Invited Speakers

Pavlos Motakis, York University, Toronto

Title: Methods for constructing Banach spaces with prescribed properties in their operator spaces

Abstract: One of the aspects of the classical fields of Banach space theory is the construction of Banach spaces X such that L(X), the algebra of bounded linear operators on X, satisfies a prescribed property. Main examples of this are the Lοο-space by Argyros and Haydon on which every bounded linear operator is a scalar multiple of the identity plus a compact operator and the reflexive space by Argyros and the speaker on which every bounded linear operator has a non-trivial closed invariant subspace. Other recent developments show that the quotient algebra of L(X) over its compact operator ideal can be any separable C(K) space and even isomorphic to a Hilbert space. These achievements are based on a long and rich history of construction techniques developed by Tsirelson, Maurey, Rosenthal, Bourgain, Delbaen, Schlumprecht, Gowers, Argyros, Deliyanni, and others. In this mini-course, we will explore the main milestones in the development of this theory and focus on some of the details of certain representative examples.

Invited Talks

George Androulakis, University of South Carolina

Title: Connections between classical and quantum information theory

Abstract: In this introductory talk I will focus on the following questions: How can classical information be quantified? How can quantum information be quantified? How can two classical information sources be distinguished? How can two quantum information sources be distinguished? The goal of the talk will be to give some connections between classical and quantum information theory while exploring the answers to the above questions. No previous knowledge of classical or quantum information theory will be assumed.

Georgios Katsimpas, Harbin Engineering University

Title: Operator-algebraic aspects of Free Probability Theory

Abstract: Motivated by the (still open) free group factor isomorphism problem, Free Probability theory was originated by Voiculescu in the 1980’s as an extension of classical probability theory, and views bounded linear operators on Hilbert spaces as non-commutative analogs of classical random variables. Free Probability shares central connections with diverse mathematical fields, including operator algebras, random matrix theory and combinatorics. In this talk, we will present an overview of the fundamental concepts encountered in the theory of Free Probability, with particular emphasis on the notion of free independence, and present various applications in the field of operator algebras. A major modern research area in this field concerns the development of notions of entropy in the non-commutative context, which gave answers to longstanding open problems regarding the structure of the free group factors. In this direction, we will discuss the development of non-microstate notions of entropy within the field of bi-free probability theory.

Elias Katsoulis, East Carolina University

Title: Fell’s absorption principle for semigroup operator algebras

Abstract: Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper we explain how to extend Fell’s absorption principle to an arbitrary submonoid P of a group G by using an enhanced version of the left regular representation. Li’s semigroup C*-algebra C*(P) and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for C*(P) but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. Other applications will also be discussed.

Sophocles Mercourakis, National and Kapodistrian University of Athens

Τίτλος: Ομοιόμορφη Κατανομή ακολουθιών και η αλληλεπίδρασή της με τη Συναρτησιακή Ανάλυση

Περίληψη: Στην ομιλία αυτή θα παρουσιάσουμε πρώτα κάποιες εφαρμογές της Θεωρίας της Ομοιόμορφης Κατανομής ακολουθιών (Uniform Distribution of sequences) στη Συναρτησιακή Ανάλυση και κατόπιν έννοιες και αποτελέσματα της ίδιας Θεωρίας, τα οποία θέτουμε σε ένα ευρύτερο πλαίσιο. Στο πλαίσιο αυτό εντάσσεται και η γενίκευση ενός κλασικού αποτελέσματος αυτής της Θεωρίας, το οποίο ισχύει για αύξουσες συναρτήσεις, για την κλάση των συναρτήσεων φραγμένης κύμανσης.

Mihalis Mourgoglou, University of the Basque Country & Ikerbasque

Title: Varopoulos extensions and Applications to Boundary Value Problems in rough domains

Abstract: In this talk I will talk about some recent advances in solvability of Boundary Value Problems for second order elliptic operators in rough domains. The main ingredient is the construction of Varopoulos’ type extensions. Namely, those are extensions u of boundary data f in L^p and w of boundary data g in the Hajlasz-Sobolev space W^{1,p} so that the Carleson functional of grad(u) and Lw are in L^p and the non-tangential maximal function of u and grad(w) are in L^p with norms bounded by the L^p norm of f and the W^{1,p} norm of g respectively. My plan is to show how those extensions appear in a natural way and sketch their construction.

Dimitrios Ntalampekos, Stony Brook University

Τίτλος: Παραμετροποίηση μετρικών επιφανειών πεπερασμένου εμβαδού

Περίληψη: Θα παρουσιάσουμε το ιστορικό της παραμετροποίησης επιφανειών από τα κλασικά αποτελέσματα για λείες επιφάνειες έως την σύγχρονη έρευνα σε μη λείες επιφάνειες. Το πρόβλημα της παραμετροποίησης λείων επιφανειών με σύμμορφες απεικονίσεις επιλύθηκε το 1907 από τους Koebe και Poincare. Σε μη λείες επιφάνειες όπου δεν υπάρχει εφαπτόμενος χώρος χρησιμοποιούμε ημισύμμορφες, ημισυμμετρικές, ή Lipschitz απεικονίσεις. Θα περιγράψουμε κοινή εργασία με τον Matthew Romney όπου επιλύουμε το πρόβλημα της παραμετροποίησης για όλες τις μη λείες επιφάνειες πεπερασμένου εμβαδού. Συγκεκριμένα, αποδεικνύουμε ότι κάθε μετρική σφαίρα πεπερασμένου εμβαδού μπορεί να παραμετροποιηθεί από την Ευκλείδεια σφαίρα με μία ασθενώς ημισύμμορφη απεικόνιση.

Aristotelis Panagiotopoulos, Kurt Gödel Research Center, University of Vienna

Title: Classifying irreducible unitary representations is hard

Abstract: One of the leading questions in many mathematical research programs is whether a certain collection of objects can be classified under some notion of equivalence using “simple” invariants. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of classification problems and for establishing negative anti-classification results. In this talk, I will survey this anti-classification “toolkit”, using the problem of classifying the unitary representations of a countable discrete group as a study case. In the process, I will cover some classical results (Glimm, Effros, Hjorth) as well as some more recent developments coming from my joint work with Shaun Allison as well as with David Kerr. 

Yiannis Sakellaridis, Johns Hopkins University

Title: Riemann’s zeta function, period integrals, and quantization

Abstract: It has been known since Riemann’s 1859 report to the Berlin Academy that the zeta function, and subsequently its generalizations (called L-functions), often admit a presentation as integrals of certain highly symmetric functions (automorphic forms), whose symmetries give rise to properties of zeta such as the meromorphic continuation and the functional equation. Such integral presentations, however, had been haphazard and poorly understood, with no obvious connection between the definition of the L-function and the integral representing it. I will present an explanation of this method that has surprising connections to quantum field theory. The explanation suggests that there is a duality between certain “nice” symplectic manifolds (with group actions), such that the “quantization” of one manifold gives rise to an L-function, while the “quantization” of its dual gives rise to the period integral. (All the heavy terms in this abstract will be explained.) This is joint work with David Ben-Zvi and Akshay Venkatesh.

Bunyamin Sari, University of North Texas

Title: Banach spaces with the Lebesgue property of Riemann integrability

Abstract: A classical theorem often covered in introductory Analysis classes is the Lebesgue criterion of Riemann integrability, stating that a bounded function f on [0,1] is Riemann integrable if and only if the set of discontinuities of f has Lebesgue measure zero. The Riemann integral of a function f from unit interval into a Banach space X is defined in a similar fashion. The value of the integral of f is a vector x(f) in the space X which is arbitrarily well approximated in norm by Riemann sums. However, the Lebesgue criterion fails for every classical Banach space except for X=l_1. Which classes of infinite dimensional Banach spaces satisfy the Lebesgue criterion?
This talk will introduce a recent solution to this problem, unveiling a novel notion of sequential asymptotic structure in Banach spaces. We’ll explore how this concept contrasts with other asymptotic structures, a line of inquiry pioneered by the Greek school through the construction of intriguing examples of exotic Banach spaces.

Based on two joint works with Harrison Gaebler, and H. Gaebler and Pavlos Motakis.

Konstantinos Tyros, National and Kapodistrian University of Athens

Title: Berry-Esseen Bounds for random tensors

Abstract: pdf

Petros Valettas, University of Missouri, Columbia

Title: Gaussian Methods in Linear Dvoretzky Theory

Abstract: The Dvoretzky theorem is a fundamental concept in the local theory of normed spaces. In its simplest form, it states that high-dimensional normed spaces contain relatively large subspaces that are almost Euclidean. Determining its optimal quantitative form is an important open question dating back to Grothendieck in the 1950s. This talk has two main purposes: first, to provide a brief historical overview of the problem, and second, to discuss the current standing by explaining how probabilistic dichotomies, based on dipole randomness and structure, are a crucial element in resolving the random version of the theorem. Based on joint work(s) with Grigoris Paouris (Texas A&M University).