Mini Courses
Alexandros Eskenazis, Sorbonne Université and Trinity College, Cambridge
Τίτλος: Μετρικές εμφυτεύσεις
Περίληψη: Θα παρουσιάσουμε μια εισαγωγή στην θεωρία εμφυτεύσεων πεπερασμένων μετρικών χώρων. Ανάμεσα στα θέματα που θα συζητηθούν περιλαμβάνονται οι bi-Lipschitz εμφυτεύσεις σε χώρους Hilbert, οι εμφυτεύσεις doubling μετρικών χώρων και επίπεδων γραφημάτων, η μετρική ελάττωση της διάστασης, και η θεωρία των μετρικών type και cotype. Αν υπάρξει χρόνος, μπορεί να αναφερθούμε και σε εμφυτεύσεις σε χώρους Alexandrov, σε μετρικά θεωρήματα τύπου Kwapień, και σε άλλα σύγχρονα θέματα.
Marina Iliopoulou, National and Kapodistrian University of Athens
Title: The polynomial method
Abstract: In the last 15 years, algebraic techniques have emerged in the intimately connected areas of incidence geometry and harmonic analysis, revealing a deep algebraic-geometric nature underlying major problems in the fields. This all started in 2008, when Dvir employed the polynomial method, an old technique from number theory, to solve the Kakeya problem in finite fields. In this series of talks, we will introduce Dvir’s solution, as well as the refined ‘polynomial partitioning’ method by Guth and Katz (developed a little later to solve the Erdős distinct distances problem on the plane). We will then discuss applications in incidence geometry (joints problem, Szemerédi-Trotter theorem) and in harmonic analysis (Fourier restriction problem).
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.
Dimitrios Ntalampekos, Stony Brook University
Title: Hausdorff dimension and fractal sets
Abstract: We will present the basic theory of Hausdorff measure and dimension. Then we will study various fractal sets and we will compute their dimensions, such as the Cantor set, the von Koch snowflake, the Sierpinski gasket and the Sierpinski carpet.
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.
Elias Katsoulis, East Carolina University
Title: TBA
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Sophocles Mercourakis, National and Kapodistrian University of Athens
Title: TBA
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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: TBA
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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: TBA
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Petros Valettas, University of Missouri, Columbia
Title: TBA
Abstract: