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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.

TBD

TBA

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