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""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.

"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 L_{2}-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 r^{2} 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 traveling salesman problem is easy to state: given a number of cities along with the cost of travel between each pair of them, find the cheapest way to visit them all and return to your starting point. Easy to state, but difficult to solve. It is a real possibility that there may never exist an efficient method that is guaranteed to solve every instance of the problem. This is a deep mathematical question: Is there an efficient solution method or not? The topic goes to the core of complexity theory concerning what computers can and cannot solve. For the stout-hearted who would like to tackle the problem, the Clay Mathematics Institute will hand over a $1,000,000 prize to anyone who can either produce an efficient method or prove it cannot be done. In this talk we discuss the history, applications, and computation of this fascinating problem. TBD

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.TBD

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

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 fractionl 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 determinded 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).

Quadratic Diophantine equations (sums of integer squares ) have fascinated mathematicians for centuries however even today some of the finer local to global questions are not understood, and the complexity of finding solutions is challenging. We will explain and review these features and highlight some applications, for example to quantum computation with the construction of optimal universal quantum gates. TBD

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.TBD

TBA

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 R^{3} 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.

TBA