Past

Primitive elements are elements that are part of a basis for a free group. We present the classical Whitehead algorithm for the recognition of such elements, and discuss the ideas behind the proof. We also present a second algorithm, more recent and completely different in the approach.

I will describe a general method for comparing the counting functions of determinantal point processes in terms of trace class norm distances between their kernels (and review what all of those words mean). Then I will outline joint work with Elizabeth Meckes using this method to prove a version of a self-similarity property of eigenvalues of Haar-distributed unitary matrices conjectured by Coram and Diaconis.  Finally, I will discuss ongoing work by my PhD student Kyle Taljan, bounding the rate of convergence for counting functions of GUE eigenvalues to the Sine or Airy process counting functions.

Smectic A liquid crystals are of great interest in physics for their striking defect structures, including curvature walls and focal conics. However, the mathematical modeling of smectic liquid crystals has not been extensively studied. This work takes a step forward in understanding these fascinating topological defects from both mathematical and numerical viewpoints. In this talk, we propose a new (two- and three-dimensional) mathematical continuum model for the transition between the smectic A and nematic phases, based on a real-valued smectic order parameter for the density perturbation and a tensor-valued nematic order parameter for the orientation. Our work expands on an idea mentioned by Ball & Bedford (2015). By doing so, physical head-to-tail symmetry in half charge defects is respected, which is not possible with vector-valued nematic order parameter.

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 send email to @email.

I will explain how the representation theory of rational Cherednik algebras interacts with the commutative algebra of certain subspace arrangements arising from the reflection arrangement of a complex reflection group. Potentially, the representation theory allows one to study both qualitative questions (e.g., is the arrangement Cohen-Macaulay or not?) and quantitative questions (e.g., what is the Hilbert series of the ideal of the arrangement, or even, what are its graded Betti numbers?), by applying the tools (such as orthogonal polynomials, Kazhdan-Lusztig characters, and Dirac cohomology) that representation theory provides. This talk is partly based on joint work with Susanna Fishel and Elizabeth Manosalva.

Modular and hierarchical structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular, or "community", structures have included identifying fundamental limits of detectability by formally defining community structure using probabilistic generative models. Detecting hierarchical community structure introduces additional challenges alongside those inherited from community detection. Here we present a theoretical study on hierarchical community structure in networks, which has thus far not received the same rigorous attention. We address the following questions: 1) How should we define a valid hierarchy of communities? 2) How should we determine if a hierarchical structure exists in a network? and 3) how can we detect hierarchical structure efficiently? We approach these questions by introducing a definition of hierarchy based on the concept of stochastic externally equitable partitions and their relation to probabilistic models, such as the popular stochastic block model. We enumerate the challenges involved in detecting hierarchies and, by studying the spectral properties of hierarchical structure, present an efficient and principled method for detecting them.

https://arxiv.org/abs/2009.07196 (15 sept.)

Motivated by the advent of machine learning, the last few years saw the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely susceptible to rounding errors. An application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs), but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. In this talk we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite difference methods using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the numerical scheme to reduce rounding errors and we present \emph{a priori} estimates for local and global rounding errors. Let $u$ be the roundoff unit. While the worst-case local errors are $O(u)$ with respect to the discretization parameters, the RtN and SR error behaviour is substantially different. We show that the RtN solution is discretization, initial condition and precision dependent, and always stagnates for small enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, the leading order errors introduced by SR are zero-mean, independent in space and mean-independent in time, making SR resilient to stagnation and rounding error accumulation. In fact, we prove that for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are essentially bounded (up to logarithmic factors) in higher dimensions.

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 send email to @email.

We propose a subspace Gauss-Newton method for nonlinear least squares problems that builds a sketch of the Jacobian on each iteration. We provide global rates of convergence for regularization and trust-region variants, both in expectation and as a tail bound, for diverse choices of the sketching matrix that are suitable for dense and sparse problems. We also have encouraging computational results on machine learning problems.

We construct a class of Cauchy initial data without (marginally) trapped surfaces whose future evolution is a trapped region bounded by an apparent horizon, i.e., a smooth hypersurface foliated by MOTS. The estimates obtained in the evolution lead to the following conditional statement: if Kerr Stability holds, then this kind of initial data yields a class of scale critical vacuum examples of Weak Cosmic Censorship and the Final State Conjecture. Moreover, owing to estimates for the ADM mass of the data and the area of the MOTS, the construction gives a fully dynamical vacuum setting in which to study the Spacetime Penrose Inequality. We show that the inequality is satisfied for an open region in the Cauchy development of this kind of initial data, which itself is controllable by the initial data. This is joint work with Nikos Athanasiou https://arxiv.org/abs/2009.03704.

Let $r>3$ be an integer and consider the following game on the complete graph $K_n$ for $n$ a multiple of $r$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one and Breaker claims $b$ edges. Maker wins if her graph contains a $K_r$-factor, that is a collection of $n/r$ vertex-disjoint copies of $K_r$, and Breaker wins otherwise. In other words, we consider the $b$-biased $K_r$-factor Maker-Breaker game. We show that the threshold bias for this game is of order $n^2/(r+2)$. This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen, Böttcher, Kohayakawa, Naves and Person who resolved the case $r=3$ or $4$ up to a logarithmic factor.
    Joint work with Rajko Nenadov.

I will introduce recent work on the two- and three-dimensional uniform spanning trees (USTs) that establish the laws of these random objects converge under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. (In the three-dimensional case, the scaling result is currently only known along a particular scaling sequence.) I will also discuss various properties of the intrinsic metrics and measures of the limiting spaces, including their Hausdorff dimension, as well as the scaling limits of the random walks on the two- and three-dimensional USTs. In the talk, I will attempt to emphasise where the differences lie between the two cases, and in particular the additional challenges that arise when it comes to the three-dimensional model.
    The two-dimensional results are joint with Martin Barlow (UBC) and Takashi Kumagai (Kyoto). The three-dimensional results are joint with Omer Angel (UBC) and Sarai Hernandez-Torres (UBC).

A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. However, neither the polynomial nor what sufficiently large means are understood in general. We obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}$ whose convex hull is a simplex; previously such results were only available for $d = 1$. Our approach also gives information about the structure of $hA$, answering a recent question posed by Granville and Shakan. The work is joint with Leo Goldmakher at Williams College.

In Joint work with Modj Shokrian-Zini we study (numerically) our proposal that interacting physics can arise from single particle quantum Mechanics through spontaneous symmetry breaking SSB. The staring point is the claim the difference between single and many particle physics amounts to the probability distribution on the space of Hamiltonians. Hamiltonians for interacting systems seem to know about some local, say qubit, structure, on the Hilbert space, whereas typical QM systems need not have such internal structure. I will discuss how the former might arise from the latter in a toy model. This story is intended as a “prequel” to the decades old reductionist story in which low energy standard model physics is supposed to arise from something quite different at high energy. We ask the question: Can interacting physics itself can arise from something simpler.

Abstract: A recent paradigm views deep neural networks as discretizations of certain controlled ordinary differential equations, sometimes called neural ordinary differential equations. We make use of this perspective to link expressiveness of deep networks to the notion of controllability of dynamical systems. Using this connection, we study an expressiveness property that we call universal interpolation, and show that it is generic in a certain sense. The universal interpolation property is slightly weaker than universal approximation, and disentangles supervised learning on finite training sets from generalization properties. We also show that universal interpolation holds for certain deep neural networks even if large numbers of parameters are left untrained, and are instead chosen randomly. This lends theoretical support to the observation that training with random initialization can be successful even when most parameters are largely unchanged through the training. Our results also explore what a minimal amount of trainable parameters in neural ordinary differential equations could be without giving up on expressiveness.

Joint work with Martin Larsson, Josef Teichmann.

The Spin(7) and SU(4) structures on a Calabi-Yau 4-fold give rise to certain first order PDEs defining special Yang-Mills connections: the Spin(7) instanton equations and the Hermitian Yang-Mills (HYM) equations respectively. The latter are stronger than the former. In 1998 C. Lewis proved that -over a compact base space- the existence of an HYM connection implies the converse. In this talk we demonstrate that the equivalence of the two gauge-theoretic problems fails to hold in generality. We do this by studying the invariant solutions on a highly symmetric noncompact Calabi-Yau 4-fold: the Stenzel manifold. We give a complete description of the moduli space of irreducible invariant Spin(7) instantons with structure group SO(3) on this space and find that the HYM connections are properly embedded in it. This moduli space reveals an explicit example of a sequence of Spin(7) instantons bubbling off near a Cayley submanifold. The missing limit is an HYM connection, revealing a potential relationship between the two equation systems.

 I will review what is known and not known about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives. I will then explain some recent results which relate the joint moments to an interesting class of measures, known as Hua-Pickrell measures. This leads to the proof of a conjecture, due to Chris Hughes in 2000, concerning the asymptotics of the joint moments, as well as establishing a connection between the measures in question and one of the Painlevé equations.

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

 Inherent fluctuations may play an important role in biological and chemical systems when the copy number of some chemical species is small. This talk will present the recent work on the stochastic modeling of reaction-diffusion processes in biochemical systems. First, I will introduce several stochastic models, which describe system features at different scales of interest. Then, model reduction and coarse-graining methods will be discussed to reduce model complexity. Next, I will show multiscale algorithms for stochastic simulation of reaction-diffusion processes that couple different modeling schemes for better efficiency of the simulation. The algorithms apply to the systems whose domain is partitioned into two regions with a few molecules and a large number of molecules.

Topological data analysis has proven to be an effective tool in machine learning, supporting the analysis of neural networks, but also driving the development of new algorithms that make use of topological features. Graph classification is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will briefly summarise recent advances in topology-based graph classification, focussing equally on ’shallow’ and ‘deep’ approaches. Starting from an intuitive description of persistent homology, we will discuss how to incorporate topological features into the Weisfeiler–Lehman colour refinement scheme, thus obtaining a simple feature-based graph classification algorithm. We will then build a bridge to graph neural networks and demonstrate a topological variant of ‘readout’ functions, which can be learned in an end-to-end fashion. Care has been taken to make the talk accessible to an audience that might not have been exposed to machine learning or topological data analysis.
 

We consider continuous time financial models with continuous paths, in a pathwise setting using functional Ito calculus. We look at applications of optimal transport duality in context of robust pricing and hedging and that of calibration. First, we explore exntesions of the discrete-time results in Aksamit et al. [Math. Fin. 29(3), 2019] to a continuous time setting. Second, we addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton--Jacobi--Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.

Based on joint works with Ivan Guo, Gregoire Loeper, Shiyi Wang.
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Tissue folding during animal development involves an intricate interplay
of cell shape changes, cell division, cell migration, cell
intercalation, and cell differentiation that obfuscates the underlying
mechanical principles. However, a simpler instance of tissue folding
arises in the green alga Volvox: its spherical embryos turn themselves
inside out at the close of their development. This inversion arises from
cell shape changes only.

In this talk, I will present a model of tissue folding in which these
cell shape changes appear as variations of the intrinsic stretches and
curvatures of an elastic shell. I will show how this model reproduces
Volvox inversion quantitatively, explains mechanically the arrest of
inversion observed in mutants, and reveals the spatio-temporal
regulation of different biological driving processes. I will close with
two examples illustrating the challenges of nonlinearity in tissue
folding: (i) constitutive nonlinearity leading to nonlocal elasticity in
the continuum limit of discrete cell sheet models; (ii) geometric
nonlinearity in large bending deformations of morphoelastic shells.
 

One of the standard methods for the solution of elliptic boundary value problems calls for reformulating them as systems of integral equations.  The integral operators that arise in this fashion typically have singular kernels, and, in many cases of interest, the solutions of these equations are themselves singular.  This makes the accurate discretization of the systems of integral equations arising from elliptic boundary value problems challenging.

Over the last decade, Generalized Gaussian quadrature rules, which are n-point quadrature rules that are exact for a collection of 2n functions, have emerged as one of the most effective tools for discretizing singular integral equations. Among other things, they have been used to accelerate the discretization of singular integral operators on curves, to enable the accurate discretization of singular integral operators on complex surfaces and to greatly reduce the cost of representing the (singular) solutions of integral equations given on planar domains with corners.

We will first briefly outline a standard method for the discretization of integral operators given on curves which is highly amenable to acceleration through generalized Gaussian quadratures. We will then describe a numerical procedure for the construction of Generalized Gaussian quadrature rules.

Much of this is joint work with Zydrunas Gimbutas (NIST Boulder) and Vladimir Rokhlin (Yale University).

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 send email to @email.

In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk we will introduce this problem of uniqueness of asymptotic limits and discuss joint work with Melanie Rupflin and Peter M. Topping in which we examined the situation for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface.

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The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with Jose Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

I will explain what it means for a manifold to have an affine structure and give an introduction to Benzecri's theorem stating that a closed surface admits an affine structure if and only if its Euler characteristic vanishes. I will also talk about an algebraic-topological generalization, due to Milnor and Wood, that bounds the Euler class of a flat circle bundle. No prior familiarity with the concepts is necessary.

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer n to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

Classical random matrix theory begins with a random matrix model and analyzes the distribution of the resulting eigenvalues.  In this work, we treat the reverse question: if the eigenvalues are specified but the matrix is "otherwise random", what do the entries typically look like?  I will describe a natural model of random matrices with prescribed eigenvalues and discuss a central limit theorem for projections, which in particular shows that relatively large subcollections of entries are jointly Gaussian, no matter what the eigenvalue distribution looks like.  I will discuss various applications and interpretations of this result, in particular to a probabilistic version of the Schur--Horn theorem and to models of quantum systems in random states.  This work is joint with Mark Meckes.