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Plenary Titles and Abstracts
Spectral theory for Dirac operators with singular potentials
Speaker: Jussi Behrndt, Graz University of Technology
Abstract: In this talk, we discuss qualitative spectral properties of self-adjoint Dirac operators. We first briefly review some of the standard results for regular potentials from the literature and turn to more recent developments afterward. Our main objective in this lecture is to discuss singular potentials supported on curves or hyperplanes, where it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties.
This talk is based on joint some recent works with P. Exner, M. Holzmann, V. Lotoreichik, T. Ourmieres-Bonafos, and K. Pankrashkin.
Hyperholomorphic spectral theories and applications
Speaker: Fabrizio Colombo, Politecnico di Milano
Abstract: The aim this talk is to give an overview of the spectral theories associated with the notions of holomorphic- ity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators (A1,...,An). A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic func- tions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholo- morphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.
Principles of Energy Harvesting in Stochastic Thermodynamic Engines
Speaker: Tryphon Georgiou, University of California, Irvine
Abstract: The recent confluence of three subjects, Stochastic Control, Optimal Mass Transport, and Stochastic Thermodynamics, has allowed deeper understanding of the mechanism by which physical contraptions (whether engineered or biological) can transform heat differentials or, as in the age-long conundrum of Maxwell's demon, information into useful work. Our goal in the talk is to overview some of these developments and highlight the geometric framework that allows quantitive assessments on the performance that stochastic thermodynamic engines are capable of. We will then specifically focus on Brownian gyrating engines that consist of over-damped particles that are fed by sources of stochastic excitation and reside in a controlled potential.
The talk is based on joint works with Rui Fu (UCI), Olga Movilla (UCI), Amir Taghvaei (UCI) and Yongxin Chen (GaTech). Research funding by NSF and AFOSR is gratefully acknowledged.
Time-and-band limiting: finding differential operators that commute with naturally appearing integral operators
Speaker: F. Alberto Grünbaum, University of California, Berkeley
Abstract: A series of remarkable papers by D. Slepian, H. Landau and H. Pollak (Bell Labs around 1960) show that the operators of time-band limiting admit commuting differential operators. A search for the reason behind this unexpected and extremely useful accident has produced over the year a large collection of new such examples. This has deep connections with nonlinear integrable systems such as the Kadomtsev-Petviashvili equations and its master symmetries and many other areas of mathematics. Natural areas of applications include several areas of signal processing as well as random matrix theory.
Semigroups arising in third order in time dynamics with applications to nonlinear acoustics
Speaker: Irena Lasiecka, University of Memphis
Abstract: A third-order (in time) Partial Differential Equation (PDE) sys- tems arise naturally in a variety of second order PDE models where time relaxation parameter accounts for an extra derivative, which then leads to a singularly perturbed dynamics. It has been known since the sixties that such models, even in linear case, may be ill-posed in the sense of semigroups. This has motivated an extensive studies of third order dynamics from the point of view of semi- group theory. A class of third order models arising in nonlinear acoustics will be discussed. Such nonlinear (quasi-linear) Partial Differential Equation (PDE) describes nonlinear propagations of high frequency acoustic waves and it is motivated by an array of applications in engineering and medical sciences-including high intensity focused ultrasound [HIFU] technologies. The important feature is that the model resolves the infinite speed of propagation paradox associated with a classical second order in time equation. Replacing a classical heat trans- fer by heat waves gives rise to the third order in time derivative scaled by a small parameter τ > 0, the latter represents the thermal relaxation time parameter and is intrinsic to the properties of the medium where the dynamics occurs.
The aim of the lecture is to provide a brief overview of recent results in the area which are pertinent to generation of both linear and non-linear semigroups and their asymptotic behavior with vanishing relaxation parameter τ ≥ 0.
Peculiar features associated with the third order dynamics lead to novel phenomenological behaviors.
An Overview of the Mathematics of Superoscillations.
Speaker: Daniele Struppa, Chapman University
Abstract: Yakir Aharonov first identified an apparently odd phenomenon that he called superoscillations in the context of his theory of weak values. Roughly speaking a superoscillatory sequence (or a superoscillation in brief) is a band-limited function that can oscillate faster than the highest frequency that it contains. In this talk, based on a series of papers over the last ten years or so, I will introduce the notion of superoscillations, I will explain the nature of their surprising behavior, and I will discuss in some detail one of the most important question regarding such functions. Specifically, suppose we are considering an initial value problem for the Schrodinger equation, and suppose the initial value is a superoscillating function. The question is whether the solution to this initial value problem is still superoscillating. We usually refer to this question as the longevity question for superoscillations. In the talk I will show in detail the answer for the simple case of the free particle, and I will highlight the general principles of the theory that is necessary to study more complex cases of longevity. I will conclude the presentation with a fairly large literature review and ideas for further explorations.
Spectral Inclusions for Operators with Spectral Gaps
Speaker: Christiane Tretter, University of Bern
Abstract: Analytical information about the spectra and resolvents of non-selfadjoint operators is of great importance for applications and numerical analysis. However, even for perturbations of selfadjoint operators there are only a few classical results. In this talk relatively bounded, not necessarily symmetric perturbations of selfadjoint operators with spectral gaps are considered. We present new spectral inclusion results and various modifications e.g. for gaps of the essential spectrum or for infinitely many gaps, and several applications.
Boundary Feedback Stabilization of Fluids in Besov Spaces of Low Regularity
by Means of Finite Dimensional Controllers: 3D Navier-Stokes Equations and
Boussinesq Systems
Speaker: Roberto Triggiani, University of Memphis, University of Virginia
Abstract: We shall present two main recent results. First, the 3D-Navier-Stokes equations can be uniformly stabilized in the vicinity of an unstable equilibrium solution by means of a ’minimally’ invasive, localized, boundary-based, tangential, static, feedback control strategy, which moreover is finite dimensional. Finite dimensionality in 3D was an open problem. Its solution required a new, suitable, tight Besov space setting of low regularity. Next, an analogous result for the Boussinesq system, coupling the N-S equations with a heat equation. In both cases, unique continuation properties of suitably over-determined adjoint eigen-problems play a critical role.
Rigidity for II_1 factors
Speaker: Stefaan Vaes, University of Leuven
Abstract: Discrete groups and their actions on probability spaces give rise to II_1 factors. When the group is amenable, by Connes' theorem, we essentially always get the unique hyperfinite II_1 factor. In the nonamenable case, Popa's deformation/rigidity theory has led to striking rigidity theorems, including W*-superrigidity results where the group and its action can be entirely recovered from the ambient II_1 factor. I will give a survey of some of these results, including the computation of invariants of II_1 factors and the challenging problem of deciding when II_1 factors can be embedded one into the other.
Invariance of absolutely continuous spectra and quasicentral modulus
Speaker: Dan-Virgil Voiculescu, University of California, Berkeley
Abstract: The quasicentral modulus is a numerical invariant for n-tuples of operators which appears to play a key role in normed ideal perturbations of operators and multivariable generalizations of the theorems of Kato-Rosenblum and Weyl-von Neumann-Kuroda. I will discuss recent advances and some open problems.
Semi-Plenary Titles and Abstracts
Linear systems and differential equations in random matrix theory
Speaker: Gordon Blower, Lancaster University, UK
Abstract:The aim of this talk is to solve certain nonlinear differential equations from random matrix theory using linear systems.
A linear system (-A,B,C) with state space Hilbert space H can be used to define a Hankel integral operator on L^2 which has a Fredholm determinant. An alternative description is in terms of a class of operators studied by Howland, who observed an analogy between Schroedinger differential operators on the real line and the Hankel integral operator with kernel 1/(x+y). The Fredholm determinant determines a tau function, which depends upon various parameters in the linear system. As an illustration, the talk give solutions to the sinh-Gordon PDE and Painleve III' transcendental ordinary differential equation. These differential equations arise in random matrix theory, and have applications to MIMO in wireless communications.
The work arises in collaboration with Yang Chen (Macau), and Ian Doust (UNSW, Australia).
Quaternionic non-selfadjoint operators and their spectral theory
Speaker: Uwe Kähler, University of Aveiro
Abstract: One of the principal problems in studying spectral theory for quaternionic or Clifford-algebra-valued operators lies in the fact that due to the noncommutativity many methods from classic spectral theory are not working anymore in this setting. For instance, even in the simplest case of finite rank operators there are different notions of a left and right spectrum. Hereby, the notion of a left spectrum has little practical use while the notion of a right spectrum is based on a nonlinear eigenvalue problem. In the present talk we will recall the notion of S-spectrum as a natural way to consider a spectrum in a noncommutative setting and use it to study quaternionic non-selfadjoint operators. To this end we will discuss quaternionic Volterra operators and triangular representation of quaternionic operators similar to the classic approaches by Gohberg, Krein, Livsic, Brodskii and de Branges. Hereby we introduce spectral integral representations with respect to quaternionic chains and discuss the concept of P-triangular operators in the quaternionic setting. This will allow us to study the localization of spectra of non-selfadjoint quaternionic operators.
Shift Operators on Harmonic Hilbert Function Spaces on Real Balls and von Neumann Inequality
Speaker: H. Turgay Kaptanoğlu, Bilkent University, Ankara, Turkey
Abstract: On harmonic function spaces, we define shift operators using zonal harmonics and partial derivatives, and develop their basic properties. These operators turn out to be multiplications by the coordinate variables followed by projections on harmonic subspaces. This duality gives rise to a new identity for zonal harmonics. We introduce large families of reproducing kernel Hilbert spaces of harmonic functions on the unit ball of $\mathbb R^n$ and investigate the action of the shift operators on them. We prove a dilation result for a commuting row contraction which is also what we call harmonic type. As a consequence, we show that the norm of one of our spaces $\breve{\mathcal G}$ is maximal among those spaces with contractive norms on harmonic polynomials. We then obtain a von Neumann inequality for harmonic polynomials of a commuting harmonic-type row contraction. This yields the maximality of the operator norm of a harmonic polynomial of the shift on $\breve{\mathcal G}$ making this space a natural harmonic counterpart of the Drury-Arveson space.
This is joint work with Daniel Alpay of Chapman University, Orange, CA.
Certification of quantum devices via operator-algebraic techniques
Speaker: Laura Mancinska, University of Copenhagen
Abstract: In this talk, I will introduce the concept of self-testing which aims to answer the fundamental question of how do we certify proper functioning of black-box quantum devices. We will see that operator-algebraic techniques can be applied to this area and that there is a close link between self-testing and stability of algebraic relations. We will leverage this link to propose a family of protocols capable of certifying quantum states and measurements of arbitrarily large dimension with just four binary-outcome measurements. One of our main proof ingredients is a certain algebraic analogue of Gowers-Hatami stability theorem for group representations.
This is a joint work with Chris Schafhauser and Jitendra Prakash.
Bianalytic mappings between free spectrahedra
Speaker: Scott McCullough, University of Florida
Abstract: Many optimization problems in systems and control engineering can be formulated in terms of Linear Matrix Inequalities, LMIs. The solution set of an LMI is a spectrahedron. Polydiscs and Matrix balls are examples of spectrahedra. The fully matricial solution set of an LMI, known synonymously as an LMI domain or free spectrahedron, has close ties to operator systems and related topics such as quantum information theory. The natural class of mappings between free spectrahedra are free analytic maps. This talk will discuss the problem of classifying the free bianalytic maps between free a pair of spectrahedra, with some emphasis on automorphisms.
On a Function of G.H. Hardy and J.E. Littlewood
Speaker: Ahmed Sebbar, Chapman University
Abstract: Notes on the Theory of Series (XX); On Lambert Series, Proceedings of the London Mathematical Society, Ser. (2), 61 (1936), 257-270, Hardy and Littlewood considered an important series and its expansion in terms of a Bessel function. We show that this formula is actually a part of a larger construction and we explain the link to a theorem of Beurling on Riemann zeta function and to the zeta function of some ternary quadratic forms.
This is joint work with Roger Gay.
Lipschitzness of operator functions
Speaker: Anna Skripka, University of New Mexico
Abstract: We will discuss Lipschitzness of operator functions with respect to Schat- ten norms in the case of both compact and noncompact perturbations. The latter naturally arise in problems of mathematical physics and noncommuta-tive geometry. We will consider Lipschitz-type bounds for operator functions and characterizations of operator Lipschitzness in terms of familiar properties of the respective scalar functions. Both the celebrated results for compact perturbations and new results for noncompact perturbations rest on multi-linear operator integration, a powerful technical method with a long history in noncommutative analysis.
Perturbations of periodic Sturm–Liouville operators
Speaker: Carsten Trunk, TU Ilmenau, Germany
Abstract: The work by G.W. Hill in 1886 has led to the ‘Hill’s equation’ for the linear second-order ordinary differential equation with periodic coefficients. This time-independent Schr ̈odinger equation in one spatial dimension with a periodic potential is used within the description of certain effects of atomic nuclei in a crystal. Here the spectral parameter λ has a physical interpretation as the total energy of an electron, and the band structure of the essential spectrum to regions of admissible and forbidden energies. Moreover, impurities (i.e. perturbations) can lead to additional discrete energy levels in the forbidden regions (i.e. eigenvalues in the gap of the essential spectrum). Here we investigate the change of the spectrum under L1-assumptions on the differences of the coefficients. We describe the essential spectrum and the absolutely continuous spectrum of the perturbed operator. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by the Ukrainian mathematician Rofe-Beketov from the 1960ies.
This is based on joint works with J. Behrndt (Graz), P. Schmitz (Ilmenau), and G. Teschl (Vienna).
Toeplitz operators on the Bergman space
Speaker: Nikolai L. Vasilevski, CINVESTAV, Mexico City
Abstract: The talk is intended for a wide audience, not necessarily consisting of experts in the theory of Toeplitz operators, and is a review of the results on the description of algebras generated by Toeplitz operators. We begin with a somewhat surprising and unpredictable result on the existence of a large class of non-isomorphic commutative C∗-algebras generated by Toeplitz operators. As it turned out, their symbols must be invariant under the action of maximal Abelian subgroups of the biholomorphisms of the unit ball.
The next surprise was the discovery of a large number of Banach (not C∗) algebras, which turned out to be, as a rule, not semisimple. The problem here is to find a compact set of maximal ideals and to describe the radical.
Finally we consider non-commutative C∗-algebras generated by Toeplitz operators whose symbols are invariant under the action of a subgroup of some maximal Abelian group of biholomorphisms. It turned out that different types of action of the same subgroup lead to completely different properties of the corresponding algebras.
Integral representation formulae and residue calculus with applications to interpolation
Speaker: Alain Yger, University of Bordeaux
Abstract: Integral representation formulae with weights (Bochner-Martinelli, Cauchy-Weil, Cauchy-Fantappié, etc.) have been extensively developed since more than 20 years. The same for what concerns multivariate residue theory, which revealed to be quite an efficient tool to provide closed formulae of the Kronecker-Jacobi type solving explicitly the Bézout identity in the algebraic setting, more generally in weighted algebras of entire functions such that the Paley-Wiener algebra. Despite the fact that such residue calculus highly relies on commutativity, it seems that some technics which support it could be transposed to operator theory (residues being from the beginning defined as traces of operators!). Also the crucial role played by distributions or currents is not so well known outside the world of multivariate complex analysis. I will present in this talk a selection of examples which motivate the use of such residue theory from a concrete point of view and, at the same time, suggest some transposition to non-commutative horizons. My collaboration with D. Alpay since five years motivated indeed the topics I will discuss in this talk.