Past

I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.
 

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.
 

Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. I will also discuss an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities.

In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am going to illustrate the construction on the example of the sl_2 invariant for the Hopf link. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

My talk will attempt to capture the imperfect state of the art in high-resolution ice sheet modelling, aiming to expose the core performance-limiting issues.  The essential equations for modeling ice flow in a changing climate will be presented, assuming no prior knowledge of the problem.  These geophysical/climate problems are of both free-boundary and algebraic-equation-constrained character.  Current-technology models usually solve non-linear Stokes equations, or approximations thereof, at every explicit time-step.  Scale analysis shows why this current design paradigm is expensive, but building significantly faster high-resolution ice sheet models requires new techniques.  I'll survey some recently-arrived tools, some near-term improvements, and sketch some new ideas.

The dynamics of a tissue in development or regeneration arises from the behaviour of its constituent cells and their interactions. We use mathematical models and inference from experimental data to to infer the likely cellular behaviours underlying changing tissue states. In this talk I will show examples of how we apply canonical birth-death process models to novel experimental data, how we are extending such models with volume exclusion and multistate dynamics, and how we attempt to more generally learn cell-cell interaction models directly from data in interpretable ways. The applications range from in vitro models of embryo development to in vivo blood regeneration that is disrupted with ageing.

Let $G$ be a $p$-adic Lie group. From the perspective of $p$-adic manifolds, possibly the most natural $p$-adic representations of $G$ to consider are the locally analytic ones.  Motivated by work of Pan, when $G$ acts on a rigid analytic variety $X$ (e.g., the flag variety), we would like to geometrise locally analytic $G$-representations, via a covariant localisation theory which should intertwine Schneider-Teitelbaum's duality with the $p$-adic Beilinson-Bernstein localisation. I will report some partial progress in the simplified situation when we replace $G$ by its germ at $1$. The main ingredient is an infinite jet bundle $\mathcal{J}^\omega_X$ which is dual to $\widehat{\mathcal{D}}_X$. Our "co"localisation functor is given by a coinduction to $\mathcal{J}^\omega_X$. Work in progress.

I will present results on short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases or more general nilsequence phases. These results imply the Gowers uniformity of suitably normalised versions of these functions in intervals of length X^c around X for suitable values of c (depending on the function and on whether one considers all or almost all short sums). I will also discuss an application to an averaged form of the Hardy-Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.

We conduct a systematic large-scale analysis of order book-driven predictability in high-frequency returns by leveraging deep learning techniques. First, we introduce a new and robust representation of the order book, the volume representation. Next, we carry out an extensive empirical experiment to address various questions regarding predictability. We investigate if and how far ahead there is predictability, the importance of a robust data representation, the advantages of multi-horizon modeling, and the presence of universal trading patterns. We use model confidence sets, which provide a formalized statistical inference framework particularly well suited to answer these questions. Our findings show that at high frequencies predictability in mid-price returns is not just present, but ubiquitous. The performance of the deep learning models is strongly dependent on the choice of order book representation, and in this respect, the volume representation appears to have multiple practical advantages.

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a poset, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. 

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects, thus generalising the HKKP filtration to other moduli problems of non-linear origin. In particular, we will make sense of the notion of a filtration labelled by sequence of numbers for a point of an algebraic stack.

There is a deep analogy between Kleinaian groups and subgroups of the mapping class group. Inspired by this, Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles.  In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged.  Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in the mapping class group might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.

I will start with two very brief surveys.  First is a class of problems, namely variational inequalities (VIs), which generalize PDE problems, and second is a class of solver algorithms, namely full approximation storage (FAS) nonlinear multigrid for PDEs.  Motivation for applying FAS to VIs is demonstrated in the standard mathematical model for glacier surface evolution, a very general VI problem relevant to climate modeling.  (Residuals for this nonlinear and non-local VI problem are computed by solving a Stokes model.)  Some existing nonlinear multilevel VI schemes, based on global (Newton) linearization would seem to be less suited to such general VI problems.  From this context I will sketch some work-in-progress toward the scalable solutions of nonlinear and nonlocal VIs by an FAS-type multilevel method.

We explore the dynamics of a compressible fluid bubble surrounded by an incompressible fluid of infinite extent in three-dimensions, constructing bubble solutions with finite time blowup under this framework when the equation of state relating pressure and volume is soft (e.g., with volume singularities that are locally weaker than that in the Boyle-Mariotte law), resulting in a finite time blowup of the surrounding incompressible fluid, as well. We focus on two families of solutions, corresponding to a soft polytropic process (with the bubble decreasing in size until eventual collapse, resulting in velocity and pressure blowup) and a cavitation equation of state (with the bubble expanding until it reaches a critical cavitation volume, at which pressure blows up to negative infinity, indicating a vacuum). Interestingly, the kinetic energy of these solutions remains bounded up to the finite blowup time, making these solutions more physically plausible than those developing infinite energy. For all cases considered, we construct exact solutions for specific parameter sets, as well as analytical and numerical solutions which show the robustness of the qualitative blowup behaviors for more generic parameter sets. Our approach suggests novel -- and perhaps physical -- routes to the finite time blowup of fluid equations.

A fundamental result in 3-manifold topology is the loop theorem: Given a null-homotopic simple closed curve in the boundary of a compact 3-manifold $M$, it bounds an embedded disk in $M$. The standard topological proof of this uses the tower construction due to Papakyriakopoulos. In this talk, I will introduce basic existence and regularity results on minimal surfaces, and show how to use the tower construction to prove a geometric version of the loop theorem due to Meeks--Yau: Given a null-homotopic simple closed curve in the boundary of a compact Riemannian 3-manifold $M$ with convex boundary, it bounds an embedded disk of least area. This also gives an independent proof of the (topological) loop theorem.

Wreath-like product groups were introduced recently and used to construct the first positive examples of rigidity conjectures of Connes and Jones. In this talk, I will review those examples, as well as discuss some ideas to construct examples with other rigidity phenomena by modifying the wreath-like product construction.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's $K$-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the $K$-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry ​in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemerédi's theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

We consider the problem of solving nonconvex quadratic optimization problems, potentially with additional integrality constraints on the variables.  Gurobi takes a branch-and-bound approach to solve such problems to global optimality, and in this talk we will review the three main algorithmic components that Gurobi uses:  Convex relaxations based on local linearization, globally valid cutting planes, and nonlinear local optimization heuristics.  We will explain how these parts play together, and discuss some of the implementation details.

In this talk, I will review some recent progress on random graph matching problems, that is, to recover the vertex correspondence between a pair of correlated random graphs from the observation of two unlabelled graphs. In this talk, I will touch issues of information threshold, efficient algorithms as well as complexity theory. This is based on joint works with Hang Du, Shuyang Gong and Zhangsong Li.

I will explain how bornological and condensed structures can both be described as algebraic theories. I will also show how this permits the construction of functors between bornological and condensed structures. If time permits I will also briefly describe how to compare condensed derived geometry and bornological derived geometry and sketch how they relate to analytic geometry and Arakelov geometry

The numerical solution of elliptic PDEs is often the most computationally intensive task in large-scale continuum mechanics simulations.  High-order finite element methods can efficiently exploit modern parallel hardware while offering very rapid convergence properties.  As the polynomial degree is increased, the efficient solution of such PDEs becomes difficult. In this talk we introduce preconditioners for high-order discretizations. We build upon the pioneering work of Pavarino, who proved in 1993 that the additive Schwarz method with vertex patches and a low-order coarse space gives a  solver for symmetric and coercive problems that is robust to the polynomial degree. 

However, for very high polynomial degrees it is not feasible to assemble or factorize the matrices for each vertex patch, as the patch matrices contain dense blocks, which couple together all degrees of freedom within a cell. The central novelty of the preconditioners we develop is that they have the same time and space complexity as sum-factorized operator application on unstructured meshes of tensor-product cells, i.e., we can solve $A x=b$ with the same complexity as evaluating $b-A x$. Our solver relies on new finite elements for the de Rham complex that enable the blocks in the stiffness matrix corresponding to the cell interiors to become diagonal for scalar PDEs or block diagonal for vector-valued PDEs.  With these new elements, the patch problems are as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. In the non-separable case, themethod can be applied as a preconditioner by approximating the problem with a separable surrogate.  Through the careful use of incomplete factorizations and choice of space decomposition we achieve optimal fill-in in the patch factors, ultimately allowing for optimal-complexity storage and computational cost across the setup and solution stages.

We demonstrate the approach by solving the Riesz maps of $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$ in three dimensions at $p = 15$.

Data sets of self-reported online resumes are a valuable tool to understand workers' career trajectories and how workers may adapt to the changing demands of employers. However, the sample of workers that choose to upload their resumes online may not be representative of a nation's workforce. To understand the advantages and limitations of these datasets, we analyze a data set of more than 1 Million online resumes and compare the findings with a administrative data from the Current Population Survey (CPS).
 

I will report on a research program which uses ideas from stochastic analysis in the context of constructive Euclidean quantum field theory. Stochastic analysis is the study of measures on path spaces via push-forward from Gaussian measures. The foundational example is the map, introduced by Itô, which sends Brownian motion to a diffusion process solution to a stochastic differential equation. Parisi–Wu's stochastic quantisation is the stochastic analysis of an Euclidean quantum field, in the above sense. In this introductory talk, I will put these ideas in context and illustrate various stochastic quantisation procedures and some of the rigorous results one can obtain from them.