Past

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ''shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated with them. In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

The amazing results of DeepMind's AlphaFold2 in the last CASP experiment  caused a huge stir in both the AI and biology fields, and this was of 
course widely reported in the general media. The claim is that the  protein folding problem has finally been solved, but has it really? Not 
to spoil the ending, but of course not. In this talk I will not be  talking (much) about AlphaFold2 itself, but instead what inspiration we 
can take from it about future directions we might want to take in protein structure bioinformatics research using modern AI techniques. 
Along the way, I'll illustrate my thoughts with some recent and current  machine-learning-based projects from my own lab in the area of protein 
structure and folding.
 

From latent variable models in machine learning to inverse problems in computational imaging, tensors pervade the data sciences.  Often, the goal is to decompose a tensor into a particular low-rank representation, thereby recovering quantities of interest about the application at hand.  In this talk, I will present a recent method for low-rank CP symmetric tensor decomposition.  The key ingredients are Sylvester’s catalecticant method from classical algebraic geometry and the power method from numerical multilinear algebra.  In simulations, the method is roughly one order of magnitude faster than existing CP decomposition algorithms, with similar accuracy.  I will state guarantees for the relevant non-convex optimization problem, and robustness results when the tensor is only approximately low-rank (assuming an appropriate random model).  Finally, if the tensor being decomposed is a higher-order moment of data points (as in multivariate statistics), our method may be performed without explicitly forming the moment tensor, opening the door to high-dimensional decompositions.  This talk is based on joint works with João Pereira, Timo Klock and Tammy Kolda. 

(Joint with Y. Halevi and A. Hasson.) We consider two kinds of expansions of a valued field $K$:

(1) A $T$-convex expansion of real closed field, for $T$ a polynomially bounded o-minimal expansion of $K$.

(2) A $P$-minimal field $K$ in which definable functions are PW differentiable.

We prove that any interpretable infinite field $F$ in $K$ is definably isomorphic to a finite extension of either $K$ or, in case (1), its residue field $k$. The method we use bypasses general elimination of imaginaries and is based on analysis of one dimensional quotients of the form $I=K/E$ inside $F$ and their connection to one of 4 possible sorts: $K$, $k$ (in case (1)), the value group, or the quotient of $K$ by its valuation ring. The last two cases turn out to be impossible and in the first two cases we use local differentiability to embed $F$ into the matrix ring over $K$ (or $k$).

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.

In its simplest instance, kinetic Brownian in R^d is a C¹ random path (m_t, v_t) with unit velocity v_t a Brownian motion on the unit sphere run at speed a > 0. Properly time rescaled as a function of the parameter a, its position process converges to a Brownian motion in R^d as a tends to infinity. On the other side the motion converges to the straight line motion (= geodesic motion) when a goes to 0. Kinetic Brownian motion provides thus an interpolation between geodesic and Brownian flows in this setting. Think now about changing R^d for the diffeomorphism group of a fluid domain, with a velocity vector now a vector field on the domain. I will explain how one can prove in this setting an interpolation result similar to the previous one, giving an interpolation between Euler’s equations of incompressible flows and a Brownian-like flow on the diffeomorphism group.

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

In this talk, we present some effective methods for distributing points for approximating analytic functions with prescribed decay on a strip region including the real axis. Such functions appear when we use numerical methods with variable transformations. Typical examples of such methods are provided by single-exponential (SE) or double-exponential (DE) sinc formulas, in which variable transformations yield single- or double-exponential decay of functions on the real axis. It has been known that the formulas are nearly optimal on a Hardy space with a single- or double-exponential weight on the strip region, which is regarded as a space of transformed functions by the variable transformations.

Recently, we have proposed new approximation formulas that outperform the sinc formulas. For constructing them, we use an expression of the error norm (a.k.a. worst-case error) of an n-point interpolation operator in the weighted Hardy space. The expression is closely related to potential theory, and optimal points for interpolation correspond to an equilibrium measure of an energy functional with an external field. Since a discrete version of the energy becomes convex in the points under a mild condition about the weight, we can obtain good points with a standard optimization technique. Furthermore, with the aid of the formulation with the energy, we can find approximate distributions of the points theoretically.

[References]
- K. Tanaka, T. Okayama, M. Sugihara: Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces, IMA Journal of Numerical Analysis Vol. 37 (2017), pp. 861-904.

- K. Tanaka, M. Sugihara: Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization, IMA Journal of Numerical Analysis Vol. 39 (2019), pp. 1957-1984.

- S. Hayakawa, K. Tanaka: Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization, arXiv:1906.03133.

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 contact @email.

Eva Antonopoulou

Surfactants in drop-on-demand inkjet printing

The rapid development of new applications for inkjet printing and increasing complexity of the inks has created a demand for in silico optimisation of the ink jetting performance. Surfactants are often added to aqueous inks to modify the surface tension. However, the time-scales for drop formation in inkjet printing are short compared to the time-scales of the surfactant diffusion resulting a non-uniform surfactant distribution along the interface leading to surface tension gradients. We present both experiments and numerical simulations of inkjet break-up and drop formation in the presence of surfactants investigating both the surfactant transport on the interface and the influence of Marangoni forces on break-up dynamics. The numerical simulations were conducted using a modified version of the Lagrangian finite element developed by our previous work by including the solution for the transport equation for the surfactants over the free surface. During the initial phase of a “pull-push-pull” drive waveform, surfactants are concentrated at the front of the main drop with the trailing ligament being almost surfactant free. The resulting Marangoni stresses act to delay and can even prevent the break-off of the main drop from the ligament. We also examine and present some initial results on the effects of surfactants on the shape oscillations  of the main drop. Although there is little change to the oscillation frequency, the presence of surfactants significantly increases the rate of decay due to the rigidification of the surface, by modifying the internal flow within the droplet and enhancing the viscous dissipation.

Hadrien Oliveri

An optic ray theory for nerve durotaxis

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell’s law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).
The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method.
We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The core argument relies on the analysis of the asymptotic behavior of large random determinants in the exponential scale.

In this talk I will discuss a striking duality, T-duality, we discovered between two seemingly unrelated objects: the hypersimplex and the m=2 amplituhedron. We draw novel connections between them and prove many new properties. We exploit T-duality to relate their triangulations and generalised triangles (maximal cells in a triangulation). We subdivide the amplituhedron into chambers as the hypersimplex can be subdivided into simplices - both enumerated by Eulerian numbers. Along the way, we prove several conjectures on the amplituhedron and find novel cluster-algebraic structures, e.g. a generalisation of cluster adjacency.

This is based on the joint work with Lauren Williams and Melissa Sherman-Bennett https://arxiv.org/abs/2104.08254.

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).

The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method. 

We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The core argument relies on the analysis of the asymptotic behavior of large random determinants in the exponential scale.

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision while retaining good accuracy. In this talk we focus on explicit stabilised Runge-Kutta (ESRK) methods for parabolic PDEs as they are especially amenable to a mixed-precision treatment. However, some of the concepts we present can be extended more generally to Runge-Kutta (RK) methods in general.

Consider the problem $y' = f(t,y)$ and let $u$ be the roundoff unit of the low-precision used. Standard mixed-precision schemes perform all evaluations of $f$ in reduced-precision to improve efficiency. We show that while this approach has many benefits, it harms the convergence order of the method leading to a limiting accuracy of $O(u)$.

In this talk we present a more accurate alternative: a scheme, which we call $q$-order-preserving, that is unaffected by this limiting behaviour. The idea is simple: by using $q$ high-precision evaluations of $f$ we can hope to retain a limiting convergence order of $O(\Delta t^{q})$. However, the practical design of these order-preserving schemes is less straight-forward.

We specifically focus on ESRK schemes as these are low-order schemes that employ a much larger number of stages than dictated by their convergence order so as to maximise stability. As such, these methods require most of the computational effort to be spent for stability rather than for accuracy purposes. We present new $s$-stage order $1$ and $2$ RK-Chebyshev and RK-Legendre methods that are provably full-order preserving. These methods are essentially as cheap as their fully low-precision equivalent and they are as accurate and (almost) as stable as their high-precision counterpart.

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Consider an $n$ by $n$ random matrix $A$ with i.i.d entries. In this talk, we discuss some results on the magnitude of the smallest singular value of $A$, and, in particular, the problem of estimating the singularity probability when the entries of $A$ are discrete.

Riche—Williamson recently proved that the characters of tilting modules for GL_h are given by non-singular p-Kazhdan—Lusztig polynomials providing p>h.  This is equivalent to calculating the decomposition numbers for symmetric groups labelled by partitions with at most h columns.  We discuss how this result can be generalised to all cyclotomic quiver Hecke algebras via a new and explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson’s diagrammatic endomorphism algebras of Bott–Samelson bimodules. 

This allows us to give an elementary and explicit proof of the main theorem of Riche–Williamson’s recent monograph and extend their categorical equivalence to all cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza’s categorical blob conjecture.  Furthermore, it allows us to classify and construct the homogeneous simple modules of quiver Hecke algebras via BGG resolutions.   
 
This is joint work with A. Cox, A. Hazi, D.Michailidis, E. Norton, and J. Simental.  
 

Online social networks such asTwitter, Facebook, Instagram and TikTokserve as mediafor the spread of information between their users.We areinterested in developing models forthis information diffusion to gain a greater understanding of its drivers. Some models forthe spread ofonlinebehaviour and informationassume that the information behaves similarly to a virus, where infection is equally likely after each exposure, these dynamics are known as a simple contagion. In a simple contagion, the exposures are independent of each other.However,online adoption of some behaviour and content has been empirically observed to be more likely after multiple exposures from their network neighbours, the exposures are not independent of each other, we refer to this as a complex contagion.Analytically tractable descriptions of complex contagions havebeendeveloped for continuous-time dynamics. These extend mean-field and pair approximation methods to account for clustering in the network topologies; however, no such analogous treatments for discrete-time cascade processes exist using branching processes. We describe a novel definition of complex contagion adoption dynamics and show how to construct multi-type branching processeswhichaccount for clustering on networks. We achieve this by tracking the evolution of a cascade via different classes of clique motifs whichaccount for the different numbers of active, inactive and removed nodes. This description allows for extensive MonteCarlo simulations (which are faster than network-based simulations), accurate analytical calculation of cascade sizes, determination of critical behaviour and other quantities of interest

Consider an n by n random matrix A with i.i.d entries. In this talk, we discuss some results on the magnitude of the smallest singular value of A, and, in particular, the problem of estimating the singularity probability when the entries of A are discrete.

Standard worst-case rounding error bounds of most numerical linear algebra algorithms grow linearly with the problem size and the machine precision. These bounds suggest that numerical algorithms could be inaccurate at large scale and/or at low precisions, but fortunately they are pessimistic. We will review recent advances in probabilistic rounding error analyses, which have attracted renewed interest due to the emergence of low precisions on modern hardware as well as the rise of stochastic rounding.

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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 contact @email.

Neural controlled differential equations (Neural CDEs) are a continuous-time extension of recurrent neural networks (RNNs). They are considered SOTA for modelling functions on irregular time series, outperforming other ODE benchmarks (ODE-RNN, GRU-ODE-Bayes) in offline prediction tasks. However, current implementations are not suitable to be used in online prediction tasks, severely restricting the domains of applicability of this powerful modeling framework. We identify such limitations with previous implementations and show how said limitations may be addressed, most notably to allow for online predictions. We benchmark our online Neural CDE model on three continuous monitoring tasks from the MIMIC-IV ICU database, demonstrating improved performance on two of the three tasks against state-of-the-art (SOTA) non-ODE benchmarks, and improved performance on all tasks against our ODE benchmark.

Joint work with Patrick Kidger, Lingyi Yang, and Terry Lyons.

I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos--Holzegel--Rodnianski on the linear stability of the Schwarzschild family.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

 Given a solution of the Einstein equations, a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. I will then focus in the talk on recent results showing the locally Lipschitz inextendibility of FLRW models with particle horizons and spherically symmetric weak null singularities. The latter in particular apply to the spherically symmetric spacetimes constructed by Luk and Oh, improving their C^2-formulation of strong cosmic censorship to a locally Lipschitz formulation.