In the first part of the talk, we analyze the first left-definite space associated with the one-dimensional Krein Laplacian operator. We show results regarding polynomial density in the space, we prove the existence and uniqueness of a sequence of orthogonal polynomials (OP) in the space, and present some properties of the sequence.

For the second part of the talk, we give some background on the Riemann-Hilbert problem (RHp) for OP, and the Krall-Legendre OP sequence, which is known for satisfying a fourth order differential equation. Then, we formulate the RHp for the Krall-Legendre OP, we prove the existence and uniqueness of its solution, and show how to obtain the first order matrix ODE, and the second order ODE for the Krall-Legendre OP, as a first approach to find the fourth order ODE from the Riemann-Hilbert formulation.

The so-called Angelesco-Jacobi polynomials (generalizing classical Jacobi polynomials) are an instance of the multiple or Hermite–Pade orthogonal polynomials, and they depend on four real parameters, α, β, γ>-1, and a<0. Their zeros are known to be real and simple. In the first part of the talk, the behavior (monotonicity and interlacing) of their zeros as functions of these parameters is addressed, showing that classical tools allow us to extend some known results in a non-trivial way. The second part of the talk deals with the newly discovered applications of the notion of the free finite convolution of polynomials (that is being developed in the framework of free probability theory) to the study of properties of zeros of some hypergeometric polynomials. In particular, we will discuss possible applications to multiple orthogonal polynomials.

Every ten years, starting in 1790, the U S has decided how many representatives h should be in the House of Representatives, representing the s states. In 1790, h=90 and s=15. Today h=435 and s=51. (Yes, 51.). The fair share, or quota, for state i, q_i, is calculated using the census done every 10 years. The sum of the s values of q_i is equal to h. Thus, after the 2020 census, Tennessee might deserve 7.843972… representatives. But the Constitution demands that Tennessee gets an integral number of representatives. An apportionment is a function (or rule) which assigns a positive integer a_i of representatives to state i, such that these a_i‘s sum to h. In our history we have used many apportionments. “Mathematicians” contributing to this theory included G. Washington, J. and J. Q. Adams, T. Jefferson, B. Franklin, J. von Neuman, G. Birkhoff, M. Morse, and T. van Karmann. Some of these “mathematicians” made valuable contributions, while others did not. Can you guess who were the good ones and who were otherwise? Would you be shocked to learn that the current apportionment method used is unconstitutional? These and other questions will be raised, and some will be answered. This talk should be accessible to undergraduates as well as professors in any departments (who understand at least some calculus).

Let du/dt = Au be the heat equation for t>0 and x in Rⁿ involving a potential V(x) that is singular in some sense. Fix the initial value u(x,0) = f(x) ≥0, f not zero. Let V_n(x) be an increasing sequence of bounded potentials converging pointwise to V(x), so that dv/dt = An v, v(0)=f, and v(x,t) exists for all n, x, t. Instantaneous blowup (IBU) refers to the case when v(x,t) goes to infinity as n does for all x in Rⁿ and t>0. The 1984 result of P. Baras and J. Goldstein established IBU for V(x) being the inverse square potential: V(x) = c/r², where r is the length of x and c> [(n-2)/2]². We published two proofs of this result. A third proof, based entirely on scaling, was published recently. The IBU result with Rⁿ replaced by the Heisenberg group Hⁿ, was published recently in the Annali di Pisa by G. Goldstein, J. Goldstein, A. Kogoj, A. Rhandi, and C. Tacelli. But the proof is like the 1984 paper, being based on the DeGiorgi-Nash-Moser theory recently adapted to Hⁿ. Our efforts to find a “scaling proof” have not yet been successful. We also mention some related results (involving the Ornstein-Uhlenbeck equation, etc.).

In this talk we will review the development of Gaussian Singular from the Gaussian Riesz transforms until the recent development of the generalized Gaussian singular integrals and from boundedness on Lebesgue L^p spaces until variable Lebesgue L^p(.) spaces.

Topological full groups, as an algebraic invariant, were introduced to study continuous orbit equivalence relations by Giordano, Putnam, and Skau. Then, these groups have been found applications to geometric group theory by providing interesting examples with certain properties such as simple-ness, soficity, amenability, and LEF-ness. In this talk, we will show new methods of establishing the soficity and LEF-ness for topological full groups. Moreover, we will explain how can one obtain amenability from the sofic approximations when the acting group is amenable and the action is free and distal.

The weighted Bergman space A²_h for a weight h on the unit disk D:={z:|z|<1} consists of all functions f analytic in D such that
∫_D |f(z)|²h(z)dA(z)<∞,
where dA is the area measure. The reproducing kernel K_h(z,ζ) of A²_h is a function analytic in z and ζ, uniquely determined by the property

f(z)=∫_D f(ζ)K_h(z,ζ)h(ζ)dA(ζ),         z∈ D,         f∈ A²_h.

The first part of this talk discusses a canonical formula for the reproducing kernel corresponding to a weight of the form

h(z)=∏_1≤k≤s|^(z-a_k)⁄_(1-akz)|^m_k,

with m_k>-2 and |a_k|<1 for all 1≤k≤s. These weights are of importance, in part, because the canonical divisor of the Bergman space A^p for a finite zero set a₁,…, a_s (a_k repeated m_k times) can be directly expressed via the reproducing kernel for the weight above.

The second part of this talk concerns the sequence (p_n)_n≥0 of polynomials orthonormal over D with respect to a weight w of the form w(z)=|v(z)|²h(z), with v analytic and free of zeros in D, and h as in the above equation. Equipped with the canonical formula for the kernel K_h(z,ζ), we will be able to derive an integral representation for p_n that allows for a fine description of its asymptotic behavior as n→∞.

We will introduce the finite free additive and multiplicative convolutions of two polynomials, and explain how it is related to free probability in the limit. We will provide several equivalent ways to define or rephrase this convolution, Some of these definitions include taking the expected characteristic polynomial after conjugating by orthogonal, unitary or signed permutations, as well as precise formulas in terms of the coefficients of the polynomials. We will also study some elementary combinatorial tools such as the finite free cumulants. We will then explain how finite free probability is a very effective tool to study the effect of differentiating a sequence of polynomials several times and then looking at the resulting limiting root distribution.

A Sjölinâ-Soria-Antonov type extrapolation theorem for locally compact σ-compact non-discrete Hausdorff groups is proved.

Applying the extrapolation result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class L log⁺L log⁺log⁺log⁺L.

Another application deals with the halo conjecture in the theory of differentiation of integrals which states that if the maximal operator M_B corresponding to a translation invariant differentiation basis B in a locally compact group is of restricted weak type ϕ, then the basis B differentiates the integrals of functions from the class ϕ. Namely, the result has been established that gives an approximation to the conjecture for functions ϕ(u) close to u, while for the case of ϕ(u)=u, this implies the validity of the conjecture.

I will talk about an intriguing connection between geometric complex analysis and a problem of stability in discrete dynamical systems. The talk will be accessible to graduate students and non-experts.

The torsional rigidity of a region in the complex plane quantifies the resistance to twisting of an infinite beam with that region as cross-section. The approximate values of torsional rigidity for some common shapes are well-known, while the exact value is known for only a small number of regions. We are interested in the effect of symmetrization on torsional rigidity and how this can be used as a tool in some related extremal problems. I will present several mathematical models for torsional rigidity and indicate directions for future research with these extremal problems in mind. We will expand on the earlier work of G. Pólya, A. McNabb, and others.

Maximal operators are central objects in harmonic analysis and their boundedness on different spaces of functions may be used to prove differentiation theorems such as the Lebesgue Differentiation Theorem. With this motivation, in this talk I will present a couple of tools involving covering properties that will help us to obtain weak-type estimates of the Hardy-Littlewood Maximal Operator, the Lacunary Maximal Operator, and the maximal operator associated with a collection B. This is part of my Ph.D. thesis, under the guidance of Dr. Paul Hagelstein. 

Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem. The so-called geometric approach may help in many cases. In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.

The scaling behavior of the entanglement entropy of a state ψ contains information about the correlation structure of a quantum mechanical system when in state ψ. We present results on the XXZ model in higher dimensions, and discuss in particular which states satisfy a logarithmically corrected area law.

In this talk we present recent results in the direction of solving boundary value problems for general second-order systems on rough domains with boundary data taken in non-standard function spaces, thus expanding the scope of the work of B. Dahlberg, E. Fabes, and C. Kenig, among others, going back to the early 1980’s. More specifically, we study the Neumann problem with boundary data in Herz-type Hardy spaces, and the Dirichlet problem with boundary data in the space of functions of bounded central mean oscillations (in the spirit of the John-Nirenberg BMO space, with a twist). We develop a comprehensive Calderón-Zygmund theory for the relevant class of singular integral operators acting on (and from) these brands of spaces, and succeed in employing the method of boundary layer potentials to establish solvability results for the aforementioned boundary problems. This is part of my Ph.D thesis, under the guidance of Professor Marius Mitrea.

I will discuss a version of the Divergence Theorem for vector fields which may lack any type of continuity and for which the boundary trace is taken in a strong, nontangential pointwise sense. These features of our brand of Divergence Theorem make it an effective tool in dealing with problems arising in various areas of mathematics, including Harmonic Analysis, Complex Analysis, Potential Analysis, and Partial Differential Equations.

We investigate L_p-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (motivated by its consequences in metric geometry) to discrete groups. As application, we establish new forms of metric X_p inequalities in group von Neumann algebras. Joint work with I. Cano-Marmol and J. Conde-Alonso.

Continued fractions have a long history of use in number theory and analysis. In this honors thesis defense, we provide many of the historical theorems that made them important to mathematics, along with some more recent developments. Specifically, the results from Edward Burger’s article “A Tail of Two Palindromes” are made analogous to spectral m-functions. The main theorem characterizes all discrete m-functions whose Jacobi parameters are eventually doubly palindromic.

Using the variational characterization of the smallest eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension T_F of the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result we characterize all lower bounds of T_F as those λ∈R for which the differential equation τu = λu has a strictly positive solution u > 0 on (a,b). 

This talk is based on joint work with Fritz Gesztesy.

We show that Dirac and CMV operators with limit-periodic coefficients generically have spectra that are Cantor sets of zero Lebesgue measure. The proof relies on a perturbative construction due to Avila for discrete Schr\”odinger operators together with a new argument based on analyticity and noncommutation. [Joint work with Benjamin Eichinger, Ethan Gwaltney, and Milivoje Lukic]

We investigate complete non-selfadjointness for all maximally dissipative extensions of a Schröodinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We
show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterization of these extensions and the corresponding subspaces and present a specific example. 

(Joint work with Sergey Naboko and Ian Wood)

I will consider problems of stability in discrete dynamical systems. The talk will be accessible to non-experts and graduate students.

In this talk, we will discuss a relation between the Maslov index, a topological invariant counting signed intersections of paths of Lagrangian planes with a fixed reference plane, and the spectral flow, the net number of eigenvalues of a path of Fredholm operators crossing zero in the positive direction. Applications will be given to several parameter-dependent boundary value problems ranging from Rohleder-type inequalities for Robin Laplacians to Hadamard-type formulas for quantum graphs. In addition, we will discuss second order Taylor expansions for reslovents, spectral projections, and eigenvalues of one-parameter families of self-adjoint extensions of symmetric operators.

Suppose that M is a finite-dimensional C^*-algebra, equipped with its Plancherel trace. A lot of study has gone into the “quantum automorphism group” Aut⁺(M) of M, and what kinds of properties it has. For example, if M is abelian and with dimension n, then Aut⁺(M)=S_n⁺ is the quantum permutation group on n letters. The latter object has received recent attention in quantum information, due to a specific class of two player, non-local games related to graphs. In this talk, we will explore this link between quantum information and quantum groups, and see some new results about Aut⁺(M) that were inspired by a new example of a graph-related non-local game. In particular, we’ll see that the von Neumann algebra L^∞(Aut⁺(M)) always satisfies Connes’ embedding problem, and is not isomorphic to any free group factor when dim(M) is a perfect square. (This is based on joint work with Michael Brannan, Floris Elzinga and Makoto Yamashita.)

Differential Galois Theory is the Galois Theory in the context of linear differential equations. In this talk we present some links between the theory of orthogonal polynomials and classical/differential Galois theory. Some results involve integrability of differential equations in a similar sense as for solvability of polynomials. This talk is based on my previous papers available on arxiv: arXiv:1906.09764, arXiv:1012.4796 and arXiv:1008.3445. 

In group theory, the Zappa-Szep product generalizes the semi-direct product by encoding a two-way action between two groups. Similar constructions have appeared in the context of semigroups and self-similar action on graphs. In this talk, I will briefly survey these constructions and their applications in operator algebras. The focus is on the self-similar actions on groupoids and Fell bundles, which generalizes the semi-direct product constructions. This leads to a way of constructing equivalent groupoids and Fell bundles from certain symmetric actions, generalizing several imprimitivity theorems arising from semi-direct products. This is a joint work with Anna Duwenig.​

Let G be a locally compact quantum group with dual Ĝ. Suppose that the left Haar weight a and the dual left Haar weight â are tracial, e.g. G is a unimodular Kac algebra. We prove that for 1<p≤ 2 ≤ q<∞, the Fourier multiplier m_x is bounded from L_p(Ĝ,â) to L_q(Ĝ,â) whenever the symbol x lies in L_r,∞(G,a), where 1/r=1/p-1/q. Moreover, we have 

 

||m_x:L_p(Ĝ,â)⟶ L_q(Ĝ,â)||≤ c_p,q ||x||_{Lr,∞,(G,a))},

where c_p,q is a constant depending only on p and q. This was first proved by Hörmander (Acta. Math. 1960) for Rⁿ, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of L_p-Fourier multipliers over discrete group von Neumann algebras. A similar result for S_p-S_q Schur multipliers is also proved. The talk is based on arXiv:2201.08346.

We present new aspects of the solvability of the classical Neumann boundary value problem in a graph Lipschitz domain in the plane. When the domain is the upper half-plane and the boundary data is assumed to belong to weighted Lebesgue or weighted Lorentz spaces, we show that the solvability of the Neumann problem in these settings may be characterized in terms of Muckenhoupt weights and related weights, respectively. For a general graph Lipschitz domain Ω, as proved in an unpublished work by E. Fabes and C. Kenig, there exists ε_Ω>0 such that the Neumann problem is solvable with data in L^p(∂Ω) for 1<p<2+ε_Ω; we show that the Neumann problem is solvable at the endpoint 2+ε_Ω with data in the Lorentz space L^2+ε_Ω,1(∂Ω). We present examples of the results in Schwarz-Christoffel Lipschitz domains and related domains.

This talk is about the spectral theory associated with the differential equation Ju’+qu=wf on the real interval (a,b) when J is an n×n constant, invertible skew-Hermitian matrix and q and w are n×n matrices whose entries are distributions of order zero (local measures) with q Hermitian and w non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact, we are able to show that the deficiency indices of the corresponding minimal relation are still bounded by n.

The notion of type I, hailing from the very origins of operator algebras and representation theory, can be seen as a rigorous way to define the class of groups for which unitary representations can be classified in any meaningful manner. By a celebrated result of Thoma, a discrete group is type I if and only if it is virtually abelian. In the non-discrete case, the current state of the art is not nearly as complete, despite numerous results ensuring that various important families of groups (e.g. every connected semisimple Lie group) are type I. What is completely lacking, in contrast to Thoma’s theorem, is a definite structural consequence of type I. This talk is around the following conjecture: Every second countable locally compact group of type I admits a cocompact amenable subgroup. We motivate the conjecture, provide some supporting evidence for it, and prove it for type I hyperbolic locally compact groups admitting a cocompact lattice.
This is joint work with Pierre-Emmanuel Caprace and Nicolas Monod.

I will talk about my recent work on solutions to some Gromov’s conjectures and open questions on positive scalar curvature. These include positive solutions to Gromov’s cube inequality on positive scalar curvature, and Gromov’s open question on λ-Lipschitz rigidity of positive scalar curvature metrics on hemispheres or more generally spheres with certain subsets removed. If time permits, I will also talk about my recent joint work with Jinmin Wang and Guoliang Yu on an index theoretic proof for Gromov’s (strict) cube inequality on positive scalar curvature with the optimal constant.

Logarithmic Sobolev inequalities (LSI) were first introduced by L. Gross in 70’s, and later found rich connections to geometry, graph theory, optimal transport and information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attention in quantum information theory and quantum many-body system. For a classical Markov semigroup, an important advantage of log-Sobolev inequalities is the tensorization property that if two semigroups each satisfy an LSI, so does their tensor product semigroup. Nevertheless, the tensorization property fails in the quantum cases. In this talk, I’ll present some recent progresses on tensor stable log-Sobelev inequalities for finite dimensional quantum Markov semigroups. This talk is based on joint work with Cambyse Rouze.

In this talk I will report on recent progress in the direction of employing boundary layer potential operators in proving the unique solvability of boundary value problems in general rough settings, best described in the language of geometric measure theory. Using a blend of techniques from Harmonic Analysis, Calderon-Zygmund Theory, Functional Analysis, we succeed in proving well-posedness results in the case when the boundary data is arbitrarily prescribed in Muckenhoupt weighted Morrey spaces. This work is part of my Ph.D. thesis, carried out under the guidance of Professor Marius Mitrea.

In this talk, I will discuss recent work examining the transport properties of a subclass of uniform limit-periodic, continuum Schrodinger operators, specifically those operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. Specifically, for H one of these operators and X_H(t) the Heisenberg evolution of the position operator, we show the limit of t^-1X_H(t)ψ as t→∞ exists and is nonzero for ψ belonging to a large subspace of initial states localized in position and momentum. Since this implies these operators exhibit a particularly strong form of ballistic transport, and it is known that operators in the subclass considered have absolutely continuous spectrum, this work provides more examples of almost periodic operators for which there is absolutely continuous spectrum and ballistic transport in this sense.

A droplet of perfectly conducting fluid is placed into an external electric field. If the droplet is not annihilated, the forces of pressure and surface tension will balance with the field, and the droplet will come to equilibrium, resulting in a free boundary problem. This problem was first addressed by E.B. McLeod (1955), who found a single solution to the equation describing equilibrium. The problem was readdressed by D. Khavinson, A. Solynin, and D. Vassilev (2005), where some progress was made, but no new `physical’ solutions were discovered. In this talk, I will discuss some recent work of myself and F. Wang, where we found a new 1-parameter family of physical solutions, using methods of quadratic differentials. 

In this talk, we will utilize Clark theory to obtain spectral information about symmetric differential operators. This will begin with describing the self-adjoint extensions of a symmetric differential expression using two methods. Then, various tools of Clark Theory will be established. Lastly, we will compute (Clark) spectral measures for the self-adjoint extensions of specific examples of symmetric differential expressions.

This talk concerns asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. First, we will discuss a symplectic version of the celebrated Krein formula for resolvent difference. Then we will switch to an asymptotic analysis of resolvent operators via first order expansion for the path of Lagrangian planes associated with perturbed operators. This asymptotic perturbation theory yields an Hadamard–Rellich-type variational formula for multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. Applications will be given to quantum graphs, periodic Kronig–Penney model, elliptic second order partial differential operators with Robin boundary conditions, and heat equations with thermal conductivity. This talk is based on various joint projects with G. Berkolaiko (Texas A&M) and Y. Latushkin (Missouri/NYU).

The generators of finite dimensional self-adjoint quantum systems are fully characterized as Lindblad operators. I will use the geometric approach to prove the entropy decay for semigroups generated by such Lindblad operators. I will also mention some related open problems (classical and quantum) that I am currently working on. This is joint work with Marius Junge and Li Gao.

In this talk, I will introduce a theory of heat semigroup based Besov spaces in abstract Dirichlet spaces which include Riemannian manifolds, sub-Riemannian manifolds and some fractals like Sierpinski gasket. I will then discuss its applications on Sobolev and isoperimetric inequalities with sharp exponents. The talk is based on joint works with Patricia Alonso-Ruiz, Fabrice Baudoin, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev.

TBA

TBA

The classical Khintchine inequality implies that the L¹ norm of the function Σ_k a_k z^{2^k} on the torus is equivalent to the l² norm of its Fourier coefficients. Paley’s inequality is an improvement of the classical Khintchine inequality, which says that the Fourier coefficients of the the function in the Hardy space on torus is square summmable on the support of the lacunary set. We will introduce a similar result for the countable ordered nonabelian group, e.g. free group.

The main objective of this talk is to illustrate how one can construct Schrödinger operators on a bounded interval with predefined essential and discrete spectra. The required structure of the spectrum is realized by a special choice of a sequence of δ-interactions. Our construction is inspired by a celebrated paper [R. Hempel, L. Seco, B. Simon, J. Funct. Anal. 102 (1991), 448-483] and its sequel [R. Hempel, T. Kriecherbauer, P. Plankensteiner, Math. Nachr. 188 (1997), 141-168], where a similar problem was treated for Neumann Laplacians on bounded domains. 

This is joint work with Jussi Behrndt (TU Graz).

Quantum Markov semigroups models the time evolution of dissipative open quantum systems. Mathematically, they are noncommutative version of Markov semigroups where the underlying function space is replaced by matrix or operator algebras. For a Markov semigroup, the modified logarithmic Sobolev inequality (MLSI) describes the convergence property in terms of relative entropy. It was a famous result by Bakry and Emery that the heat semigroup on a compact Riemannian manifold satisfies MLSI if the Ricci curvature admits a strictly positive lower bound. Recently Carlen and Maas introduce the notation of Ricci curvature lower bound for quantum Markov semigroup and show that a positive Ricci lower bound implies MLSI in the noncommutative setting. In this talk, I will present an approach to MLSI for quantum Markov semigroup via Ricci curvature bounded below but not necessarily positive. We will show that “central” semigroups on groups and quantum groups admits nonnegative curvature and MLSI. This approach gives new examples of MLSI in both operator algebras and quantum information theory. This talk is based on joint work with Michael Brannan and Marius Junge.

We consider the Schroedinger operator on a finite interval with an L¹-potential. Borg’s two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional Schroedinger operators, we will discuss a complex analytic approach to the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?

In this talk I will discuss upper and lower bounds on the norms of monic Lp-extremal polynomials. In particular, I will present a sharp universal lower bound for the extremal polynomials with respect to a probability measure on a compact subset of the complex plane and its improvement for the orthogonal polynomials with respect to the equilibrium measure of a compact subset of the real line.

One of the most efficient approaches to solving boundary value problems for elliptic partial differential equations is the method of boundary layer potentials.
In this talk I will survey some of the recent progress made in understanding the nature of integral operators of boundary layer type in optimal geometrical settings.

To obtain matrix-valued Aleksandrov–Clark (AC) measures, fix a matrix-valued pure contraction b on the unit disk D. Here b may be non-inner and/or non-extreme. For each unitary matrix α, the linear fractional transformation (I+b(z))α^*)(I-b(z))α^*)^-1 has non-negative real part. So, Herglotz’s representation theorem associates a matrix-valued measure μ^α*. The collection {μ^α: α unitary} forms the family of matrix-valued AC measures, which stand in bijection with matrix-valued pure contractions. A description of the measures’ absolutely continuous parts is easily obtained in terms of non-tangential boundary values of b. 

 

The singular parts μ^α_s are harder. We present a matrix-valued version of Nevanlinna’s result relating non-tangential boundary limits with the measures’ point masses. The connection to Carathéodory angular derivatives is more subtle than in the scalar setting. Aleksandrov spectral averaging yields restrictions on the singular parts. We have a directional version of the mutual singularity of μ^α_s and μ^β_s, α≠β both unitary. This presentation is based on joint work with R.T.W. Martin and S. Treil.
If time permits, we may consider some applications (joint work with my grad student M. Bush and R.T.W. Martin).

This will be a summary of my research activities undertaken during my research leave in Spring 2019.

 

 

Let M₁ and M₂ be two Riemannian manifolds each of which have theboundary N. Consider the Laplacian on M₁ and M₂ augmented with Dirichlet boundary conditions on N. A natural question to ask is if there is any relation between spectral properties of the Laplacian on M₁, M₂, and the Laplacian on the manifold M (without boundary) obtained gluing together M₁ and M₂, namely M=M₁∪M₂. Using spectral zeta functions, a partial answer is given by the Burghelea-Friedlander-Kappeler-gluing formula for zeta-determinants. This formula contains an (in general) unknown polynomial which is completely determined by some data on a collar neigh-borhood of the hypersurface N. I will use conformal transformations to understand the geometric content of this polynomial. The understanding obtained will pave theway for a fairly straightforward computation of the polynomial (at least for low dimensions of M). Furthermore it leads to a partial understanding of the heat invariant for the Dirichlet-to-Neumann map, that is for the Steklov problem.

H.Stahl’s theorem on convergence of Pade approximats for analytic functions with branch points [St86-1]- [St86-1] is one of the fundamental results in the theory of rational approximations of analytic functions. It is also one of the basic facts in the theory of orthogonal pokynomilas. Denominators of Pade approximats are complex (nonhermittian) orthogonal polynomilas and Stahl’s created an original method of investigating asymptotics of complex orthogonal polynomials based directly on complex orthogonality. 

 

The method has been further developed by A. Gonchar and E. Rakhmanov in [GR87] for the case of orthogonality with varying weight which is equivalnt to introduction of external field in associated equilibrium problem. Incorporation of the external field in the method makes circle of its application essentially larger. The GRS-method may be applied to general rational interpolation, Hermite – Pade approximants (systems of interpolation conditions) and other constructions of free-pole approximation by rational functions. Also the geometry of the problem in the presence of the external field becomes much more sophisticated.

We generalize the method for the case of interpolation by meromorphic functions on a closed Riemann surface R which are, hence, algebraic functions. It turns out that basic results on zero distribution of poles of interpolating functions remain valid in essentially the same form.

Both discrete and continuous Dirac-type systems play an important role in several domains of mathematics. We discuss Weyl theory, the corresponding direct and inverse spectral problems, and their explicit and general solutions. There is also a close connection between spectral and scattering problems.

Many interacting random particle systems, in a suitable limit, can be described in terms of integrable systems. For instance, it is now a classical result that the one-point fluctuations of the height function in the TASEP (with step initial data) converge, in a suitable scaling, to the Tracy-Widom distribution F₂. In addition to remarkable universality features of F₂, appearing in dozens of different models that are seemingly unrelated, this distribution also enjoys several nice different characterizations, for instance it can be given interms of a somewhat simple Fredholm determinant for an integral operator, or yet in terms of a solution to the Painléve II differential equation. 

 

If, on one hand, one can learn about the particle systems by studying the corresponding integrable system,on the other hand these different characterizations have intrinsict operator-theoretic value on their own, and a large class of integral operators can actually be connected to integrable systems in a nice systematic and, somehow, easy way via Riemann-Hilbert problems.

In this talk, we plan to discuss how this general methodology recently produced a nice connection between the limiting distribution for the periodic TASEP and a nonlocal-nonlinear PDE.

This is based on work in progress with Jinho Baik (University of Michigan) and Zhipeng Liu (Universityof Kansas).

A classical problem that goes back to Chebyshev and his study of mechanisms is to find the best uniform approximation of a continuous function f: [a,b]→R by algebraic polynomials of degree ≤n. It is well known that the monic degree n polynomial that deviates the least from zero on [-1, 1] is given by
T_n(x)=2^-n+1cos(nθ) with x=cos(θ). This polynomial oscillates for x between -1 and 1 and grows faster than any other monic polynomial of the same degree outside [-1, 1]. But how can we describe the monic polynomials of least deviation from zero on E⊆R when E is the union of, say, k intervals or a Cantor-type set? 

 

In the talk, I shall discuss the theory for these polynomials that also bear the name of Chebyshev. Since explicit representations are rare, I’ll focus on their asymptotic behavior and the asymptotics of the approximation error. One may ask how this depends on the size and geometry of E. As we shall see, potential theory enters the field and part of the analysis relies on studying the zeros in gaps of E. Towards the end, I shall also explain how relatively little is known when E is a closed region in the complex plane and discuss some open problems in the field.

The talk is based on joint work with B. Simon (Caltech), P. Yuditskii (JKU Linz), and M. Zinchenko (UNM).

We extend the classical boundary values for (general, three-coefficient) regular Sturm-Liouville operators on compact intervals to the singular case as long as the associated minimal operator is bounded from below, utilizing principal and nonprincipal solutions of the underlying differential equation. 

 

We derive the singular Weyl-Titchmarsh-Kodaira m-function and illustrate the theory with the examples of the Bessel, Legendre, and Laguerre (resp., Kummer) operators.

This is based on joint work with Lance Littlejohn and Roger Nichols.

The sequence of Laguerre polynomials {L_n^(α)(x)}_n=0^∞ is orthogonal on (0,∞) with respect to the weight function e^-x x^α provided α > -1. The classical result that for each natural number n, the zeros of L_n-1^(α)(x) interlace with the zeros of L_n^(α)(x) can (and has) been extended by several authors in various ways.

In 2014, we proved that for each natural number n, the zeros of L_n^(α)(x) and L_n-k^(α+t)(x) are interlacing for each t with 0 < t ≤ 2k and the t-interval 0 < t ≤ 2k is sharp in order for interlacing to hold for every k in {1,2,…, n-1} and each natural number n. Recently, this was extended to zeros of Laguerre polynomials of equal degree, namely, for each natural number n, the zeros of L_n^(α)(x) and L_n^(α+t)(x) are interlacing for each t with 0 < t ≤ 2 and the t interval 0 < t ≤ 2 is sharp in order for interlacing to hold for every natural number n.

For α in the parameter range -2 < α < -1, the polynomial L_n^(α)(x) has n real zeros with n-1 positive zeros and 1 negative zero. However, for α lying in this range -2 < α < -1, the sequence {L_n^(α)(x)}_n=0^∞ is NOT orthogonal with respect to any positive measure and, for each natural number n, the zeros of L_n-1^(α)(x) DO NOT interlace with the zeros of L_n^(α)(x) for any natural number n. However, for each natural number n, the zeros of xL_n-1^(α)(x) interlace with the zeros of L_n^(α)(x).

The aim of the talk is to give some background to these results and raise open problems and questions that arise naturally.

 

We consider various families of random unitary band matrices (CMV and Joye) and study their spectral properties. In particular, we investigate the distribution of the eigenvalues of these matrices and use them to construct matrix models for the circular beta ensemble. We also study spectral properties of these matrix models for beta approaching zero and infinity.

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the nodal sets is obtained for the Robin eigenfunctions. For the analytic domains, we show a sharp upper bound for the nodal sets on the boundary of the Robin and Neumann eigenfunctions. Furthermore, the sharp doubling inequality and vanishing order are obtained.

An approach based on a Riemann-Hilbert problem associated to a simultaneous rational approximation problem involving the Riemann zeta function will be discussed. As a consequence of the proposed approach infinitely many rational approximants (Diophantine approximations) to ζ(3) proving its irrationality can be generated. Moreover, this approach unifies the most famous “independent” proofs of the irrationality of ζ(3). In addition, rational approximants for ζ(5), ζ(7),…, ζ(41),… will be presented.

This talk will focus on the location of zeros of orthogonal polynomials on the unit circle. We will begin with an introduction of some basic results and then briefly review some more recent results of Nevai-Totik and Mhaskar-Saff. We will conclude with new results connecting the zeros of orthogonal polynomials to the zeros of paraorthogonal polynomials and highlight some interesting connections with complex function theory and even algebraic geometry. Some open problems will be discussed along the way. This talk is based on joint work with Barry Simon and Andrei Martinez-Finkelshtein.

In this talk I would like to give you a glimpse into the history of the H^p Hardy spaces starting from its origins in complex analysis, moving into the realm of real analysis first on Euclidean space, then into one-parameter spaces of homogeneous type, and finally into two-parameter spaces of homogeneous type. The tools we will need are “dyadic cubes” and orthonormal “wavelets” in these exotic settings that will allow us to introduce appropriate square functions used in the definition of the Hardy spaces. To show independence of these Hardy spaces on the reference wavelets and dyadic grids, we will introduce atomic Hardy spaces (defined independently of wavelets and their reference grids) and use a crucial tool in the multi-parameter world, a Journé-type lemma adapted to this setting. We are following in the foot steps of R. Coifman, G. Weiss, R. Fefferman, and J. Pipher to mention just a few. This is work in progress joint with Yongsheng Hahn, Ji Li, and Lesley Ward.

Products of random matrices have been introduced in the literature a long while ago, but only in the past few years they received attention from the mathematical physics’ community, much in virtue of the development of new techniques that can be applied to their study. 

 

In this talk we plan to briefly explain some motivations on why we should care about these products, discuss some recent developments when the matrices in the product are independent, and share some of the most recent findings, amongst others by the speaker together with Lun Zhang (Fudan University), in a model when the matrices being multiplied are not independent, but instead coupled in a useful way.

We will focus on asymptotic results obtained when the size of the matrices involved gets large, discussing both universal and non-universal features of the model, and in particular the emergence of the recently found Meijer-G kernel. Some key elements in our analysis are a family of mixed type multiple orthogonal polynomials with respect to Bessel weights, and their zero asymptotic distribution which is described in terms of a constrained vector equilibrium problem.

The topic of Fourier multipliers has been studied for many years since the late 19th century in order to understand the convergence of Fourier series. Following the success in this field, the study of harmonic analysis has been expanded to the class of abelian groups. Recently, there has been a lot of new progress in the area of non-commutative harmonic analysis. As a result, much effort has been made to generalize all the theory into the realm of non-commutative groups. In this talk, we will discuss about non-commutative version of Fourier multipliers on free groups and some results and questions regarding it.

The virtual levels, also known as the threshold resonances and zero energy resonances, admit several equivalent definitions:
(1) there are corresponding virtual states from the space slightly larger than L2;
(2) absence of the limiting absorption principle in the vicinity of the threshold point;
(3) bifurcation of eigenvalues from thresholds under a small perturbation.
We prove the equivalence of these definitions and study properties of corresponding virtual states. Once there is no virtual level, we study the limiting absorption principle near the threshold points, proving that, in the vicinity of a threshold, the resolvent is uniformly bounded in particular weighted spaces. We apply the theory to the Schroedinger operators, although the approach works for general non-selfadjoint operators (and in arbitrary dimension). 

 

This is a joint work with Nabile Boussaid (Besançon) and Fritz Gesztesy (Baylor).

We consider a method of embedding eigenvalues in a band of absolutely continuous spectrum of a periodic Jacobi operator by adding a potential. We first discuss embedding a single eigenvalue and then show that the method can be extended to allow embedding infinitely many eigenvalues into the band.

In this talk we report on a two-step reduction method for spectral problems on a star graph with n+1 edges and a self-adjoint matching condition at the central vertex. 

 

The first step is a reduction to the problem on a single edge, but with an energy depending boundary condition at the vertex. In the second step, by means of an abstract inverse result for m-functions, a reduction to a problem on a path graph with two edges, joined by continuity and Kirchhoff conditions, is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realisations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and M. Krein, we answer the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?

This is joint work with Heinz Langer and Christiane Tretter.

Can one distinguish groups by their group algebras? In this talk, I will try to convince you that the answer is yes, for infinite Vilenkin groups of bounded exponents. 

 

We prove that if G₁ and G₂ are two infinite Vielenkin groups of bounded exponents such that G₁ is not a subgroup of G₂ then there are finite dimensional invariant convolution subalgebras of L¹(G₁) distant from any invariant convolution subalgebras of L¹(G₂). We show that the norm of certain class of algebraic isomorphims between them grows to infinity with the dimension.

Motivated by the numerical analysis of highly oscillatory integrals, we discuss a family of polynomials known as the Kissing Polynomials, which satisfy a non-Hermitian orthogonality condition on the interval [-1,1]. We first discuss how these polynomials can be used to bridge the gap between traditional numerical methods, such as Gaussian quadrature, and the asymptotic methods that are commonly used to handle oscillatory integrals. Next, we discuss various types of asymptotics of these polynomials, where the connection with the asymptotic theory of highly oscillatory integrals becomes apparent. We will end with numerics of the zeros (part of an ongoing work with Guilherme Silva at the University of Michigan) and some open problems concerning the Kissing Polynomials.

 

 

The theory of nonperiodic homogenisation problems for elliptic type equations in divergence form has been conveniently described by the notion of (local) H-convergence for multiplication operators as coefficients. In this talk, we introduce the notion of “nonlocal H-convergence”, which serves as an abstract description of divergence form homogenisation problems, where the coefficients are allowed to be nonlocal. This new notion is a direct generalisation of local H-convergence. The topology induced by nonlocal H-convergence is both compact and Hausdorff. It turns out that this topology is weaker than the strong operator topology and cannot be compared with the weak operator topology. Time permitting, we shall sketch an application to electromagnetic theory. This will lead to new homogenisation results for nonperiodic, nonlocal, fully 3D time-dependent Maxwell’s equations. The talk is based in arXiv:1804.02026.

We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L²→L² and L²→H¹ operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina.

We say that m+1 decreasing numbers h₀, h₁, …, h_m are alternation points for a finite sequence p₀, p₁, …, p_m of orthogonal polynomials if

p_m-n(h_j)=(-1)^jp_n(h_j)
for all j,n=0,…,m. For example, the Chebyshev points h_j=cos(jπ/m), for j=0,…,m are alternation points for the Chebyshev polynomials T₀, T₁,…,T_m. 

 

In this talk, we show that every finite decreasing sequence is a set of alternation points for some finite sequence of orthogonal polynomials and apply this to construct Lagrange polynomials and cubature formulas for the even and odd nodes of the Cartesian product of the points.

I will give a general introduction to quantum information science and, in particular, to quantum information theory (which are not the same!). No prior knowledge of quantum mechanics, computer science, information theory is necessary, although the enthusiasm to hear about them is needed. The talk will be accessible to a graduate student and will be aimed at a general audience. Three main questions will be discussed: “What is a quantum computer?” “Why don’t we yet have one? Or do we?” and “How does one build a quantum computer?”. Everyone who is eager to glimpse into the future of science and our society is welcome to attend.

The Heisenberg uncertainty principle states that it is impossible to prepare a quantum particle for which both position and momentum are sharply defined. A natural measure of uncertainty is entropy. The first entropic formulation of uncertainty principle was proved by Hirschman in 1957 and since then entropic uncertainty relations have been obtained for many scenarios, including some recent advances with quantum memory. In this talk, I will present an approach to entropic uncertainty relations using noncommutative L^p norms. We prove a general entropic uncertainty relations for two quantum channels (completely positive trace preserving maps). We will also discuss the connection between noncommutative L^p Spaces and Renyi information measure sand show how this gives a quantum information approach to von Neumann algebra index. This talk is based on joint works with Marius Junge and Nicholas LaRacuente.

In 1999, Iftimovici and Mantoiu proved a global limiting absorption principle for massive Dirac operators in three dimensions. The first limiting absorption principle for the three-dimensional massless Dirac operator was given by Saito and Umeda in 2008. In this talk, we establish a global limiting absorption principle for the n-dimensional massless Dirac operator based on Kato’s inequality. The spectral theoretic consequences of the global limiting absorption principle for the Dirac operator will be explored.

Let (M,g) be a compact smooth Riemannian manifold. I will discuss some recent off-diagonal estimates for the remainder in the Weyl Law and their applications to the study of random waves, a popular Gaussian model for eigenfunctions of the Laplacian. This is joint work with Y. Canzani.

A binary operator c on an algebra of sets can be considered “a conditional” provided that it satisfies a number of plausible “logical” axioms, such as that A ∩ c(A,B) ⊆ B (if A and A–>B), then B) and c(A,B∩ C)=c(A,B)∩ c(A,C) (A–> BC iff A–> B and A–> C). 

 

If the algebra of sets has a probability measure, a plausible further axiom on conditionals is the Adams Thesis connecting conditionals and conditional probabilities: P(c(A,B)) = P(B|A). There are a number of simple theorems showing that in the presence of various auxiliary assumptions, the Adams Thesis can only hold in trivial cases.

I show, however, that there is a natural collection of logical axioms on conditionals such that any probability space can be extended to a probability space on which there is a conditional satisfying the axioms and the Adams Thesis. The proof uses deep results in measure theory from the middle of the last century: the Maharam classification theorem and the von Neumann-Maharam lifting theorem.

This talk is a brief introduction on the so-called noncommutative L^p spaces. Two main examples will be those associated with matrices and discrete groups.

The Kato-Rosenblum theorem and Aronszajn-Donoghue theory provide us with reasonably good understanding of the subtle theory of rank one d=1 perturbations. We will briefly discuss these statements. When d>1, the situation is different. While the Kato-Rosenblum theorem still ensures the stability of the absolutely continuous part of the spectrum, the singular parts can behave more complicated. We demonstrate this using simple examples. Nonetheless, some positive results prevail in the finite rank setting.

There are operators such as the backward shift operator on various analytic function spaces that cannot be understood by spectral theory. M. J. Cowen and R. G. Douglas, in 1978, defined an important class of operators to study these kinds of operators. They introduced concepts from complex geometry to classify operators up to unitary equivalence. I will discuss some recent progress on the similarity problem of Cowen-Douglas operators. A well-known method of solving various versions of the corona problem is used to come up with a similarity characterization.

We will explain an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules. This is joint work with R. Douglas, M. Jabbari, and G. Yu.

We give a sharp lower bound to the fundamental frequency of a vibrating clamped plate under compression in the context of plates of different shapes of fixed area. Mathematically, the problem is that of bounding the first eigenvalue of a certain 4th-order partial differential operator with leading term the bi-Laplacian from below by a positive constant over the square of the area of the domain. We give a Rayleigh-Faber-Krahn-type result for this problem for small compressions. Thus, our lower bound is saturated for a disk, and the constant appearing in the inequality is that for the disk under the given compression. Our results apply only in two dimensions. Time permitting, possibilities and impediments for the analogs of our results in higher dimensions will be discussed. 

 

This is joint work with Rafael Benguria (P. Universidad Catolica de Chile, Santiago) and Rajesh Mahadevan (Universidad de Concepcion, Chile).

In 1961, Birman proved a sequence of inequalities I_n, for n∈N, valid for functions in C₀ⁿ((0,∞))⊆ L²((0,∞)). In particular, I₁ is the classical (integral) Hardy inequality and I₂ is the well-known Rellich inequality. 

 

In this talk, we give a proof of this sequence of inequalities valid on a certain Hilbert space H_n([0,∞)) of functions defined on [0,∞). Moreover, f∈H_n([0,∞)) implies f^‘∈H_n-1([0,∞)); as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman’s sequence. We also show that for any finite b>0, these inequalities hold on the standard Sobolev space Hⁿ((0,b)). Furthermore, the Birman constants ((2n-1)!!)²/2²ⁿ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L²((0,∞)). We also show that these Birman constants are related to the norm of a generalized continuous Cesaro averaging operator.

We show an improvement of the classical Kolomogorov-Riesz theorem. This is joint work with Harald Hanche-Olsen and Eugenia Malinnikova (both at NTNU).

We will discuss several aspects of mathematical theory designed for recognition of two-dimensional images or “shapes.” An idea to use the so-called “finger-prints” to study two-dimensional ”shapes” goes back to a paper Kahler structure on the K-orbits of a group of diffeomorphisms of the circle of Alexander Kirillov published in 1987. But it was Eiten Sharon and David Mumford who turned this idea (in their 2004 paper 2d-shape analysis using conformal mapping) into a tool, which can be applied for recognition of planar shapes, such as shapes on TV screens and in pictures. After that this theory became quite a popular topic in recent publications on applications of complex analysis to problems in pattern recognition. 

 

  An interesting approach to fingerprint problem was suggested by Peter Ebenfelt, Dima Khavinson and Harold Shapiro in their paper Two-dimensional shapes and lemniscates published in 2011. In this paper, the authors have shown, in particular, that fingerprints of polynomial lemniscates (which, by a classical result due to David Hilbert, are dense in the space of all two-dimensional shapes) are generated by solutions of functional equations, which involve Blaschke products. A simpler proof of the main result of Ebenfelt, Khavinson and Shapiro and its generalization to the case of rational lemniscates was presented in a nice short paper Shapes, fingerprints and rational lemniscates by Malik Younsi published in 2016.

  The first goal of this talk is to discuss how methods of Complex Analysis can be applied to the problems of pattern recognition. In particular, I will discuss the main results on fingerprints obtained by Ebenfelt, Khavinson and Shapiro and by Younsi. In addition, I will also mention a different approach to fingerprints via circle packing which was used by Brock Williams.

  My second goal here is to present my recent results, which include as special cases the Ebenfelt-Khavinson-Shapiro characterization of fingerprints of polynomial lemniscates as well as Younsi’s characterization of rational lemniscates. My main intention here is to emphasize the role of quadratic differentials in this developing theory.

In 1988 Carl Cowen completely characterized hyponormal Toeplitz operators acting on the Hardy space. In the Bergman space setting, similar studies have mostly focused on Toeplitz operators with harmonic symbols. In this talk, we will examine these results and give some new results on the hyponormality of Toeplitz operators acting on the Bergman space with non-harmonic symbols. This is joint work with Conni Liaw.

The boundedness of Fourier multipliers is a central topic in analysis. I will review the so-called Mikhlin condition for Fourier multipliers to be bounded on Euclidean L^p spaces in the first half of the seminar. In the second half, I will explain an extension of the classical Mikhlin multiplier theorem to Free groups (recent joint work with Q. Xu).

This is a three-part series. The goal of the lecture series is to show that the Hardy-Littlewood maximal operator is bounded on L^p(w) if and only if w lies in the Muckenhoupt A_p class, via the reverse-Holder inequality.

TBA.

We introduce the notion of stationary actions in the context of C^*-algebras. As an application of this concept we prove a new characterization of C^*-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic and conceptual understanding of why C^*-simplicity is stronger than the unique trace property. In addition it allows us to conclude C^*-simplicity of new classes of examples, including recurrent subgroups of C^*-simple groups. This is joint work with Yair Hartman.

None

The Vitali Covering Lemma states that, given a finite collection of balls in ℝⁿ, there exists a disjoint subcollection that fills at least 3⁻ⁿ of the measure of the union of the original collection. We present classical proofs of this lemma due to Banach and Garnett. Subsequently, we provide a new proof of this lemma that utilizes probabilistic “Erdös” type techniques and Padovan numbers.

In this talk I will discuss a formula relating the spectral flow of the one-parameter families of second order elliptic operators to the Maslov index, the topological invariant counting the signed number of conjugate points of certain paths of Lagrangian planes. In addition, I will present formulas expressing the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of second order differential operators. The talk is based on joint work with Yuri Latushkin.

We study relations between probability measures μ on the unit circle and their sequences of Verblunsky coefficients (coefficients in the recurrence relation obeyed by orthogonal polynomials with respect to μ). The Szego theorem is a celebrated result proving that the logarithm of the absolutely continuous part of μ is integrable if and only if the sequence of Verblunsky coefficients is square-summable. We will discuss some recent results on higher-order Szego theorems, which are similar equivalence statements relating weaker integrability conditions on the measure to weaker decay, and bounded variation, conditions on the coefficients. We will also discuss a related problem of determining the asymptotic behavior of the weight at the critical point for power-law decaying coefficients.

We extend Schoenberg’s classical theorem, which relates conditionally positive definite functions F:Rⁿ → C, n ∈ N, to their positive semidefinite exponentials exp(tF), t>0, to matrix-valued conditionally positive semidefinite functions F:Rⁿ → C^mxm, m ∈ N. Moreover, we study the closely related property that the multiplier operators exp(tF)(-i∇), t>0, is positivity preserving and its failure to extend directly to the matrix-valued context. If time permits, we will discuss some of the main tools used in the proofs. (Joint work with Fritz Gesztesy.)

In the early 2000’s, Littlejohn and Wellman developed a general left-definite theory for certain self-adjoint operators which explicitly determined their domains. However, the description of these domains do not contain boundary conditions. We present characterizations of these domains given by the left-definite theory for all operators which possess a complete system of orthogonal eigenfunctions, in terms of classical boundary conditions.

This talk centers on Krein-type resolvent identities and boundary data maps associated to self-adjoint extensions of minimal three-coefficient Sturm–Liouville operators on a finite interval. We consider various parameterizations of the self-adjoint extensions, give the explicit form of Krein’s resolvent identity in terms of boundary conditions, and identify the Krein-von Neumann extension in the special case when the underlying minimal operator is strictly positive. Boundary data maps for general self-adjoint extensions, and their application to Krein’s resolvent identity, trace formulas, and symmetrized perturbation determinants, are discussed. This talk is based on joint work with Stephen Clark, Fritz Gesztesy, and Maxim Zinchenko.

I will discuss a class of von Neumann algebras introduced by Hiai, called the q-Araki-Woods algebras. These operator algebras are constructed by combining two well-known deformations of free group von Neumann algebras: the q-Gaussian deformations of Bożejko and Speicher, and the type III deformations of Shlyakhtenko. The main focus of this talk will be on the problem of characterizing (and computing norms of) of a simple class of linear maps on these algebras, called completely bounded radial multipliers. I will describe some joint work with Steve Avsec and Mateusz Wasilewski in this direction. Although we are still very far from any sort of full classification of the cb radial multipliers on q-Araki-Woods algebras, our work on this problem allow us to establish an extremely useful finite-rank approximation property for these algebras: they always have the complete metric approximation property.

For h-periodic coefficients and any integer k>2 it is well known that the eigenvalues of some self-adjoint complex boundary condition on the interval [a,a+h] are the same as the periodic eigenvalues on the interval [a,a+kh]. For each k we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. In addition, we prove an analogous result for semi-periodic eigenvalues.

The story will be on a joint work with Brian Simanek and Tim Ferguson. I will review the meaning and connections between the three objects in the title and will explain the main result that every Markov semigroup generator satisfying Γ₂ criterion has a bound H\infty calculus on BMO with an optimal angle.

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Let D be a bounded domain of the complex plane and let {P_n} be the sequence of orthonormal polynomials over D, that is, for each n=0,1,2,… P_n has degree n and positive leading coefficient, and

∫_D P_n(z)P_m(z)dA(z)=δ_n,m.

The asymptotic behavior of P_n(z) as n→∞ has been investigated to a considerable extent, particularly when the region D is a simply connected domain bounded by a Jordan curve. In this work we investigate the situation when the domain of orthogonality is no longer simply connected but a multiply connected domain, trying to understand how the holes in the domain influence the asymptotic behavior of P_n inside D. Of particular interest is the canonical case of a circular multiply connected domain (CMCD) consisting of a disk with a finite number of mutually disjoint subdisks removed. We will also consider domains D that are conformally equivalent in a strong sense to a CMCD. This is a joint work with James Henegan.

Hardy and Littlewood studied L^p-l^p multiplier on circle and there have been some progress in this direction. In particular, the inequality was studied on compact homogeneous spaces recently. In this seminar, I will talk about such inequalities on compact quantum groups, mainly on reduced group C^* algebras and free quantum groups.

In this talk we consider a certain class of identities involving determinants of Hermite polynomials. The prototypical example of such a determinant is the Wronskian of a finite set of Hermite polynomials. Remarkably, all such Wronskians may be re-expressed as an infinity of other determinants of a certain structure, determinants that we refer to as “pseudo-Wronskians”. The entire theory can be easily understood in terms of Maya diagrams and partitions. This approach allows us to find the optimal determinantal representation of any given Hermite Wronskian. Time permitting, we will also discuss applications to exceptional Hermite polynomials and to rational solutions of the Painleve IV equation.

At the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications, held in Patras, Greece, 1999, W.N. Everitt, K.H. Kwon, and L.L. Littlejohn presented ten open problems. Problem 10 called for a GKN-type theory for identifying self-adjoint operator representations of a Lagrangian symmetric differential expression in a Hilbert space L²([a,b];dμ) where dμ is a discrete Sobolev measure of the form 

 

 

 

dμ = w + Σ_j=0ⁿ(α_jδ_a^(j)+β_jδ_b^(j)) α_j,β_j≥ 0; n ∈ N₀.
In particular Problem 10 asked for a ‘recipe’ for finding self-adjoint differential operators generated by a Bochner-Krall spectral differential expression having the associated orthogonal polynomial sequence as eigenfunctions. 

 

In this lecture we will discuss the history and motivation of this question and present a new GKN-EM type theory that resolves this long-standing problem.

We study the asymptotic distribution of zeros for the random polynomials P_n=Σξ_kB_k(z), where {ξ_k}_k=0^∞ are non-trivial i.i.d. complex random variables. The polynomials {B_k}_k=0^∞ are deterministic, and are selected from a standard basis such as Bergman or Szego polynomials associated with a Jordan domain G bounded by an analytic curve. We show that the zero counting measures of P_n converge almost surely to the equilibrium measure on the boundary of G if and only if the expected value of log⁺|ξ₀| is finite. This talk is based on joint work with Igor Pritsker.

Many PDEs can be usefully rewritten as abstract ODEs of the form u’ = Au, where A is a typically unbounded operator on a Banach space. In the linear case, the Hille-Yosida theorem allows us to determine precisely when A generates a strongly continuous semigroup of operators, from which one can deduce that the PDE is well-posed. In the nonlinear case, an analogy to Hille-Yosida can be found for so-called “maximal monotone” operators. In this talk I will outline the theory of monotone operators and nonlinear semigroups, give important examples, and briefly describe how to apply the theory to certain nonlinear PDEs.

Two of the most important spaces in modern function theory are the Hardy and Bergman spaces. In this talk, I will give a brief survey of some of the similar differences in the structure of these spaces, as well as the flavor of solving problems in these different settings. In particular we will look at the differences between the zeros of Hardy and Bergman functions, and look at the differences in the Invariant Subspace Problem in both Hardy and Bergman settings.

In this talk, I will discuss extremal problems in spaces of analytic functions and connections between such problems and other areas of analysis. The main focus will be the problem of maximizing linear functionals on Bergman spaces, which are spaces of analytic functions of finite L^p norm. I will speak about the connections of this problem to topics in analysis such as partial differential equations, quasiconformal mappings, and uniform convexity.

An old problem that goes back to Halmos asked whether a pair of unitary matrices U and V that almost commute (in the sense that the operator norm of the commutator UV – VU is small) can always be slightly perturbed to obtain a pair of commuting unitary matrices. Voiculescu produced examples in the 80s that gave a negative answer, and commented that they seemed to depend on the cohomology of the two-torus. Later, Exel and Loring provided a direct link by giving a formula that related the K-theory of the two-torus to these almost commuting unitaries. 

 

We will discuss previous and ongoing work with Marius Dadarlat that elucidates this connection further. In fact, regarded in the right light, the answer comes from an index theory calculation that relates deformations of the fundamental group Z² of the two-torus with “almost flat” bundles over the two-torus. In this context, the theory generalizes to other surface groups, and beyond.

There are many classical results about operator-theoretic properties of the compressed shift on one-variable model spaces, especially spaces associated to finite Blaschke products. In this talk, we will discuss generalizations of such results to the setting of two-variable model spaces associated to rational inner functions on the bidisk. Among other things, we will discuss characterizations and properties of the numerical range and radius of compressed shifts on two variable model spaces as well as when the commutator of a compressed shift with its adjoint has finite rank. This is joint work with Constanze Liaw and Pam Gorkin.

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The n^th root asymptotic behavior of some discrete multiple orthogonal polynomials is presented. Two main ingredients of the proposed approach for the study of the aforementioned asymptotic behavior are discussed; namely, an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution and the limiting behavior of the coefficients of the recurrence relations for multiple orthogonal polynomials.

I will give a brief introduction of the notion of finite dynamic asymptotic dimension and its applications to algebraic K-theory. In particular, for a large class of groups, we prove the Farrell-Jones conjecture with coefficients in the algebra of Schatten class operators. It is based on joint work with Guoliang Yu.

Let B be a collection of open sets in ℝⁿ. Associated to B is the geometric maximal operator M_B defined by

M_Bf(x) = sup_{x ∈ R ∈ B}∫_R|f|.
For 0 <α< 1, the associated Tauberian constant C_B(α) is given by

C_B(α) = sup_{E ⊆ ℝⁿ : 0 <|E|< ∞}|{x ∈ ℝⁿ : M_Bχ_E(x)>α}|/|E|.
A maximal operator M_B such that lim_{α → 1^–}C_B(α) = 1 is said to satisfy a Solyanik estimate.
In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates. This research is joint with Ioannis Parissis.

A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is showen. No special geometrical condition are imposed on the inaccessible part of the boundary of the domain, apart from that that the boundary of the domain is C1,1. The coefficients are assumed to coincide on a neighbourhood of the boundary: a natural property in many applications.

I’ll discuss a geometric approach to graph partitioning where the optimality criterion is given by the sum of the first Dirichlet-Laplacian eigenvalues of the partition components. This eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective. I’ll give a consistency result for geometric graphs, stating convergence of graph partitions to an appropriate continuum partition. The model has a semi-supervised extension, provides natural representatives for the clusters, and is related to an interesting random process.

Dirichlet types spaces D_α, α∈R, are a well-studied range of spaces; each consisting of analytic functions f on the unit disk D with the property that a certain weighted sum of Fourier coefficients of f is bounded. These weights depend on α. The classical Hardy space furnishes the most prominent example. 

 

A function f(z) is said to be cyclic in D_α if the sequence of forward orbits z^k f(z), k∈{0,1,2,…}, has a dense span in D_α. Since the constant function g=1 is cyclic, we set out on an endeavor to learn about the approximation of the function 1 by functions of the form p_n f for fixed f∈ D_α and polynomials p_n of varying degrees n.

We look for hypercontractivity results for the Poisson semi-group associated to free groups. We rely mainly on probabilistic methods combined with a noncommutative two-points inequality.

One of the basic spectral transformations of orthogonal polynomials on the real line corresponds to the multiplication of the orthogonality measure dµ(t) by t provided that all orthogonal polynomials for the original and resulting measures exist. This transformation is called Christoffel transformation. Intuitively, the inverse transformation to Christoffel transformation can be gotten by dividing the measure by t and this is another basic spectral transformation. The latter one is known as Geronimus transformation. In fact, both of them are discrete versions of the famous Darboux transformations and it turns out that the consistency of Christoffel and Geronimus transformations leads to the discrete-time Toda equation, which is basically the qd-algorithm, one of the most important tools in numerical analysis.
In my talk I’m going to discuss some nonclassical situations in regard to spectral transformations of orthogonal polynomials on the real line. Namely, I’ll start with showing how spectral transformations help to deal with signed measures with one sign change and with the corresponding non-symmetric tridiagonal matrices. In particular, the theory of such matrices brings us to the context of Padé approximation and, more importantly, the behavior of spurious poles of Padé approximants at infinity for Cauchy transforms of signed measures with one sign change is fully characterized. Then, we’ll see that to get such results for signed measures with several sign changes and other singularities one needs to consider multiple Christoffel and Geronimus transformations. Finally, it’ll be demonstrated that multiple Geronimus transformations could lead to Sobolev orthogonal polynomials.

We construct an Anderson–type Hamiltonian in its spectral representation via the iterative introduction of rank one perturbations. We remain in control of the expected value for the total mass of the resulting operators’ absolutely continuous part.

The Circular Beta Ensemble is a family of random unitary matrices whose eigenvalue distribution plays an important role in statistical physics. The spectral measure is a canonical way of describing the unitary matrix that takes into account the full operator, not just its eigenvalues. When the matrix is infinitely large (i.e. an operator on some infinite-dimensional Hilbert space) the spectral measure is supported on a fractal set and has a rough geometry on all scales. This talk will describe the analysis of these fractal properties. Joint work with Raoul Normand and Balint Virag.

In multivariable operator theory, joint spectra are defined to gauge the interaction of operators. If a tuple of operators is commuting, then the Taylor spectrum, which is defined through a Koszul complex, played a fundamental role in the theory. However, for noncommuting tuples spectral theory remains a mystery. In 2008, the notion of projective joint spectrum was introduced by the speaker. It appears to be the simplest kind of spectrum one can define for non-commuting tuples. Yet, this simplicity turns out to be the very source of its richness.

Geometric inequalities provide insight into the structure of manifolds. Our objective is to develop deeper understanding for how sharp constants for function-space inequalities encode information about the geometric structure of the manifold. Functional forms that characterize smoothness lie at the heart of understanding and rigorously describing the many-body interactions that determine the behavior of dynamical phenomena. Smoothing estimates provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space.

With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C^*-algebraic results and theories to more general algebras. We have continued this work together with Read, and also with Matthew Neal, and with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. We describe some of this work, and the ideas behind it.

Riesz transforms and Hormander Fourier multipliers are central objects in the classical Fourier analysis. I will explain recent work on Riesz transforms associated with group 1-cocycles. It is a new point of view of P. A. Meyer and D. Bakry’s Riesz transforms associated with Markov semigroups. From this viewpoint, all classical Hormonder Fourier multipliers are essentially Riesz transforms. The talk is based on a joint work with Junge and Parcet.

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential
operator A in L²(-1,1) having the Legendre polynomials {P_n}_n=0^∞ as eigenfunctions. As a particular consequence, they explicitly determine the domain D(A²) of the self-adjoint operator A². However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L²(-1,1) meaning that D(A²) should exhibit four boundary conditions. In this talk, after a gentle ‘crash course’ on left-definite theory and the classical Glazman-Krein-Naimark (GKN) theory, we show that D(A²) can, in fact, be expressed using four (separated) boundary conditions. In addition, we determine a new characterization of D(A²) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory.

I will briefly introduce the theory of thick distributions. I will also introduce the asymptotic expansions of distributions and of thick distributions. As an example, I will show how recent observations of Kolomeisky et al. fit into the established framework of the distributional asymptotics of spectral functions. A common tool in Casimir physics (and many other areas) is the asymptotic (high-frequency) expansion of eigenvalue densities, employed as either input or output of calculations of the asymptotic behavior of various Green functions.

We will use Smale’s 7th problem to motivate the polarization problem. This is a classic problem in potential theory that provides the crudest bounds on the effectiveness of a greedy algorithm for finding optimal energy configurations. I will mention some of my own recent work on this problem along with some directions for future research. Many open problems will be provided throughout this talk.

This will be an introduction to the research field of minimum energy problems. We will start by motivating the topic with some big open problems like sphere packing problems and Smale’s 7th problem for the 21st century. A large portion of the talk will focus on equilibrium measures and leading order asymptotic of minimal energies. This talk will be accessible to non-specialists and serve as a primer for the higher-level research talk on September 23.