Past

"Colouring" a tournament means partitioning its vertex set into acylic subsets; and the "domination number" is the size of the smallest set of vertices with no common in-neighbour. In some ways these are like the corresponding concepts for graphs, but in some ways they are very different. We give a survey of some recent results and open questions on these topics.

Joint with Tung Nguyen and Alex Scott.

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

Diffusion Limited Aggregation is a very simple mathematical model which describes a wide range of natural phenomena. Despite its simplicity, there is very little progress in understanding its large-scale structure. Since its introduction by Witten and Sander over 40 years ago, there was only one mathematical result. In 1987 Kesten obtained an upper bound on the growth rate. In this talk I will discuss DLA and some related models and the recent progress in understanding DLA. In particular, a new simpler proof of Kesten result which generalizes to other aggregation models.

Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 

• 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 

• 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? 

In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. 
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

The affinity between statistical physics and machine learning has a long history. Theoretical physics often proceeds in terms of solvable synthetic models; I will describe the related line of work on solvable models of simple feed-forward neural networks. I will then discuss how this approach allows us to analyze uncertainty quantification in neural networks, a topic that gained urgency in the dawn of widely deployed artificial intelligence. I will conclude with what I perceive as important specific open questions in the field.

 

In this lecture, I will discuss the resolution of the Arthur-Barbasch-Vogan conjecture on the unitarity of special unipotent representations for any real form of a connected reductive complex Lie group, with contributions by several groups of authors (Barbasch-Ma-Sun-Zhu, Adams-Arancibia-Mezo, and Adams-Miller-van Leeuwen-Vogan). The main part of the lecture will be on the approach of the first group of authors for the case of real classical groups: counting by coherent families (combinatorial aspect), construction by theta lifting (analytic aspect), and distinguishing by invariants (algebraic-geometric aspect), resulting in a full classification, and with unitarity as a direct consequence of the construction.

Multi-parameter persistence is a main research topic in topological data analysis. Major questions involve the computation and the structural properties
of persistence modules. In this context, I will sketch two very recent results:

(1) We define a natural bifiltration called the localized union-of-balls bifiltration that contains filtrations studied in the context of local persistent homology as slices. This bifiltration is not k-critical for any finite k. Still, we show that a representation of it (involving algebraic curves of low degree) can be computed exactly and efficiently. This is joint work with Matthias Soels (TU Graz).

(2) Every persistence modules permits a unique decomposition into indecomposable summands. Intervals are the simplest type of summands, but more complicated indecomposables can appear, and usually do appear in examples. We prove that for homology-dimension 0 and density-Rips bifiltration, at least a quarter of the indecomposables are intervals in expectation for a rather general class of point samples. Moreover, these intervals can be ``peeled off'' the module efficiently. This is joint work with Angel Alonso (TU Graz).

Critical levels appear as singularities of waves propagating in shear flows. When magnetic field exists, critical levels are located where the phase velocity of the wave relative to the basic flow matches the velocity of Alfvén waves. Critical levels are known for locally strong wave amplitude in its vicinity, known as the critical layers. In this talk, I will demonstrate the situation where magnetic critical layers can contribute to the instability of the MHD flow.  We consider two different flow configurations. One is the shallow water flow, and the other is the 2D flow on a sphere. Asymptotic analysis has been used to explore deeper insights of the instability mechanism.

Data-driven modeling and deep learning present powerful tools that are opening up new paradigms and opportunities in the understanding, discovery, and design of soft and biological materials. I will describe our recent applications of deep representational learning to expose the sequence-function relationship within homologous protein families and to use these principles for the data-driven design and experimental testing of synthetic proteins with elevated function. I will then describe an approach based on latent space simulators to learn ultra-fast surrogate models of protein folding and biomolecular assembly by stacking three specialized deep learning networks to (i) encode a molecular system into a slow latent space, (ii) propagate dynamics in this latent space, and (iii) generatively decode a synthetic molecular trajectory.

Given a group G and a field k, we can "twist" the multiplication of the group algebra kG by a 2-cocycle, and the result is a twisted group algebra. Twisted group algebras arise as direct sums of blocks of group algebras, and so are of interest in representation and block theory. In this talk we will discuss some recently obtained results on the first Hochschild cohomology of twisted group algebras, in particular that these cohomology groups are nontrivial whenever G is a finite simple group.

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an “open market” setup consisting of only assets with high capitalizations. In this work we relax previously studied notions of open markets and develop a tractable framework for them under mild structural conditions on the market.

Within this framework we also introduce a large parametric class of processes, which we call rank Jacobi processes. They produce a stable capital distribution curve consistent with empirical observations. Moreover, there are explicit expressions for the growth-optimal portfolio, and they are also shown to serve as worst-case models for a robust asymptotic growth problem under model ambiguity.

Time permitting, I will also present an extended class of models and illustrate calibration results to CRSP Equity Data.

This talk is based on joint work with Martin Larsson.

The pioneering work of Murray and von Neumann shows that any countable discrete group G gives rise in a canonical way to a group von Neumann algebra, denoted L(G). A main theme in operator algebras is to classify group von Neumann algebras, and hence, to understand how much information does L(G) remember of the underlying group G. In the amenable case, the classification problem is completed by the work of Connes from 1970s asserting that for all infinite conjugacy classes amenable groups, their von Neumann algebras are isomorphic.

In sharp contrast, in the non-amenable case, Popa's deformation rigidity/theory (2001) has led to the discovery of several instances when various properties of the group G are remembered by L(G). The goal of this talk is to survey some recent progress in this direction.

Lightweight compliant surfaces are commonly used as roofs (awnings), filtration systems or propulsive appendages, that operate in a fluid environment. Their flexibility allows for shape to change in fluid flows, to better endure harsh or fluctuating conditions, or enhance flight performance of insect wings for example. The way the structure deforms is however key to fulfill its function, prompting the need for control levers. In this talk, we will consider two ways to tailor the deformation of surfaces in a flow, making use of the properties of origami (folded sheet) and kirigami (sheet with a network of cuts). Previous literature showed that the substructure of folds or cuts allows for sophisticated shape morphing, and produces tunable mechanical properties. We will discuss how those original features impact the way the structure interacts with a flow, through combined experiments and theory. We will notably show that a sheet with a symmetric cutting pattern can produce an asymmetric deformation, and study the underlying fluid-structure couplings to further program shape morphing through the cuts arrangement. We will also show that extreme shape reconfiguration through origami folding can cap fluid drag.

We will give an introduction to hyperbolic cusps and their Dehn fillings. In particular, we will give a brief survey of quantitive results in the field. To motivate this work, we will sketch how these techniques are used for studying the classical question of characteristic slopes on knots.

The notion of amenable actions by discrete groups on C*-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed until 2020. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also present far reaching results towards the weak containment problem which asks wether an action $\alpha:G\to \Aut(A)$ is amenable if and only if the maximal and reduced crossed products coincide.

In this lecture we report on joint work with Alcides Buss and Rufus Willett.

Given a group G, one can seek to understand (some of) its subgroups. Centralisers of elements are easy to define, but maybe not so easy to understand: even in such well studied groups as Out(Fn) they are not yet understood in general. I'll discuss recent work with Armando Martino where we extend what is known in Out(Fn), involving a (surprising?) connection to free-by-cyclic groups and their automorphisms as well as working with actions on trees. The strategies seem like they should apply in many more cases, and if time allows I'll discuss ongoing work (with Gilbert Levitt and Armando Martino) exploring these possibilities.

In this talk, we introduce Newton-MR variants for solving nonconvex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient method as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. With certain useful monotonicity properties of MINRES as well as its inherent ability to detect non-positive curvature directions as soon as they arise, we show that our algorithms come with desirable properties including the optimal first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]