Past

Andrea Giudici: Multiple shapes from one elastomer sheet

Active soft materials, such as Liquid Crystal Elastomers (LCEs), possess a unique property: the ability to change shape in response to thermal or optical stimuli. This makes them attractive for various applications, including bioengineering, biomimetics, and soft robotics. The classic example of a shape change in LCEs is the transformation of a flat sheet into a complex curved surface through the imprinting of a spatially varying deformation field. Despite its effectiveness, this approach has one important limitation: once the deformation field is imprinted in the material, it cannot be amended, hindering the ability to achieve multiple target shapes.

In this talk, I present a solution to this challenge and discuss how modulating the degree of actuation using light intensity offers a route towards programming multiple shapes. Moreover, I introduce a theoretical framework that allows us to sculpt any surface of revolution using light.

Edwina Yeo: Modelling the onset of arterial blood clotting

Arterial blood clot formation (thrombosis) is the leading cause of both stroke and heart attack. The blood protein Von Willebrand Factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates the protein unfolds and rapidly captures platelets from the flow.

I will present two pieces of modelling to predict the location of clot formation in a diseased artery. Firstly a continuum model to describe the mechanosensitive protein VWF and secondly a model for platelet transport and deposition to VWF. We interface this model with in vitro data of thrombosis in a long, thin rectangular microfluidic geometry. Using a reduced model, the unknown model parameters which determine platelet deposition can be calibrated.

Although Green's theorem, currently considered one of the cornerstones of multivariate calculus, was published in 1828, its widespread introduction into calculus textbooks can be traced back to the first decades of the twentieth century, when vector calculus emerged as a slightly autonomous discipline. In addition, its contemporary version (and its demonstration), currently found in several calculus textbooks, is the result of some adaptations during its almost 200 years of life. Comparing some books and articles from this long period, I would like to discuss in this lecture the didactic adaptations, the editorial strategies and visual representations that shaped the theorem in its current form.

Although tensor products and their symmetrisation have appeared in mathematical literature since at least the mid-nineteenth century, they rarely appear in the function-theoretic operator theory literature. In this talk, I will introduce the symmetric and antisymmetric tensor products from an operator theoretic point of view. I will present results concerning some of the most fundamental operator-theoretic questions in this area, such as finding the norm and spectrum of the symmetric tensor products of operators. I will then work through some examples of symmetric tensor products of familiar operators, such as the unilateral shift, the adjoint of the shift, and diagonal operators.

The talk is based on joint work with Uri Bader. We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. If time permits, we describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger).

Let $p \ge 3$ be a prime integer. The density of a non-empty solution set of a system of affine equations $L_i(x) = b_i$, $i=1,\dots,k$ on a vector space over the field $\mathbb{F}_p$ is determined by the dimension of the linear subspace $\langle L_1,\dots,L_k \rangle$, and tends to $0$ with the dimension of that subspace. In particular, if the solution set is dense, then the system of equations contains at most boundedly many pairwise distinct linear forms. In the more general setting of systems of affine conditions $L_i(x) \in E_i$ for some strict subsets $E_i$ of $\mathbb{F}_p$ and where the solution set and its density are taken inside $S^n$ for some non-empty subset $S$ of $\mathbb{F}_p$ (such as $\{0,1\}$), we can however usually find subsets of $S^n$ with density at least $1/2$ but such that the linear subspace $\langle L_1,\dots,L_k \rangle$ has arbitrarily high dimension. We shall nonetheless establish an approximate boundedness result: if the solution set of a system of affine conditions is dense, then it contains the solution set of a system of boundedly many affine conditions and which has approximately the same density as the original solution set. Using a recent generalisation by Gowers and the speaker of a result of Green and Tao on the equidistribution of high-rank polynomials on finite prime fields we shall furthermore prove a weaker analogous result for polynomials of small degree.

Based on joint work with Timothy Gowers (College de France and University of Cambridge).

***CANCELLED*** In this talk, I will discuss recent results related to the mathematical justification of PDEs which model multi-phase flows at the macroscopic level from mesoscopic descriptions with jump conditions at interfaces. We will also present interesting and difficult open problems.

The notion of a pseudocharacter was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the
deformation theory of a residually reducible Galois representation when the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast. 

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.