result on the inverse eigenvalue problem for symmetric doubly stochastic matrices. 
Namely, we provide a new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix. 
In our construction of such matrices, we employ the eigenvectors of the transition probability matrix of a simple symmetric random walk on the circle. 
We also demonstrate a simple algorithm for generating random doubly stochastic matrices based on our construction. Examples will be provided.

I will present work in progress with D. Gayet and F. Pouran (Grenoble) to establish Russo-Seymour-Welsh (RSW) estimates for 2d statistical mechanics models that do not satisfy the FKG inequality. RSW states that critical percolation has no characteristic length, in the sense that large rectangles are crossed by an open path with a probability that is bounded below by a function of their shape, but uniformly in their size; this ensures the polynomial decay of many relevant quantities and opens the way to deeper understanding of the critical features of the model. All the standard proofs of RSW rely on the FKG inequality, i.e. on the positive correlation between increasing events; we establish the stability of RSW under small perturbations that do not preserve FKG, which extends it for instance to the high-temperature anti-ferromagnetic Ising model.

Subfactors and K-theory are useful mechanisms for understanding modular tensor categories and conformal field theories. As part of this programme, one issue to try and construct or reconstruct a conformal field theory as the representation theory of a conformal net of algebras, or as a vertex operator algebra from a given abstractly presented modular tensor category. Orbifold models play an important role and orbifolds of Tambara-Yamagami systems are relevant to understanding the double of the Haagerup as a conformal field theory. This is joint work with Andreas Aaserud, Terry Gannon and Ulrich Pennig.

I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality (CoD) in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an  approximation precision $\varepsilon >0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed  and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the CoD in the sense that  the number of real parameters used to describe the approximating DNNs  grows at most polynomially in both the PDE dimension $d \in  \N$ and the reciprocal of the prescribed approximation accuracy $\varepsilon >0$. There are now also a few rigorous mathematical results in the scientific literature which  substantiate this conjecture by proving that  DNNs overcome the CoD in approximating solutions of PDEs.  Each of these results establishes that DNNs overcome the CoD in approximating suitable PDE solutions  at a fixed time point $T >0$ and on a compact cube $[a, b]^d$ but none of these results provides an answer to the question whether the entire PDE solution on $[0, T] \times [a, b]^d$ can be approximated by DNNs without the CoD. 
In this talk we show that for every $a \in \R$, $ b \in (a, \infty)$ solutions of  suitable  Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0, T] \times [a, b]^d$ without the CoD. 

We discuss the problem in representation theory of decomposing restricted representations. We start classically with the symmetric groups via Young diagrams and Young tableaux, and then move into the world of Lie groups. These problems have connections with both physics and number theory, and if there is time I will discuss the Gan-Gross-Prasad conjectures which predict results on restrictions for algebraic groups over both local and global fields. The pre-requisites will build throughout the talk, but it should be accessible to anyone with some knowedge of both finite groups and Lie groups.

Hardt-Simon proved that every area-minimizing hypercone $C$ having only an isolated singularity fits into a foliation of $R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypersurface $M$ in the unit ball $B_1 \subset R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then $M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in $R^{n+1}$ asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.  This is joint work with Luca Spolaor.

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of locally finite affine A_n buildings. From this data, we will show how to construct new fiber functors on the category of tilting modules for SL(n+1) in characteristic p (related to order of the projective geometry) using the web calculus of Cautis, Kamnitzer, Morrison and Brundan, Entova-Aizenbud, Etingof, Ostrik.

In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. 

One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration.

We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.

In view of the recent observations of gravitational-wave signals from black-hole mergers, classical black-hole scattering has received considerable interest due to its relation to the classical bound-state problem of two black holes inspiraling onto each other. In this talk I will discuss the link between classical scattering of spinning black holes and quantum scattering amplitudes for massive spin-s particles. Considering the first post-Minkowskian (PM) order, I will explain how the spin-exponentiated structure of the relevant tree-level amplitude follows from minimal coupling to Einstein's gravity and in the s → ∞ limit generates the black holes' complete series of spin-induced multipoles. The resulting scattering function will be shown to encode in a simple way the classical net changes in the black-hole momenta and spins at 1PM order and to all orders in spins. I will then comment on the results and challenges at 2PM order and beyond.
 

We propose Swampland constraints on consistent 5d N=1 supergravity theories. In particular, we focus on a special class of BPS monopole strings which arise only in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be computed using the anomaly inflow mechanism and provide constraints for the 5d supergravity using unitarity of the worldsheet CFT. In M-theory, where these theories can be realised by compactification on Calabi-Yau threefolds, the special monopole strings arise from M5 branes wrapping “semi-ample” 4-cycles in the threefolds. We further identify necessary geometric conditions that such cycles need to satisfy and translate them into constraints for the low-energy gravity theory.

In this talk, I will give a brief introduction to level-set methods for image analysis. I will then describe an application of level-sets to the construction of filtrations for persistent homology computations. I will present several case studies with various spatial data sets using this construction, including applications to voting, analyzing urban street patterns, and spiderwebs. I will conclude by discussing the types of data which I might imagine such methods to be suitable for analyzing and suggesting a few potential future applications of level-set based computations.

The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.

In this talk, I will present a series of new experimental data, supported by theoretical models, of the transport of ash, aerosols and bubbles in multiphase plumes rising through stratified environments, focussing on the structure of flow and the dispersal of the different phases. The models have relevance for the dispersal of volcanic ash in the atmosphere and ocean, the mixing of aerosols in buildings, and the fate of suspended sediment produced during deep sea mining. 

While the pathological mechanisms in COVID-19 illness are still poorly understood, it is increasingly clear that high levels of pro-inflammatory mediators play a major role in clinical deterioration in patients with severe disease. Current evidence points to a hyperinflammatory state as the driver of respiratory compromise in severe COVID-19 disease, with a clinical trajectory resembling acute respiratory distress syndrome (ARDS) but how this “runaway train” inflammatory response emergences and is maintained is not known. In this talk, we present the first mathematical model of lung hyperinflammation due to SARS- CoV-2 infection. This model is based on a network of purported mechanistic and physiological pathways linking together five distinct biochemical species involved in the inflammatory response. Simulations of our model give rise to distinct qualitative classes of COVID-19 patients: (i) individuals who naturally clear the virus, (ii) asymptomatic carriers and (iii–v) individuals who develop a case of mild, moderate, or severe illness. These findings, supported by a comprehensive sensitivity analysis, points to potential therapeutic interventions to prevent the emergence of hyperinflammation. Specifically, we suggest that early intervention with a locally-acting anti-inflammatory agent (such as inhaled corticosteroids) may effectively blockade the pathological hyperinflammatory reaction as it emerges.

Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing, and have recently attracted growing interests in other fields of applications. In this talk we will start from analyzing the connection between GANs and mean field games (MFGs) as well as optimal transport (OT). We will first show a conceptual connection between GANs and MFGs: MFGs have the structure of GANs, and GANs are MFGs under the Pareto Optimality criterion. Interpreting MFGs as GANs, on one hand, will enable a GANs-based algorithm (MFGANs) to solve MFGs: one neural network (NN) for the backward Hamilton-Jacobi-Bellman (HJB) equation and one NN for the Fokker-Planck (FP) equation, with the two NNs trained in an adversarial way. Viewing GANs as MFGs, on the other hand, will reveal a new and probabilistic aspect of GANs. This new perspective, moreover, will lead to an analytical connection between GANs and Optimal Transport (OT) problems, and sufficient conditions for the minimax games of GANs to be reformulated in the framework of OT. Building up from the probabilistic views of GANs, we will then establish the approximation of GANs training via stochastic differential equations and demonstrate the convergence of GANs training via invariant measures of SDEs under proper conditions. This stochastic analysis for GANs training can serve as an analytical tool to study its evolution and stability.

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.

Polycyclic groups form an interesting and well-studied class of groups that properly contain the finitely generated nilpotent groups. I will discuss the C*-algebras associated with virtually polycyclic groups, their maximal quotients and recent work with Jianchao Wu showing that they have finite nuclear dimension.

Artificial microswimmers are an emerging field of research, attracting
interest as testing beds for physical theories of complex biological
entities, as inspiration for the design of smart materials, and for the
sheer elegance, and often quite counterintuitive phenomena of
experimental nonlinear dynamics.

Self-propelling droplets are among the most simplified swimmer models
imaginable, requiring just three components (oil, water, surfactant). In
this talk, I will show how these inherently stupid objects can make
surprisingly smart decisions based on interactions with microfluidic
structures and self-generated and external chemical fields.

I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.

We briefly review mathematical models of viscous deformable interfaces (such as plasma membranes) leading to fluid equations posed on (evolving) 2D surfaces embedded in $R^3$. We further report on some recent advances in understanding and numerical simulation of the resulting fluid systems using an unfitted finite element method.

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