Past

We have recently proposed a new kind of neural network, called a Legendre Memory Unit (LMU) that is provably optimal for compressing streaming time series data. In this talk, I describe this network, and a variety of state-of-the-art results that have been set using the LMU. I will include recent results on speech and language applications that demonstrate significant improvements over transformers. I will discuss variants of the original LMU that permit effective scaling on current GPUs and hold promise to provide extremely efficient edge time series processing.

We forecast the realized covariance matrix of asset returns in the U.S. equity market by exploiting the predictive information of graphs in volatility and correlation. Specifically, we augment the Heterogeneous Autoregressive (HAR) model via neighborhood aggregation on these graphs. Our proposed method allows for the modeling of interdependence in volatility (also known as spillover effect) and correlation, while maintaining parsimony and interpretability. We explore various graph construction methods, including sector membership and graphical LASSO (for modeling volatility), and line graph (for modeling correlation). The results generally suggest that the augmented model incorporating graph information yields both statistically and economically significant improvements for out-of-sample performance over the traditional models. Such improvements remain significant over horizons up to one month ahead, but decay in time. The robustness tests demonstrate that the forecast improvements are obtained consistently over the different out-of-sample sub-periods, and are insensitive to measurement errors of volatilities.

Cayley submanifolds in Spin(7) geometry are an analogue and generalisation of complex submanifolds in Kähler geometry. In this talk we provide a glimpse into calibrated geometry, which encompasses both of these, and how it ties into the study of manifolds of special holonomy. We then focus on the deformation theory of compact and conically singular Cayleys. Finally we explain how to remove conical singularities via a gluing construction.

We study an infinite family of Massive Type IIA backgrounds that holographically describe the twisted compactification of N=(1,0) six-dimensional SCFTs to four dimensions. The analysis of the branes involved suggests a four dimensional linear quiver QFT, that deconstructs the theory in six dimensions. For the case in which the system reaches a strongly coupled fixed point, we calculate some observables that we compare with holographic results. Two quantities measuring the number of degrees of freedom for the flow across dimensions are studied.

Surface water waves of significant interest, such as tsunamis and solitary waves, are nonlinear and dispersive waves. Unluckily, the equations derived from first principles that describe the propagation of surface water waves, known as Euler's equations, are immensely hard to study. For this reason, several approximate systems have been proposed as mathematical alternatives. We show that among the numerous simplified systems of PDEs of water wave theory there is only one that is provably well-posed (in Hadamard’s sense) in bounded domains with slip-wall boundary conditions. We also show that the particular well-posed system obeys most of the physical laws that acceptable water wave equations must obey, and it is consistent with the Euler equations. For the numerical solution of our system we rely on a Galerkin/finite element method based on Nitsche's method for which we have proved its convergence. Validation with laboratory data is also presented.

We will finish off the term with a discussion of allies and community. You are encouraged to bring a friend who has never been to Mathematrix before!

We will show asymptotic analysis for hydrodynamic system, as Navier-Stokes-Fourier system, as a useful tool in in the situation when certain parameters in the system – called characteristic numbers – vanish or become infinite. The choice of proper scaling, namely proper system of reference units, the parameters determining the behaviour of the system under consideration allow to eliminate unwanted or unimportant for particular phenomena modes of motion. The main goal of many studies devoted to asymptotic analysis of various physical systems is to derive a simplified set of equations - simpler for mathematical or numerical analysis. Such systems may be derived in a very formal way, however we will concentrate on rigorous mathematical analysis. I will concentrate on low Mach number limits with so called ill-prepared data and I will present some results which concerns passage from compressible to incompressible models of fluid flow emphasising difficulties characteristic for particular problems. In particular we will discuss Navier-Stokes-Fourier system on varying domains, a multi-scale problem for viscous heat-conducting fluids in fast rotation and the incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids.

Hypergraphs for multiscale cycles in structured data

Iris Yoon

Understanding the spatial structure of data from complex systems is a challenge of rapidly increasing importance. Even when data is restricted to curves in three-dimensional space, the spatial structure of data provides valuable insight into many scientific disciplines, including finance, neuroscience, ecology, biophysics, and biology. Motivated by concrete examples arising in nature, I will introduce hyperTDA, a topological pipeline for analyzing the structure of spatial curves that combines persistent homology, hypergraph theory, and network science. I will show that the method highlights important segments and structural units of the data. I will demonstrate hyperTDA on both simulated and experimental data. This is joint work with Agnese Barbensi, Christian Degnbol Madsen, Deborah O. Ajayi, Michael Stumpf, and Heather Harrington.

Minmax Connectivity and Persistent Homology 

Ambrose Yim

We give a pipeline for extracting features measuring the connectivity between two points in a porous material. For a material represented by a density field f, we derive persistent homology related features by exploiting the relationship between dimension zero persistent homology of the density field and the min-max connectivity between two points. We measure how the min-max connectivity varies when spurious topological features of the porous material are removed under persistent homology guided topological simplification. Furthermore, we show how dimension one persistent homology encodes a relaxed notion of min-max connectivity, and demonstrate how we can summarise the multiplicity of connections between a pair of points by associating to the pair a sub-diagram of the dimension one persistence diagram.

Given a group of type ${\rm FP}_n$, one may ask if this property also holds for its subgroups. The BNS invariant is a subset of the character sphere that fully captures this information for subgroups that are kernels of characters. It also provides an interesting connection of finiteness properties of subgroups and group homology. In this talk I am going to give an introduction to this problem and present an attempt to generalize the BNS invariant to more subgroups than just the kernels of characters.

Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.

A coarse structure is a way of talking about "large-scale" properties.  It is encoded in a family of relations that often, but not always, come from a metric.  A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra.

In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra.  The general theory immediately applies to quantum metrics (suitably defined), but it is much richer.  We explain another source based on measure instead of metric, leading to the new, large, and easy-to-understand class of support expansion C*-algebras.

I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions.  I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

In 1973, Erdős conjectured the existence of high girth $(n,3,2)$-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős' conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth $(n,q,r)$-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph $G$ avoiding a given set of submatchings (which we view as a hypergraph $H$ where $V(H)=E(G)$). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when $H$ is a hypergraph with small codegrees (for which high girth designs is a specific instance). A number of applications in various areas follow from our main results including: Latin squares, high dimensional permutations, and rainbow matchings. This is joint work with Luke Postle.

This talk is motivated by computing correlations for domino tilings of the Aztec diamond.  It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).  

We will describe a new equivariant version of discrete Morse theory designed specially for quotient objects X/G which arise naturally in geometric group theory from actions of finite groups G on finite simplicial complexes X. Our main tools are (A) a reconstruction theorem due to Bridson and Haefliger which recovers X from X/G decorated with stabiliser data, and (B) a 2-categorical upgrade of discrete Morse theory which faithfully captures the underlying homotopy type. Both tools will be introduced during the course of the talk. This is joint work with Naya Yerolemou.

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus in semi-blind deblurring applications. In this setting, inexactness stems from the uncertainty in the parameters defining the blur, which are computed throughout the iterations. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. This is joint work with Silvia Gazzola (University of Bath).

In this talk I will discuss well-posedness of hyperbolic Cauchy problems with multiplicities and the role played by the lower order terms (Levi conditions). I will present results obtained in collaboration with Christian Jäh (Göttingen) and Michael Ruzhansky (QMUL/Ghent) on higher order equations and non-diagonalisable systems.

Isogeny-based cryptography is a candidate for post-quantum cryptography. The underlying hardness of isogeny-based protocols is the problem of computing endomorphism rings of supersingular elliptic curves, which is equivalent to the path-finding problem on the supersingular isogeny graph. Can path-finding be reduced to knowing just one endomorphism? An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this talk, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. This is joint work with Sarah Arpin, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange and Ha T. N. Tran.

Around 30 years ago, Lin defined an analog of the Casson invariant for knots. This invariant counts representations of the knot group into SU(2) which satisfy tr(ρ(m)) = c for some fixed c. As a function of c, the Casson-Lin invariant turns out to be given by the Levine-Tristram signature function.

If K is a small knot in S³, I'll describe a version of the Casson-Lin invariant which counts representations of the knot group into SL₂(R) with tr(ρ(m)) = c for c in [-2,2]. The sum of the SU(2) and SL₂(R) invariants is a constant h(K), independent of c. I'll discuss the proof of this fact and give some applications to the existence of real parabolic representations and left-orderings. This is joint work with Nathan Dunfield.

We introduce the mapping space signature, a generalization of the path signature for maps from higher dimensional cubical domains, which is motivated by the topological perspective of iterated integrals by K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to Jacobian equivalence classes of cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser.

Over the last years, two different approaches to construct symmetry algebras acting on the cohomology of Nakajima quiver varieties have been developed. The first one, due to Maulik and Okounkov, exploits certain Lagrangian correspondences, called stable envelopes, to generate R-matrices for an arbitrary quiver and hence, via the RTT formalism, an algebra called Yangian. The second one realises the cohomology of Nakajima varieties as modules over the cohomological Hall algebra (CoHA) of the preprojective algebra of the quiver Q. It is widely expected that these two approaches are equivalent, and in particular that the Maulik-Okounkov Yangian coincides with the Drinfel’d double of the CoHA.

Motivated by this conjecture, in this talk I will show how to identify the stable envelopes themselves with the multiplication map of a subalgebra of the appropriate CoHA. 

As an application, I will introduce explicit inductive formulas for the stable envelopes and use them to produce integral solutions of the elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equation associated to arbitrary quiver (ongoing project with G. Felder and K. Wang). Time permitting, I will also discuss connections with Cherkis bow varieties in relation to 3d Mirror symmetry (ongoing project with R. Rimanyi).

Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. However, little is known about the collective dynamical phenomena involving topological signals. Typically, topological signals of a given dimension are investigated and filtered using the corresponding Hodge Laplacians. In this talk, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions.  I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to Hodge Laplacians.   I will show that topological signals treated with the Hodge Laplacians or with the Dirac operator can undergo collective synchronization phenomena displaying different types of critical phenomena. Finally, I will show how the Dirac operator allows to couple the dynamics of topological signals of different dimension leading to the Dirac signal processing of signals defined on nodes, links and triangles of simplicial complexes. 

After reviewing different aspects of thermalization and chaos in holographic quantum systems, I will argue that universal aspects can be captured using an effective field theory framework that shares similarities with hydrodynamics. Focusing on the quantum butterfly effect, I will explain how to develop a simple effective theory of the 'scramblon' from path integral considerations. I will also discuss applications of this formalism to shockwave scattering in black hole backgrounds in AdS/CFT.

Arguably some of the most interesting phenomena in fluid dynamics, both from a mathematical and a physical perspective, stem from the interplay between a fluid and its boundaries. This talk will present some examples of how boundary effects lead to remarkable outcomes.  Singularities can form in finite time as a consequence of the continuum assumption when material surfaces are in smooth contact with horizontal boundaries of a fluid under gravity. For fluids with chemical solutes, the presence of boundaries impermeable to diffusion adds further dynamics which can give rise to self-induced flows and the formation of coherent structures out of scattered assemblies of immersed bodies. These effects can be analytically and numerically predicted by simple mathematical models and observed in “simple” experimental setups.