Analysis Seminar

School of Applied Mathematical and Physical Sciences, Department of Mathematics

Co-organizer: Marina Iliopoulou (from NKUA)

All talks in NTUA at the Seminar Room of the School of Applied Mathematical and Physical Sciences, 2nd Floor, Building E.

All talks in NKUA at the Department of Mathematics, Room Γ31.

News

There is a summer school at the Department of Mathematics:

"Mathematical Analysis" in honor of Spiros Argyros

Mini courses: Silouanos Brazitikos (University of Crete), Alexandros Eskenazis (Sorbonne Université),Vassilis Gregoriades (National Technical University of Athens), Marina Iliopoulou (National and Kapodistrian University of Athens), Pavlos Motakis (York University).

Invited lectures: George Androulakis (University of South Carolina), Georgios Katsimpas (Harbin Engineering University), Elias Katsoulis (East Carolina University), Sophocles Mercourakis (National and Kapodistrian University of Athens), Mihalis Mourgoglou (University of the Basque Country & Ikerbasque), Dimitrios Ntalampekos (Stony Brook University), Aristotelis Panagiotopoulos (University of Vienna), Yiannis Sakellaridis (Johns Hopkins University), Bunyamin Sari (University of North Texas), Konstantinos Tyros (National and Kapodistrian University of Athens), Petros Valettas (University of Missouri).


Upcoming Talks


Michail Sarantis, Carnegie Mellon University

On the zeroes of hypergraph independence polynomials 

Abstract: We prove that the multivariate independence polynomial of any hypergraph of maximum degree Δ has no zeroes on the complex polydisc of radius ~1/(eΔ), centered at the origin. Up to logarithmic factors in Δ, the result is optimal, even for graphs with all edge sizes greater than 2. As a corollary, we get an FPTAS for approximating the independence polynomial in this region of the complex plane. We furthermore prove the corresponding radius for the k-uniform linear hypertrees is Ω(Δ^{-1/(k-1)}), a significant discrepancy from the graph case.

Joint work with David Galvin, Gwen McKinley, Will Perkins and Prasad Tetali.  pdf


Alexandra Stavrianidi, Stanford University

The logarithmic correction for the fronts of a cascading family of Branching Brownian Motions  

Abstract: In this talk, I will introduce the connection between some systems of Fisher-KPP type reaction-diffusion equations and a cascading family of Branching Brownian Motions. The location of the median of the rightmost particle of this particle system coincides with the location of the front of the equations, so the associated long time asymptotics can be studied from both a probability and a PDE point of view. I will present results on the long time behavior of the system and analyze interesting applications and probabilistic connections.   pdf


Antonios Zitridis, University of Chicago

From entropic propagation of chaos to concentration bounds for stochastic particle systems 

Abstract: We shall discuss about weakly interacting stochastic particle systems with possibly singular pairwise interactions. In this setting, we observe a connection between entropic propagation of chaos (proved by Jabin and Wang, 2018) and exponential concentration bounds for the empirical measure of the system. In particular, we will show how to establish a variational upper bound for the probability of a certain rare event, and then use this upper bound to show that ''controlled" entropic propagation of chaos implies an exponential concentration bound for the empirical measure. 

Joint work with Joe Jackson.  pdf


Past Talks

Spring 2024

Alexandros Eskenazis, Sorbonne Université and Trinity College, Cambridge 

Resilience of cube slicing in ℓ_p

Abstract: We shall discuss the state of the art on the problem of identifying the volume maximizing and minimizing hyperplane sections of p-balls in R^n. After explaining a reduction to a sharp probabilistic estimate on moments of rotationally invariant random vectors, we will present a recent work with P. Nayar (Warsaw) and T. Tkocz (CMU) identifying the volume maximizing section for p greater than a universal constant.  pdf


Myrto Manolaki, University College Dublin

Ολόμορφες συναρτήσεις με χαοτική ακτινική συμπεριφορά 

Abstract: Είναι γνωστό οτι οι περισσότερες, με την τοπολογική έννοια, ολόμορφες συναρτήσεις στον μοναδιαίο δίσκο απεικονίζουν κάθε ακτίνα σε ένα πυκνό υποσύνολο του μιγαδικού επιπέδου. Σε αυτή την ομιλία θα επικεντρωθούμε σε μία νέα κλάση ολόμορφων συναρτήσεων που παρουσιάζουν ακόμα πιο χαοτική ακτινική συμπεριφορά. Συγκεκριμένα, για κάθε τέτοια συνάρτηση f, η οικογένεια {f_r(z):=f(rz) : 0<r<1} προσεγγίζει όλες τις συνεχείς συναρτήσεις σε κατάλληλα υποσύνολα του μοναδιαίου κύκλου καθώς το r τείνει στο 1. Θα δούμε πώς οι συναρτήσεις αυτές συνδέονται με κλασικά αποτελέσματα, θα μελετήσουμε την συνοριακή τους συμπεριφορά σε διάφορα χωρία και θα εξετάσουμε πότε παραμένουν αναλλοίωτες ως προς την σύνθεση από αριστερά και δεξιά.

Η ομιλία βασίζεται σε 2 εργασίες με τον Stéphane Charpentier και τον Κωνσταντίνο Μαρονικολάκη.  pdf 


Romanos Diogenes Malikiosis, Aristotle University of Thessaloniki

A linear programming approach to Fuglede's conjecture in Z_p^3

Abstract: Delsarte's method on linear programming bounds is a very powerful tool which provides an upper bound on the size of a set A in an additive group G, whose difference set A-A avoids a given set E. This tool may have limitations, but has been used successfully in various settings, most notably in the sphere packing problem in 8 and 24 dimensions.

Here, we will present an application of this method to Fuglede's conjecture in G=Z_p^3.  pdf


Alexandra Tzella, University of Birmingham (ΣΕΜΦΕ, at 15:30)

Diffusion in arrays of obstacles: beyond homogenisation

Abstract: We examine the diffusion of a chemical or heat released in a homogeneous medium interrupted by an infinite number of impermeable obstacles arranged in a periodic lattice. We extend classical results due to Maxwell, Rayleigh and Keller by applying ideas of large-deviation theory to describe the concentration or temperature distribution at large distances from the point of release. We use matched asymptotics to obtain explicit results in the case of nearly touching obstacles, when the transport is strongly inhibited. The technique developed can be applied to complex systems including porous media and composite materials. This is based on joint work with Y. Farah, D. Loghin and J. Vanneste.  pdf


Andreas Vikelis, University of Vienna (ΣΕΜΦΕ, at 15:30)

Λύσεις μέτρα για το σύστημα της ελαστοπλαστικότητας σε συνθήκες μεγάλων παραμορφώσεων 

Abstract: Μια σειρά από φαινόμενα που συναντάμε στη φύση και πιο συγκεκριμένα στη μηχανική των υλικών, περιγράφονται μέσα από διαφορικές σχέσεις που συχνά είναι πολύ δύσκολο να μελετηθούν λόγω της αυξημένης μαθηματικής τους πολυπλοκότητας. Στη συγκεκριμένη ομιλία θα εστιάσουμε στα εξελικτικά εκείνα φαινόμενα που περιγράφονται από ανεξάρτητα-ρυθμού συστήματα, δηλαδή για παράδειγμα συστήματα που δεν εξαρτώνται από το πόσο γρήγορα ή για πόση διάρκεια εφαρμόζονται σε αυτά εξωτερικές δυνάμεις. Εισάγοντας τις βασικές αρχές της θεωρίας αυτών των συστημάτων και πιο συγκεκριμένα την έννοια των ενεργειακών λύσεων, θα μελετήσουμε το πρόβλημα της εξελικτικής ελαστοπλαστικότητας σε συνθήκες μεγάλων παραμορφώσεων, ένα πρόβλημα που παραμένει μέχρι και σήμερα ανεξερεύνητο. Σε αυτήν την κατεύθυνση, θα παρουσιάσω ένα αποτέλεσμα ύπαρξης λύσεων-μέτρων του quasi-στατικού προβλήματος που διατηρούν τις φυσικές ιδιότητες του συστήματος, δηλαδή είναι ευσταθείς και διατηρούν την ενέργεια. Τόσο η γενικότερη θεωρία των quasi-στατικών εξελικτικών προβλημάτων, όσο και η δική μας συνεισφορά στο πεδίο αυτό, βασίζεται σε τεχνικές από τη θεωρία μεταβολών. Η δουλειά αυτή είναι σε συνεργασία με τον Ulisse Stefanelli.  pdf


Vassili Nestoridis, National and Kapodistrian University of Athens (ΕΚΠΑ, at 15:30, Room Γ21)

Προσέγγιση σε συμπαγή σύνολα συναρτήσεων και όλων των παραγώγων τους

Abstract: Σε θεωρήματα τύπου Mergelyan προσεγγίζουμε ομοιόμορφα σε συμπαγή σύνολα Κ συναρτήσεις από πολυώνυμα, ρητές συναρτήσεις ή συναρτήσεις ολόμορφες σε μεταβαλλόμενα ανοικτά που περιέχουν το Κ. Εμείς αντικαθιστούμε την ομοιόμορφη προσέγγιση στο Κ από την ομοιόμορφη στο Κ προσέγγιση όλων των παραγώγων. Η περίπτωση μιας μιγαδικής μεταβλητής βρίσκεται σε άρθρο των Αρμενιάκου, Κοτσόβολη και Νεστορίδη (arXiv:2006.02389) που δημοσιεύτηκε στο Monatchefte fur Mathematik (2022). Στην παρούσα διάλεξη παρουσιάζονται κάποιες επεκτάσεις στις πολλές μιγαδικές μεταβλητές που βασίζονται σε συνεργασία των P. M. Gauthier και Β. Ν. Νεστορίδη.  pdf


Georgios Kotsovolis, Princeton University (ΣΕΜΦΕ, at 16:00)

The infima of binary forms 

Abstract: For a binary form P(x,y) of non-zero discriminant and for a two dimensional lattice Λ  of volume 1, what is the infimum of the values P attains on the non-trivial vectors of Λ? The spectrum of a binary form P is defined to be the set of these infima as Λ ranges over all unimodular lattices. Understanding this object is a fundamental project in the geometry of numbers and even though the case of n=2 is well understood, much less is known for higher degrees. In 1940, Mordell conjectured that for a binary cubic form P,  the spectrum of P has a gap after its maximal value, a statement disproved later by Davenport, who constructed a sequence of infima converging to the top. As for n greater than 3, there has been, to our knowledge, no progress to understanding these spectra. In this talk, we show that for any binary form P, the spectrum of P is an interval, answering the problem for all degrees n.  pdf


Marina Iliopoulou, National and Kapodistrian University of Athens (ΣΕΜΦΕ, at 15:30)

On integer distance sets

Abstract: An integer distance set is a set in the Euclidean plane with the property that all pairwise distances between its points are integers. In this talk we will show that any integer distance set contains all but very few of its points on a single line or circle. This helps us address some questions by Erdős. In particular, we deduce that integer distance sets in general position (no 3 points on a line, no 4 points on a circle) are very sparse, and we derive a near-optimal lower bound on the diameter of any non-collinear integer distance set of a given size. Our proof uses existing refinements of the Bombieri-Pila determinant method. This is joint work with Rachel Greenfeld and Sarah Peluse.  pdf


Konstantinos Kavvadias, Massachusetts Institute of Technology (ΕΚΠΑ, at 15:30, Room Γ31)

Introduction to Schramm-Loewner Evolution (SLE) 

Abstract: The Schramm-Loewner Evolution (SLE_κ) is a one parameter family (κ>0) of curves which connect two boundary points of a simply connected domain. It was introduced by Schramm in 1999 as a candidate to describe the scaling limit of the interfaces that arise in discrete models at criticality from statistical mechanics on planar lattices, such as the loop erased random walk and the percolation model. In my talk, I will discuss about the intuition behind the definition of SLE_κ and I will introduce some of its basic properties obtained during the last twenty years. I will also discuss about some recent results obtained in a series of recent research works. Finally, if time permits, I will discuss about some ongoing research results.  pdf


Fall 2023

Lampros Gavalakis, Université Gustave Eiffel

Discrete entropy monotonicity for log-concave sums on Z and Z^d

Abstract:  A celebrated result of Artstein, Ball, Barthe and Naor (2004) states that the differential entropy of sums of continuous random variables increases along the central limit theorem. Although an exact analogue of this statement cannot be true for discrete random variables, Tao (2010) conjectured that an approximate version is true provided that the underlying entropies are large enough. We will present a recent proof of a special case of this conjecture for log-concave random variables on the integers and discuss current progress towards extending this result on Z^d. For the dimensional extension, we will mention some discrete analogues of results from convex analysis that may be of independent interest.

Part of the talk is based on joint work with M. Fradelizi and M. Rapaport.  pdf


Athanasios Zacharopoulos, Universidad del País Vasco

Varopoulos' extensions in domains with Ahlfors-regular boundaries

Abstract: In this talk we shall describe the construction of Varopoulos' type extensions of L^p and BMO  boundary functions in rough domains. That is, smooth extensions of functions such that the  L^p-norms of their non-tangential maximal function and the Carleson functional of their gradients can be controlled by the norm of the boundary data. After giving the geometric motivation and a brief survey of known results, we will proceed to present a new and more general approach of constructing  Varopoulos' extensions in domains with minor geometrical assumptions for the boundaries. 

This talk is based on joint work with Mihalis Mourgoglou. pdf


Marianna Chatzakou, Ghent University

Συναρτησιακές ανισότητες σε ομάδες Lie

Abstract: Η ομιλία αφορά σε μια σειρά γνωστών, στον Ευκλείδειο χώρο, κατά βάση λογαριθμικών ανισοτήτων σε ομάδες Lie. Συγκεκριμένα, θα αναφερθούμε στην επέκταση αυτών των ανισοτήτων σε ομάδες Carnot, και κάποιες φορές σε πιο ευρείες κλάσεις ομάδων Lie. Ιδιαίτερη έμφαση θα δοθεί στο πώς το "φυσικό" ανάλογο της κλασσικής λογαριθμικής ανισότητας με Gaussian μέτρο (L. Gross, 1975) εμφανίζεται στην περίπτωση των ομάδων Carnot, και συγκεκριμένα στην ομάδα Heisenberg. Στην τελευταία αυτή περίπτωση, το εμφανιζόμενο μέτρο επιτρέπει να θεωρήσουμε ένα απειροδιάστατο ανάλογο της ομάδας Heisenberg όταν αυτή εφοδιαστεί με το προβλεπόμενο μέτρο πιθανότητας.

Η ομιλία βασίζεται σε κοινές εργασίες με τους A. Kassymov και M. Ruzhansky. pdf


Alexandros Saplaouras, National Technical University of Athens

Towards the stability property of 2BSDE and of associated HJB PIDEs 

Abstract: In this talk we will deal with the probabilistic representation of viscosity solutions of integro-partial differential equations of Hamilton-Jacobi-Bellman type. We will make a pause to describe the comparison principle suitable for the required generality. Afterwards, we will explain how the stability property of 2BSDEs will enable us to obtain a Trotter-Kato type theorem.   pdf


Dimitrios Chatzakos, University of Patras

The Prime geodesic theorem in arithmetic progressions

Abstract: The Prime geodesic theorem states that the distribution of the lengths of primitive closed geodesics on Riemann surfaces has a similar asymptotic behaviour with the distribution of prime numbers.

In this talk we will discuss an analogue of Dirichlet's theorem in arithmetic progressions for the lengths of primitive closed geodesics on the  modular surface. In particular, we prove two conjectures of Golovchanskii and Smotrov  from 1999.

This is a joint work with Gergely Harcos and Ikuya Kaneko.  pdf


Georgios Moschidis, École Polytechnique Fédérale de Lausanne

Τυρβώδης συμπεριφορά βαρυτικών διαταραχών στο εξωτερικό μελανών οπών 

Abstract: Σύμφωνα με την θεωρία της γενικής σχετικότητας, η εξέλιξη των βαρυτικών κυμάτων στον χωροχρόνο διέπεται από ένα σύστημα εξισώσεων υπερβολικού χαρακτήρα, γνωστών και ως εξισώσεις του Einstein. Σε περιπτώσεις όπου τα βαρυτικά κύματα παγιδεύονται στο εσωτερικό ενός πεπερασμένου χωρίου με ανακλαστικό σύνορο, αναμένεται ότι η μη γραμμική φύση των εξισώσεων Einstein οδηγεί στην εμφάνιση τυρβωδών φαινομένων (ανάλογων με αυτά που συναντά κανείς στην ροή ρευστών σωμάτων). Ένα ενδιαφέρον ερώτημα που προκύπτει είναι εάν αντίστοιχα φαινόμενα εμφανίζονται και σε περιπτώσεις διαταραχών μελανών οπών με ασυμπτωτική γεωμετρία που προσομοιάζει αυτή του χώρου Anti de Sitter (και η οποία μπορεί να λειτουργήσει σαν ένα ιδεατό ανακλαστικό σύνορο στο "άπειρο"), καθιστώντας την μελανή οπή "ασταθή". Σε αυτήν την ομιλία, θα εξετάσουμε την τυρβώδη συμπεριφορά των λύσεων μιας μη γραμμικής κυματικής εξίσωσης (έχουσας τον ρόλο απλοποιημένου μοντέλου του συστήματος εξισώσεων Einstein) στο εξωτερικό τέτοιων μελανών οπών. Τα αποτελέσματα αυτά προέκυψαν σε συνεργασία με τον Christoph Kehle.  pdf


Polyxeni Spilioti,  University of Göttingen 

Twisted Ruelle zeta function on locally symmetric spaces, the Fried’s conjecture and further applications

Abstract: In this talk, we will present some recent results concerning with the special values of the dynamical zeta functions on locally symmetric spaces. In particular, we study the twisted Ruelle zeta function at zero and its relationship with spectral and topological invariants. These results can be viewed as extensions of previous results by Fried to the case of an arbitrary representation. Our techniques are based on the spectral theory of non-self-adjoint Laplacians and the Selberg trace formula. The results are part of joint work with Jan Frahm, Léo Bénard and Jan Frahm, and Frédéric Naud.   pdf


Odysseas Bakas, University of Patras

Endpoint bounds for certain classes of operators arising in Littlewood-Paley theory 

Abstract: In the first part of this talk we shall review certain aspects of classical Littlewood-Paley theory and briefly present some open problems in the area. Motivated in part by these problems, in the second part of talk, we will present joint work with Valentina Ciccone, Ioannis Parissis, and Marco Vitturi concerning sharp endpoint bounds for certain classes of operators arising in Littlewood-Paley theory, including Littlewood-Paley square functions and Marcinkiewicz multiplier operators of finite order.   pdf


Vagia Vlachou, University of Patras

Disjoint universality connected with  differential operators

Abstract: For a simply connected domain G, we study the problem of disjoint universality for  series of operators connected with differential operators and polynomials. The motivation for this study stems from Universal Taylor Series, if you change the role of the centre of expansion to variable instead of constant.   pdf


Michael Roysdon, Case Western Reserve University

Higher-order convex bodies and related inequalities 

Abstract: I will discuss parts of a series of joint works with J. Haddad, D. Langharst, E. Putterman, and D. Ye, which concern the examination of classical notions of convex geometry in a "higher-order" setting.  To a convex body K operators may be assigned, such as the difference body, projection body, and centroid body operators.  In this talk, I will describe methods of assigning to a convex body K in R^n convex bodies in dimension R^{nm} which extend the usual notion of difference body, projection body and centroid bodies. As a consequence, various affine-isoperimetric inequalities and Sobolev-type inequalities, one of which is stronger than the usual isoperimetric inequality, arise in this setting. Since this topic is completely new in the subject, there are still many open questions to consider concerning these new operators.   pdf


Christoforos Panagiotis, University of Bath

Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice 

Abstract: In this talk, we will consider the self-avoiding walk on the hexagonal lattice, which is one of the few lattices whose connective constant can be computed explicitly. This was proved by Duminil-Copin and Smirnov in 2012 when they introduced the parafermionic observable. In this talk, we will use the observable to show that, with high probability, a self-avoiding walk of length n does not exit a ball of radius n/logn. This improves on an earlier result of Duminil-Copin and Hammond, who obtained a non-quantitative o(n) bound. Along the way, we show that at criticality, the partition function of bridges of height T decays polynomially fast to 0. Joint work with Dmitrii Krachun.  pdf


Dimitris Gerontogiannis, Leiden University

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces 

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this new operator is intrinsically defined, its spectral properties are analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden).  pdf


Spring 2023

Giorgos Chasapis, University of Crete

Sharp moment comparison for sums of rotationally invariant random vectors and geometric applications

Abstract:   pdf


Andreas Koutsogiannis, Aristotle University of Thessaloniki

Convergence of multiple ergodic averages for totally ergodic systems 

Abstract: A collection of integer sequences is jointly ergodic if for  every ergodic measure preserving system the multiple ergodic averages,  with iterates given by the sequences, converge to “the expected limit”  in the mean, i.e., the product of the integrals. Exploiting a recent  approach of Frantzikinakis, which allows one to avoid deep tools from  ergodic theory that were previously used to establish similar results,  we study joint ergodicity in totally ergodic systems for integer parts  of suitable iterates. The motivation for this study, which is joint  work with Wenbo Sun (Virginia Tech), comes from previous work with  Dimitris Karageorgos.   pdf


Marios Georgios Stamatakis, University of Ioannina

Generalized Young measures for the hydrodynamic limit of condensing zero-range processes 

Abstract: Zero-range processes are stochastic interacting particle systems with zero range interaction. For particular choices of their parameters they exhibit phase separation with the emergence of a condensate. Such zero-range processes are referred to as condensing and their hydrodynamic limit is not known. It is expected to be given by a degenerate non-linear diffusion equation where the diffusivity vanishes at densities that exceed a critical density ρ_c. In this talk we employ an appropriate generalization of the notion of Young-measures in order to obtain the hydrodynamic limit of such equations.   pdf


Konstantinos Zemas, Universität Münster 

Rigidity estimates for isometric and conformal maps on the sphere 

Abstract: In this talk I would like to discuss stability aspects of the class of rigid motions, resp. Möbius transformations, among Sobolev maps from the standard round sphere into the ambient Euclidean space. Unlike similar in flavour results for maps defined on domains, not only an isometric, resp. conformal, deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of the sphere under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. The focus will mostly be on the case when the ambient dimension is 3, and we will also explain why, in both cases, the estimates are optimal in their corresponding settings. The adaptations needed in higher dimensions and the particular case of sphere-valued maps will also be addressed. The talk will be based on  previous works with Stephan Luckhaus and Jonas Hirsch, and an ongoing one with Xavier Lamy and Andre Guerra.    pdf


Georgios Sakellaris, Aristotle University of Thessaloniki

Το πρόβλημα του Neumann για δευτεροβάθμιες ελλειπτικές εξισώσεις με όρους χαμηλότερης τάξης

 Abstract:  pdf


There is a summer school at the Department of Mathematics:

"Mathematics of Machine and Statistical Learning"

Speakers: Constantine Caramanis (University of Texas, Austin, USA), Daniele Durante (Universitá Bocconi, Milano, Italy), Panayotis Mertikopoulos (National & Kapodistrian University of Athens), Konstantinos Spiliopoulos (Boston University, USA)


There is a summer school at the Department of Mathematics:

“Mathematical Theory of Inverse Problems and Applications” 

Speakers: Guillaume Bal (University of Chicago), Marc Bonnet (ENSTA, Paris), Fioralba Cakoni (Rutgers University), Andreas Kirsch (Karlsruhe Institute of Technology), Mourad Sini  (Ricam, Linz).


There is a summer school at the Department of Mathematics:

"Ninth Summer School in Operator Theory"

Speakers: A. Chatzinikolaou (NKUA), D. Drivaliaris (University of the Aegean), M. Ghandehari (University of Delaware), A. Giannopoulos (NTUA), V. Kanellopoulos (NTUA), A. Katavolos (NKUA), E. G. Katsoulis (East Carolina University), I. G. Todorov (University of Delaware), N. Yannakakis (NTUA).