The Khintchine inequalities are basic and fundamental inequalities at the interface of probabilties and functional analysis. We will first review some classical facts about them. Then, we will introduce what is meant by non commutative analysis by illustrating it with matrices. It took almost 30 years to get a full version of the non commutative Khintchine inequalities. Their formulation is at the heart a truly non commutative difficulty. We will explain the history that led to a satisfactory but not totally full understanding. The talk only requires very basic notions.

 

The Renaissance famously brought us amazingly realistic perspective art. Creating that art was the spark from which projective geometry caught fire and grew. This talk looks directly at projective geometry as a tool to illuminate the way we see the world around us, whether we look with our eyes, with our cameras, or with the computer (via our favorite animated movies). One of the surprising results of projective geometry is that it implies that every quadrangle (whether convex or not) is the perspective image of a square. We will describe implications of this result for computer vision, for photogrammetry, for applications of piecewise planar cones, and of course for perspective art and projective geometry.

 

In the late 1800s, Y. Zolotarev (a student of Chebyshev) posed and solved two important rational approximation problems. These problems (and variations of them) arise in many modern applications in numerical linear algebra, signal processing, and computational mathematics. This talk highlights the role of Zolotarev’s problems in modern computing and illustrates how classical ideas in approximation theory, such as conformal mapping, can be put to use in numerical contexts. We focus primarily on Zolotarev’s so-called 4th problem, the best approximation to the sign function, and use it to inspire and develop new, spectrally accurate methods for solving the spectral fractional Poisson equation.

 

This is joint with Keaton Hamm and Armenak Petrosyan. We study the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on B(L2(T)). We also provide characterizations of unconditional convergence of the Fourier series in the SOT and WOT. These considerations also give rise to some interesting questions regarding weaker versions of the rearrangement problem. Towards the end of this talk, I might indicate how these problems can be generalized from Z^d to arbitrary general groups.

 

We demonstrate how a Weyl-algebraic treatment to Truesdell’s F-equation theory published in 1948 to derive generating functions of classical special functions would allow a unified treatment of both some classical special functions and their (new) difference analogues. Our approach also illustrates that D-modules is a nature language to describe many classical special functions. In this talk we shall illustrate how the methodology can be applied to Bessel functions and Bessel polynomials, etc. Some realistic applications will be discussed.

 

For which cardinals κ is there a partition of the real line into precisely κ Borel sets? If the Continuum Hypothesis holds, then the answer to this question is fairly simple: all κ ≤ |R|. But in other models of set theory, where the Continuum Hypothesis fails, the answer to this question is surprisingly subtle, involving forcing constructions, singular cardinal combinatorics, and large cardinal axioms. In this talk, I will survey some of the history of the question, along with some recent developments.

 

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A necessary and sufficient condition on a singular kernel for the continuity of an integral operator in Hölder spaces and applications to the double layer potential

 

Abstract: Volume and layer potentials are integrals on a subset Y of the Euclidean space Rⁿ that depend on a variable in a subset X of Rⁿ. Here we follow a unified approach by assuming that X and Y are subsets of a metric space M and that Y is equipped with a measure ν that satisfies upper Ahlfors growth conditions that include non-doubling measures as done by J. García-Cuerva and A.E. Gatto in a series of papers in case X=Y and for standard kernels, and we prove a necessary and sufficient condition on the kernel for an integral operator to be bounded in Hölder spaces.

Then we present an application to the case of the double layer potential that is associated to the fundamental solution of an arbitrary constant coefficient second order elliptic operator with real principal coefficients.

 

We are going to explore two nonlocal games, CHSH (Clauser-Horne-Shimony-Holt) and magic squares. For CHSH, we will present a quantum strategy that outperforms the classical strategy. As for magic squares, a perfect (100% success rate) quantum strategy exists. (Almost) no background knowledge required.

 

A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on systems of d intervals on the real line. Classification of periodic trajectories leads to a new combinatorial object: billiard partitions.

The case study of trajectories of small periods n, d ≤ n ≤ 2d is given. A complete catalog of billiard trajectories with small periods is provided for d = 3.

The talk is based on the following papers:

V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications. Mathematical Physics, 2019, Vol. 372, p. 183-211.

G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, arXiv: 1908.01026, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

 

Let G be a complex reductive group with Lie algebra g and let V be a finite-dimensional G-module. There is a natural homogeneous quadratic moment map μ:V⊕V^∗→g^∗ whose zero fiber μ⁻¹(0) is called the (complex) shell N, and the (complex) symplectic quotient associated to V is the affine GIT quotient N//G. For many cases of G and V, it has been demonstrated by Bellamy–Schedler, Becker, Terpereau, and others that N//G has symplectic singularities; equivalently, N//G is Gorenstein with rational singularities and its smooth locus admits a holomorphic symplectic form. In some of the cases, the variety N already has rational singularities. We will present recent results demonstrating that for a given G, the shell N has rational singularities and the symplectic quotient N//G has symplectic singularities in the case of “most” G-modules V. This includes the important case where G is semisimple and V=pg with p≥2, optimizing results of Aizenbud–Avni and generalizing a theorem of Budur. We will also discuss applications indicating that the representation varieties of surface groups have rational singularities and the corresponding character varieties have symplectic singularities.

 

By now, the “Cheshire cat” description of quantum groups is well known — a quantum group is not a group at all, but something that remains when a group has faded away, leaving an algebra of functions behind. Quantum groups corresponding to a group G typically appear in parametrized families, where an algebra R of actual functions on G appears for trivial values of the parameters and “deformed” rings of functions appear for other parameters. These quantum groups are called “quantized algebras of functions” on G, and the classical function algebra R is called the “semiclassical limit” of the family of quantum groups. Traces of the quantum groups survive in such semiclassical limits in the form of Poisson structures which record, to first-order in a suitable sense, the noncommutativity of the considered quantum groups. One can (conjecturally) see the “noncommutative geometry” of the quantum groups in the Poisson geometry of the semiclassical limits.

The aim of this talk is to introduce the above ideas, present a few examples, and discuss relationships among these concepts.

 

With the rise of big data and machine learning revolutionizing countless industries, the demand for mathematical and
quantitative skills in the workforce is at an all-time high. But navigating this highly competitive and ever-evolving space presents a number of challenges that require one to broaden their conceptual understanding of mathematics and its place in the world. What has been termed “the unreasonable effectiveness of mathematics” has, in the context of data science, revolutionized the way we must think about the scientific process. And the economic pressures that are applied to this process in a fast-paced business environment necessitate that we step outside a philosophy of mathematics which relegates math to neat little silos of controlled formal truth. Mathematicians in corporate and industry positions must learn to embrace their role as mathematical story-tellers: understanding the interplay of mathematical formalism, technical pragmatism, and human desire at every juncture to weave a story of order and meaning out of the chaotic landscape of data and possibility.

 

In this talk, I shall first give a brief review of the uniform asymptotic approximation (UAA) method for solving second-order ordinary differential equations, and then apply it to the studies of gravitational waves, quasi-normal modes of black holes, and power spectra of cosmological perturbations in the framework of quantum gravity. This method provides one of the most accurate analytic computations known in the literature. In particular, when applying it to quantum cosmology, we find that the upper bound of errors is no larger than 0.15% up to the third-order approximation, which is sufficiently accurate for the current and forthcoming cosmological observations. Such analytic investigations shall also lead to much better and deeper understanding and insight of the problems, and provide possible deep machine learning platforms. At the end of my talk, I shall also mention a couple of questions that we are currently facing, and for which we thus seek new ideas.

This report is partially based on a collaboration of Drs. Gerald Cleaver (Physics), Klaus Kirsten, Tim Sheng, and Anzhong Wang (Physics), and was supported in part by Baylor University through CASPER.

 

I will survey several applications of mathematical analysis to issues arising in biology and oceanography. As time permits, these will include a model for Pulmonary Arterial Hypertension, the formation of Rogue Waves, modeling Arterial Blood Flow and propagation of Large Amplitude Internal Waves. While these may not sound closely connected, a common theme will emerge.

 

This talk focuses on the underlying mathematics of the aeroelastic phenomenon flutter—i.e., the way that an elastic structure may become unstable in the presence of an adjacent flow of air. Under certain circumstances, a feedback occurs between elastic deformations and pressure dynamics in the airflow, resulting in sustained oscillations. A canonical example was seen in the Tacoma Narrows bridge (Washington, USA), which collapsed in 1940 while fluttering in 65 kph winds. Flutter is typically discussed in the context of aero-mechanical systems: buildings and bridges in wind, and flight systems. However, applications also arise in biology (snoring and sleep apnea), and in alternative energy technologies (piezoelectric energy harvesters).

We will look at a variety of flow-structure interaction models which are partial differential equation systems coupled via an interface. After a brief discussion of relevant modeling, we will examine well-posedness and long-time behavior properties of PDE solutions for three different physical configurations that can exhibit aeroelastic flutter: (1) projectile paneling, (2) a bridge deck, (3) an elastic energy harvester. From a rigorous point of view, we attempt to capture the mechanism that gives rise to the flutter instability. Additionally, when flutter occurs, we attempt to describe its qualitative features through a dynamical systems approach, as well as how to prevent it or bring it about (stability).


 

Mean-field games (MFG) theory is a framework to model and study huge populations of agents that play non-cooperative differential games. I will discuss some of the recent developments in applying spectral methods for a numerical and possibly theoretical resolution of MFG systems with nonlocal interactions among agents. I will also draw connections with kernel methods in machine learning.

 

To any quiver, we can associate its category of finite-dimensional (nilpotent) representations over the field with one element. This category shares many basic properties with its analog over a field: in particular, a version of the Krull-Schmidt Theorem is satisfied. Inspired by the classical Tame-Wild Dichotomy for finite-dimensional algebras, we discuss a stratification of quivers based on the growth of their indecomposable F1-representations. In particular, we classify all quivers of bounded representation type over F1 and provide a functorial interpretation for unbounded quivers. As a consequence, we develop a general framework for interpreting F1-representations as certain quiver maps, which allows for a more combinatorial description of the Ringel-Hall algebras associated to these categories.

 

A mean field type control system is a mathematical model of many similar agents acting in concert. In the talk, I will consider the case when the dynamics of each agent is described by an ODE, whereas the phase space is a flat torus. This mean field type control problem can be regarded as a deterministic control problem in the space of probability measures. Its analysis with classical methods like dynamic programming requires such notions as derivative with respect to probability measure. I will discuss the approximation of the mean field type control system by the finite dimensional control system that describes the dynamics of the distribution of agents in the case when each agent moves according to continuous-time Markov chain defined on some lattice with transition rates on current distribution of agents. The points I would like to highlight are: evaluation of the approximation rate and construction of the approximating Markov chain acting on regular lattice.

 

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radial symmetry of solutions to the Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. Some generalizations of the inequality to include singular terms or more general surfaces will also be presented.

 

American education seems to be in permanent crisis. News reports tell us that schools and teachers are failing; international comparisons show American students are near the bottom; major corporations complain they cannot find qualified workers. Politicians and policymakers urge us to take immediate and radical action to address the crisis. Are things really so dire? The evidence for an education crisis is surprisingly ambiguous. What drives this apocalyptic view of education? What are the consequences of manufacturing a crisis where there isn’t one? And how can we solve education’s real problems with less melodrama and more common sense?

 

Entropy and its variants are important measures of information in both classical- and quantum information theory. For quantum systems, entropy quantities such as von Neumann entropy naturally relates to noncommutative L_p-norms. In this talk, we will discuss connection between various quantum entropies and L_p-norms. Such connections have found many applications in quantum information theory. I will talk about applications of L_p-norms in estimating quantum channel capacity and entropic uncertainty relations.

 

In this talk we discuss the topological structure of the spectrum of almost periodic Schrödinger operators, both in one dimension and in higher dimensions. The problem is quite well understood in the one-dimensional case and the talk will briefly describe some of the known results. The question is significantly less well understood in higher dimensions. The Bethe-Sommerfeld conjecture for periodic potentials serves as a guiding principle for the different mechanisms and phenomena that should be expected to play a role. Passing from periodic to non-periodic almost periodic potentials, we discuss both positive and negative results in the spirit of the Bethe-Sommerfeld conjecture.

 

Finite element exterior calculus, or FEEC, is a prime example of a structure-preserving discretization method in which key mathematical structures of the continuous problem are exactly captured at the discrete level. In the case of FEEC these structures arise from differential complexes and their cohomology, and FEEC applies geometry, topology, and analysis in order to design and analyze stable and accurate numerical methods for the differential equations related to the complexes. We will present an accessible overview of FEEC and some of its applications.

 

We give a soft introduction to several aspects of applications of operator algebras in dynamics and ergodic theory of groups. For a group G we consider several spaces including the space Sub(G) of subgroups of G, the space of positive definite functions, and state spaces of operator algebras generated by unitary representations of G, and review their connections and their applications in the study of actions of G.

I intend to make it accessible to a very general audience, only assuming advanced undergraduate algebra, linear algebra, and analysis.

 

In this talk I will give a light introduction to the theory of quantum groups by describing a concrete class of examples: the quantum symmetry groups of finite graphs. As the terminology suggests, the algebraic structure that we dub the “quantum symmetry group” of a graph ought to describe some sort of “quantized symmetries” of the given graph (in the physical sense of quantum mechanics). I will explain how recent ideas from the theory of non-local games in quantum information theory (QIT) provide this appropriate interpretation of quantum symmetry groups of graphs as “physically realizable” symmetries. Time permitting, I will also highlight some striking applications of ideas from QIT and quantum group theory to problems in graph theory and operator algebras.

 

Since this talk is aimed at a general audience we begin by reviewing some general techniques for dealing with spaces and algebras of Hilbert space operators (that is, with the `quantum analogue’ of functions and function spaces and algebras). There will be an emphasis on the theories of operator spaces, the *-algebraic approach to quantum physics, positivity, and on noncommutative measure and integration theory. We end with some new results.

 

Since the group Ext(A,B) is divisible for any torsion-free Group A, the natural question arises, when Ext(A,B) is torsion-free – especially without vanishing. While the class *B of all groups A such that Ext(A,B) is torsion-free was discussed in several former publications, there is less known about the dual class A*. We will observe some homological properties of this class of Abelian Groups and present some results in case that A is a B-solvable group.

 

F. and M. Riesz established that, in the complex plane, the harmonic measure is absolutely continuous with respect to the arc-length measure for simply connected domains (a strong connectivity condition) with rectifiable boundary (a regularity condition). In this talk we will present higher-dimensional quantitative extensions of this result and its converse for the Laplacian and also for some class of elliptic operators with variable coefficients. We will consider the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. Our results can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular L^p data case.

 

This talk is about a Borg-type inverse spectral problem for vibrating linear systems of point masses connected by springs. From the natural frequencies of vibration of the original system and a perturbation of it, we show how the masses and elastic coefficients of the springs can be reconstructed. To accomplish this, rank-three perturbations of Jacobi matrices are considered and their associated Green's functions explicitly described in terms of spectral data. We give necessary and sufficient conditions for two given sets of points to be eigenvalues (natural frequencies) of the original and modified system, respectively.

This is joint work with Luis Silva and Mikhail Kudryavtsev.

 

Finite element methods are a powerful and flexible tool for computing numerical approximations of solutions to PDEs. The theory of finite element exterior calculus, pioneered by Arnold, Falk and Winther in 2006, explores how sequences of spaces of differential forms with polynomial coefficients can inform the robust and efficient design of these methods in a wide variety of contexts. In this talk, I will explain some of the mathematical tools from this theory and how they aided in the discovery of a new family of finite element methods called “trimmed serendipity elements.” I will conclude by discussing current and future work in this area. This is joint work with Tyler Kloefkorn.

 

Partial differential equations over matrix algebras and other “noncommutative manifolds” appear naturally in theoretical physics. Powerful methods coming from harmonic analysis, like the theory of pseudodifferential operators, were introduced by Connes in 1980 to understand a quantum form the Atiyah-Singer index theorem over these algebras. Unfortunately, these techniques have been underexploited over the last 30 years due to fundamental obstructions to understand singular integral theory in this context, which constitutes a crucial technique for the most celebrated results in the theory of pseudodifferential operators.

During the talk, I will overview the core of singular integral theory as well as pseudodifferential operator theory over the archetypal algebras of noncommutative geometry. This includes the Heisenberg-Weyl algebra, quantum tori and other noncommutative deformations of Euclidean spaces of great interest in quantum field theory, string theory and quantum probability. Our Calderon-Zygmund methods in this context go much further than Connes' original results for rotation algebras. We obtain L_p-boundedness and Sobolev p-estimates for regular, exotic and forbidden symbols in the expected ranks. In the L₂-level, both Calderon-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also proved in the quantum setting. As a basic application, we prove L_p-regularity for elliptic PDEs over these algebras. Based on joint work with A.M. Gonzalez-Perez and M. Junge.

 

The familiar double soap bubble is the least-area way to enclose and separate two given volumes in Euclidean space. What if you give space a density, such as r² or e^r2 or e^-r2? The talk will include recent results and open questions. Students welcome.

 

The plan of the talk is to give a brief overview about the arguably most important spectral functions that are associated with an elliptic differential operator (in particular the Laplace operator), and elucidate the somewhat unexpected properties these functions exhibit in non-smooth situations, i.e., when considering differential operators on spaces with singularities. In particular, I will describe in greater detail what happens in the presence of conical singularities, i.e., when considering elliptic operators on spaces that are smooth outside finitely many points.

 

The area of mathematics dealing with boundary value problems is at the confluence of several major branches, including Partial Differential Equations, Harmonic Analysis, Geometric Measure Theory, and Functional Analysis. This talk is designed to bring to light some of the intricacies of this subject by focusing on the Dirichlet boundary value problem for elliptic systems in the upper-half space. The approach I will adopt, which places a particular emphasis on the role played by the Hardy-Littlewood maximal operator, {\it simultaneously} yields the well-posedness of the Dirichlet problem with boundary data in a variety of spaces of interest (including ordinary Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions). Along the way, I will derive a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of elliptic systems, and also establish the uniqueness of the Poisson kernels associated with such systems, as well as the fact that they generate strongly continuous semigroups in natural settings.

 

We obtain sign conditions and comparison theorems for Green’s functions for families of boundary value problems for both fractional differential equations and fractional difference equations. We will also discuss convergence of the Green’s functions as the length of the domain diverges to infinity.

 

In this talk we introduce the notion of operator splitting for nonlinear equations. We formulate the approach in the language of Magnus expansions in abstract spaces, allowing us to combine the language of semigroups with nonlinear operators. The focus of the talk will be extending these techniques to approximating solutions of stochastic differential equations in Hilbert spaces. These approximation techniques allow for the development of numerical methods which are of arbitrary order, yet have lower regularity conditions when compared to many existing methods. Moreover, the methods may easily be generalized to differential problems posed on smooth manifolds. If time permits, we will discus how operator splitting methods may be employed to construct approximations which respect the underlying Lie group structure of the problem at hand. There will be a thorough introduction to the considered methods and the talk will be accessible to interested graduate students.

 

The bispectral problem concerns the construction and the classification of solutions to eigenvalue problems that satisfy additional equations in the spectral parameter. It was originally posed in the context of problems related to medical imaging, but it turned out that it has interesting connections to many areas of pure and applied mathematics such as integrable systems, algebraic geometry, representation theory of Lie algebras, classical orthogonal polynomials, etc. I will review the problem and some of these connections, including recent results where the bispectrality plays a crucial role.

 

The past several decades have seen an intense study of computational tools for attacking NP-hard models in discrete optimization. We give an overview of this work, discussing current techniques, results, and research directions. The talk will highlight successful approaches adopted in the exact solution to large-scale mixed-integer programming models and the traveling salesman problem.

 

Richard Borcherds won the Fields Medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences observed by finite group theorists and arithmetic geometers in the 1970s. Weak moonshine for a finite group G is the natural generalization of this phenomenon where an infinite dimensional graded G-module

V_G=⊕_n»-∞ V_G(n)

has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by Dehority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width s∈Z⁺. We find that complete width 3 weak moonshine always determines a group up to isomorphism. Furthermore, we establish orthogonality relations for the Frobenius r-characters, which dictate the compatibility of the extension of weak moonshine for V_G to width s weak moonshine. This is joint work with Ken Ono.

 

We discuss problems where impact from optimal or equilibrium trading leads to challenging nonlinear systems and fixed point problems. These may arise from
i) Oligopolies with a small number of influential players, such as oil markets.
ii) Optimal execution where trading speed is penalized.
iii) Portfolio selection to maximize expected utility.
iv) Market impact from a significant group of portfolio optimizers in a market with clearing conditions.
The problems are addressed with computational and analytical methods, and specification of a terminal time and terminal condition has strong influence on resulting strategies. Analogous issues arise in some examples from sports.

 

In this talk, we will survey briefly the importance of some systems of particles with long-range interactions in physics. Such models appear naturally in a wide range of applications and, under suitable conditions, they may be transformed into partial differential equations with fractional derivatives. We will recall this transformation process, and consider fractional systems with a Hamiltonian structure. In particular, we will tackle the problem for solving numerically some fractional wave equations in which the energy is dissipated. A Hamiltonian finite-difference scheme will be proposed to that end, and its most important structural and numerical properties will be investigated. Illustrative examples will be provided at the end of the talk.

 

The question on how to rigorously define and prove Many-Body-Localization (MBL) phenomena has attracted significant interest over the recent years. In this talk, we will give a physical motivation for the so-called entanglement entropy (EE) and explain why an area law for the EE can be interpreted as a sign of MBL. We then introduce the Heisenberg XXZ spin Hamiltonian, which is unitarily equivalent to a direct sum of discrete many-particle Schrödinger operators with an attractive potential that energetically favors the formation of clusters of particles. After this, we present a (log-corrected) area law that works for any state corresponding to a finite but arbitrary number of such clusters. This is joint work with H. Abdul-Rahman (U of Arizona) and G. Stolz (U of Alabama at Birmingham).

 

In 2004, a research paper was published about a dog who could solve an optimization problem. Thus was a legend born. We'll tour the problem, the variations thereof, the media sensation, and the math that emerges (including a subtle bifurcation and some nifty algebraic moves).

 

Srinivasa Ramanujan (1887-1920), is one of the greatest mathematicians in history. Ramanujan, a poor uneducated Hindu in rural India, sent dozens of startlingly beautiful mathematical formulae in two letters to the British mathematician Hardy. The legend is that the Hindu Goddess Namagiri would come in his dreams and give him these formulae which revealed surprising connections between apparently disparate areas. Hardy was convinced that Ramanujan was a genius in the class of Euler and Jacobi and invited him to England. The rest is history! In this talk, after describing briefly the fascinating life story of Ramanujan with pictures of his hometown, I will provide a glimpse of his remarkable mathematical discoveries, and compare his work with some of the mathematical luminaries in history. We will also describe what is being done to foster the legacy of Ramanujan, and how his work continues to influence mainstream areas.

 

It is a classical result that for a torsion-free Abelian group A the group Ext(A,B) is divisible for any Abelian group B. Hence it is uniquely determined by some cardinals called its torsion-free rank and the p-rank. For example the natural question arises, when Ext(A,B) is torsion-free – especially without vanishing. If we concentrate on the case that A and B are rational groups, i.e. torsion-free groups of rank 1 and thus subgroups of the rational numbers, the structure of Ext(A,B) may be completely determined by the types of these rational groups.

 

It is known that monic polynomials orthogonal with respect to a compactly supported non-trivial Borel measure on the real line satisfy three-term recurrence relations with coefficients that are uniformly bounded. The coefficients then can be used to define a bounded operator on the space of square-summable sequences. This operator can be symmetrized and the spectral measure of the symmetrized operator is in fact the measure of orthogonality of the polynomials themselves. One way of arriving at the subject of orthogonal polynomials is via Padé approximation (Padé approximants are rational interpolants of a given holomorphic function; when the function is a Cauchy transform of a Borel measure on the real line, the denominators of the approximants are the orthogonal polynomials). Padé approximants can be extended to the setting of a vector of holomorphic functions and a vector of rational interpolants (this construction was introduced by Hermite to prove transcendency of e). Vector rational interpolants naturally lead to multiple orthogonal polynomials. Spectral theory of multiple orthogonal polynomials is not yet fully developed. I shall describe some of the recent advancements in this area. This is based on joint work with A. Aptekarev and S. Denisov.

 

For the majority of finite element methods for incompressible flows, the error in the discrete velocity depends on the product of the best approximation error in the pressure and the inverse of the viscosity of the flow. As a result, the smaller the viscosity, the more degrees of freedom are required to achieve a certain level of accuracy in the velocity solution. This may result in expensive simulations when the viscosity is small. In this talk, I will introduce a new class of Hybridizable Discontinuous Galerkin (HDG) finite element methods for incompressible flows. Our HDG method is constructed to result in a discrete velocity that is automatically pointwise divergence-free and divergence-conforming. An immediate consequence of these two properties is that the error in the discrete velocity computed using our HDG method does not have a dependence on the viscosity and pressure: our method is “pressure-robust”. To be of practical use we have also developed optimal preconditioners specifically for HDG methods. For this, we exploited the fact that static condensation is trivial for HDG discretizations. In this talk, I will discuss the construction of our preconditioner for the Stokes equations. Finally, I will discuss the extension of our HDG method to solve the incompressible Navier-Stokes equations on time-dependent domains. Time-dependent domain problems occur, for example, in fluid-structure interaction simulations and simulations involving free-surfaces. Constructing a space-time discretization on space-time simplices makes it possible to construct a space-time HDG method that is “pressure-robust” even on time-dependent domains.

 

Measures on the unit circle for which the logarithmic integral converges can be characterized in many different ways: e.g., through their Schur parameters or through the predictability of the future from the past in Gaussian stationary stochastic process. In this talk, we consider measures on the real line for which logarithmic integral exists and give their complete characterization in terms of the Hamiltonian in De Branges canonical system. This provides a generalization of the classical Szego theorem for polynomials orthogonal on the unit circle and complements the celebrated Krein-Wiener theorem in complex function theory. The applications to Krein strings and Gaussian processes with continuous time will be discussed (this talk is based on the joint paper with R. Bessonov).

 

A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems. Joint works with Bourgain/Gamburd and with Ghosh.

 

Interface problems are ubiquitous. They arise widely in sciences and engineering applications. Partial differential equations (PDE) are often used by mathematicians to model interface problems. Solutions to these PDE interface problems often involve kinks, singularities, discontinuities, and other non-smooth behaviors. The immersed finite element method (IFEM) is a class of numerical methods for solving PDE interface problems on unfitted meshes. In this talk, we will start by introducing the basic idea of IFEM, followed by some recent advances in developing more accurate and robust computational algorithms of IFEM. In particular, we introduce the partially penalized IFEM and nonconforming IFEM. Error analysis including a priori and a posteriori error estimates with optimal convergence rates will be shown. Finally, we will talk about applications of IFEM to more general interface problems such as elasticity systems, fluid-flow problems, moving interface problems, free boundary methods, and plasma simulation problems.

 

In his famous book “A Treatise on Electricity and Magnetism” first published in 1867 J.C. Maxwell made a claim that any configuration of N fixed point charges in R³ creates no more that (N-1)² points of equilibrium. He provided this claim with an incomplete proof containing many elements of Morse theory to be created 60 years later. We present a modern set-up and generalisations of Maxwell’s conjecture and discuss what is currently known about his original claim which is still open even in case of 3 charges. No preliminary knowledge of the topic is required.

 

New techniques of biomedical and homeland security have brought about quite a few amusing mathematical problems, involving all kind of mathematics: from pdes and spectral theory to algebraic geometry. The talk will survey briefly the hybrid imaging, inverse problems with internal information and, time permitting, Compton camera imaging. No prior knowledge of the area is required.