Past Colloquia

The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

Tensor decomposition is an unsupervised learning methodology that has applications in a wide variety of domains, including chemometrics, criminology, and neuroscience. We focus on low-rank tensor decomposition using  canonical polyadic or CANDECOMP/PARAFAC format. A low-rank tensor decomposition is the minimizer according to some nonlinear program. The usual objective function is the sum of squares error (SSE) comparing the data tensor and the low-rank model tensor. This leads to a nicely-structured problem with subproblems that are linear least squares problems which can be solved efficiently in closed form. However, the SSE metric is not always ideal. Thus, we consider using other objective functions. For instance, KL divergence is an alternative metric is useful for count data and results in a nonnegative factorization. In the context of nonnegative matrix factorization, for instance, KL divergence was popularized by Lee and Seung (1999). We can also consider various objectives such as logistic odds for binary data, beta-divergence for nonnegative data, and so on. We show the benefits of alternative objective functions on real-world data sets. We consider the computational of generalized tensor decomposition based on other objective functions, summarize the work that has been done thus far, and illuminate open problems and challenges. This talk includes joint work with David Hong and Jed Duersch.

Let $V$ be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let $[V]$ be the vector space of complex valued functions on $V$ and let $[V]_{\mathbb Z}$ be the subgroup of $[V]$ consisting of integer valued functions. We show that there exists a Z-basis of $[V]_{\mathbb Z}$ consisting of characteristic functions of certain explicit isotropic subspaces of $V$ such that the matrix of the Fourier transform from $[V]$ to $[V]$ with respect to this basis is triangular. This continues the tradition started by Hermite who described eigenvectors for the Fourier transform over real numbers.

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.

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

The strong cosmic censorship conjecture is a fundamental open problem in classical general relativity, first put forth by Roger Penrose in the early 70s. This is essentially the question of whether general relativity is a deterministic theory. Perhaps the most exciting arena where the validity of the conjecture is challenged is the interior of rotating black holes, and there has been a lot of work in the past 50 years in identifying mechanisms ensuring that at least some formulation of the conjecture be true. It turns out that when a nonzero cosmological constant Λ is added to the Einstein equations, these underlying mechanisms change in an unexpected way, and the validity of the conjecture depends on a detailed understanding of subtle aspects of black hole scattering theory, surprisingly involving, in the case of negative Λ, some number theory. Does strong cosmic censorship survive the challenge of non-zero Λ? This talk will try to address this Question!

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. 

I will report on my most recent results  with Jeremie Szeftel and Elena Giorgi which conclude the proof of the nonlinear, unconditional, stability of slowly rotating Kerr metrics. The main part of the proof, announced last year, was conditional on results concerning boundedness and decay estimates for nonlinear wave equations. I will review the old results and discuss how the conditional results can now be fully established.

In this talk, we will provide an overview of results in the setting of granular crystals, consisting of spherical beads interacting through nonlinear elastic spring-like forces. These crystals are used in numerous engineering applications including, e.g., for the production of "sound bullets'' or the examination of bone quality. In one dimension we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will analyze the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria for traveling waves in lattices. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with state-of-the-art experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

The puzzling phenonenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating medium. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.

We discuss a recent generalization of the classical 3-dimensional light bulb theorem to 4-dimensions. We connect this with fundamental questions about knotting of surfaces in 4-dimensional manifolds as well as new directions regarding knotting of 3-balls in 4-manifolds.

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to better understand how viruses form, evolve and infect their hosts, and point the way to novel antiviral solutions.

In this talk, I will present an overarching theory of virus architecture, that contains the seminal Caspar Klug theory as a special case and solves long-standing open problems in structural virology. Combining insights into virus structure with a range of different mathematical modelling techniques, such as Gillespie algorithms, I will show how viral life cycles can be better understood through the lens of viral geometry. In particular, I will discuss a recent model for genome release from the viral capsid. I will also demonstrate the instrumental role of the Hamiltonian path concept in the discovery of a virus assembly mechanism that occurs in many human pathogens, such as Picornaviruses – a family that includes the common cold virus– and Hepatitis B and C virus. I will use multi-scale models of a viral infection and implicit fitness landscapes in order to demonstrate that therapeutic interventions directed against this mechanism have advantages over conventional forms of anti-viral therapy. The talk will finish with a discussion of how the new mathematical and mechanistic insights can be exploited in bio-nanotechnology for applications in vaccination and gene therapy.

There is a very simple algorithm for the inference of posteriors for probability models on trees. This algorithm, known as "Belief Propagation" is widely used in coding theory, in machine learning, in evolutionary inference, among many other areas. The talk will be devoted to the analysis of Belief Propagation in some of the simplest probability models. We will highlight the interplay between Belief Propagation, linear estimators (statistics), the Kesten-Stigum bound (probability) and Replica Symmetry Breaking (statistical physics). We will show how the analysis of Belief Propagation allowed proof phase transitions for phylogenetic reconstruction in evolutionary biology and developed optimal algorithms for inference of block models. Finally, we will discuss the computational complexity of this 'simple' algorithm.

The subject of $p$-adic cohomologies is over fifty years old. Many new developments have recently occurred. I will mostly limit myself to discussing some pertaining to the de Rham-Witt complex. After recalling the historical background and the basic results, I will give an overview of the new approach of Bhatt, Lurie and Mathew.

Solidification processes of liquid metal alloys,  bubble dynamics (as in Taylor flows), pinch-offs of liquid-liquid jets, the formation of polymeric membranes, or the structure of high concentration photovoltaic cells are described by the dynamics of multiphase fluids. On the other hand, in applications such as mass spectrometry, lab-on-a-chip, and electro-fluidic displays, fluids on the micro-scale associated with a dielectric medium are of interest. Moreover, in many of these applications one is interested in influencing (or controlling) the underlying phenomenon in order to reach a desired goal. Examples for the latter could be the porosity structure of a polymeric membrane to achieve certain desired filtration properties of the membrane, or to optimize a microfluidic device for the transport of pharmaceutical agents.

A promising mathematical model for the behavior of multiphase flows associated with the applications mentioned above is given by a phase-field model of Cahn-Hilliard / Navier-Stokes (CHNS) type. Some strengths of phase field (or diffuse interface) approaches are due to their ability to overcome both, analytical difficulties of topological changes, such as, e.g., droplet break-ups or the coalescence of interfaces, and numerical challenges in capturing the interface dynamics between the fluid phases. Deep quenches in solidification processes of liquid alloys or rapid wall hardening in the formation of polymer membranes ask for non-smooth energies in connection with Cahn-Hilliard models. Analytically, this gives rise to a variational inequality coupled to the equations of hydrodynamics, thus yielding a non-smooth system (in the sense that the map associated with the underlying operator equation is not necessarily Frechet differentiable). In contrast to phase-field approaches,
one may consider sharp interface models. In view of this, our microfluidic applications alluded to above are formulated in terms of  sharp interface models and Hele-Shaw flows. In this context, we are particularly interested in applications of electrowetting on dielectric (EWOD) with contact line pinning. The latter phenomenon resembles friction, yields a variational inequality of the second kind, and – once again – it results in an overall nonsmooth mathematical model of the physical process.

   In both settings described above, optimal control problems are relevant in order to influence the underlying physical process to approach a desired system state.  The associated optimization problems are delicate as the respective constraints involve non-smooth structures which render the problems degenerate and prevent a direct application of sophisticated tools for the characterization of solutions. Such characterizations are, however, of paramount importance in the design of numerical solution schemes.

This talk addresses some of the analytical challenges associated with optimal control problems involving non-smooth structures, offers pathways to solutions, and it reports on numerical results for both problem classes introduced above.
 

In my talk I will discuss modular properties of false theta functions. Due to a wrong sign factor these are not directly seen to be modular, however there are ways to repair this. I will report about this in my talk.

A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis - notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions - as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0  of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.

The moments of characteristic polynomials play a central role in Random Matrix Theory.  They appear in many applications, ranging from quantum mechanics to number theory.  The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. Current techniques for gossiping are designed to deal with worst-case scenarios, which is irrelevant in applications to distributed statistical learning and denoising in sensor networks. We design second-order gossip methods tailor-made for the case where the real values are i.i.d. samples from the same distribution. In some regular network structures, we are able to prove optimality of our methods, and simulations suggest that they are efficient in a wide range of random networks. Our approach of gossip stems from a new acceleration framework using the family of orthogonal polynomials with respect to the spectral measure of the network graph (joint work with Raphaël Berthier, and Pierre Gaillard).

Mathematical models based on first principles can describe the interaction between electrical, mechanical and fluid-dynamical processes occurring in the heart. This is a classical multi-physics problem. Appropriate numerical strategies need to be devised to allow for an effective description of the fluid in large and medium size arteries, the analysis of physiological and pathological conditions, and the simulation, control and shape optimisation of assisted devices or surgical prostheses. This presentation will address some of these issues and a few representative applications of clinical interest.

It is at first sight surprising that a minimizer of an integral of the calculus of variations may make the integrand infinite somewhere.

This talk will discuss some examples of this phenomenon, how it can be related to material defects, and related open questions from nonlinear elasticity and the theory of liquid crystals.

Details to follow

The so-called moonshine phenomenon relates modular forms and finite group representations. After the celebrated monstrous moonshine, various new examples of moonshine connection have been discovered in recent years. The study of these new moonshine examples has revealed interesting connections to K3 surfaces, arithmetic geometry, and string theory.  In this colloquium I will give an overview of these recent developments.