Past

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.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Solving kinetic or related models with high-dimensional random parameters has been a challenging problem. In this talk, we will discuss how to employ the bi-fidelity stochastic collocation and choose efficient low-fidelity models in order to solve a class of multi-scale kinetic equations with uncertainties, including the Boltzmann equation, linear transport and the Vlasov-Poisson equation. In addition, some error analysis for the bi-fidelity method based on these PDEs will be presented. Finally, several numerical examples are shown to validate the efficiency and accuracy of the proposed method.

Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).

In our upcoming JSL  paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree.

The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version doi: 10.1017/jsl.2020.42.

I will discuss the notion of invariably generated groups, its importance, and some intuition. I will then present a construction of an invariably generated group that admits an index two subgroup that is not invariably generated. The construction answers questions of Wiegold and of Kantor-Lubotzky-Shalev. This is a joint work with Nir Lazarovich.

In this talk, I will study the properties of parametric estimators based on the Maximum Mean Discrepancy (MMD) defined by Briol et al. (2019). In a first time, I will show that these estimators are universal in the i.i.d setting: even in case of misspecification, they converge to the best approximation of the distribution of the data in the model, without ANY assumption on this model. This leads to very strong robustness properties. In a second time, I will show that these results remain valid when the data is not independent, but satisfy instead a weak-dependence condition. This condition is based on a new dependence coefficient, which is itself defined thanks to the MMD. I will show through examples that this new notion of dependence is actually quite general. This talk is based on published works, and works in progress, with Badr-Eddine Chérief Abdellatif (ENSAE Paris), Mathieu Gerber (University of Bristol), Jean-David Fermanian (ENSAE Paris) and Alexis Derumigny (University of Twente):

http://arxiv.org/abs/1912.05737

http://proceedings.mlr.press/v118/cherief-abdellatif20a.html

http://arxiv.org/abs/2006.00840

https://arxiv.org/abs/2010.00408

https://cran.r-project.org/web/packages/MMDCopula/

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.
    
I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

This talk is intended for a broad math and physics audience in particular including students. It will focus on the speaker’s recent contributions to the analysis of the real Ginibre ensemble consisting of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibre matrix follows a different limiting law for purely real eigenvalues than for non-real ones. We will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1 + 1 dimensions which are solvable by the inverse scattering method. The results of this talk are based on our joint work with Jinho Baik (arXiv:1808.02419 and arXiv:2008.01694)

In this talk I will discuss relations between the low-energy expansions  of open- and closed string amplitudes. At genus zero, it has been shown that the single-valued map of MZVs maps open-string amplitudes to their closed-string counterparts. After reviewing this story, I will discuss recent work at genus one which aims to define a similar mapping from the open to the closed string. Our construction is driven by the differential equations and degeneration limits of certain generating functions of string integrals and suggests a pairing of integration cycles and forms at genus one - analogous to the duality between Parke-Taylor factors and disk boundaries at genus zero. Finally, I will discuss the impact of said mapping on the elliptic MZVs and modular graph forms which arise naturally upon solving these differential equations.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.