Let $K$ be a centrally symmetric convex body of volume $1$ in ${\mathbb R}^n$. For any $s\ge 1$ consider the norm on ${\mathbb R}^s$, defined by \begin{equation*}\|{\bf t}\|_{K^s,K}=\int_{K}\cdots\int_{K}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_s\cdots dx_1.\end{equation*} A question posed by V. Milman in the 1980's is to determine if $\|\cdot\|_{K^s,K}$ is equivalent to the standard Euclidean norm up to a term which depends logarithmically (only) on the dimension. We shall discuss what is known on this question, starting with two proofs of the lower bound \begin{equation*}\|{\bf t}\|_{K^s,K}\ge c_1\,\|{\bf t}\|_2\end{equation*} where $c_1 > 0$ is an absolute constant. We are mainly interested in upper bounds for the quantity $\|{\bf t}\|_{K^s,K}$. We shall describe a recent approach which exploits the recent developments on the {\it ``slicing problem"} and leads to the estimate \begin{equation*} \|{\bf t}\|_{K^s,K}\le c_2(\log n)^{3/2}\,\sqrt{n}M(K^{\ast})L_{K^{\ast}}\|{\bf t}\|_2, \end{equation*} where $M(K):=\int_{S^{n-1}}\|\xi\|_K\,d\sigma (\xi)$ is the mean norm of $K$ on the unit sphere, $L_{K^{\ast}}=L_K$ is the isotropic constant of $K$ and $K^{\ast}$ is an isotropic linear image of $K$. Thus, we have a reduction of V. Milman's question to the problem of estimating the parameter $M(K^{\ast})$ for an isotropic centrally symmetric convex body $K^{\ast}$ in ${\mathbb R}^n$. In the last part of the talk we discuss what is known about the latter problem.

The talk is based on joint works with (and of) G. Chasapis, N. Skarmogiannis and E. Milman.

The Schwarzschild family of static black holes, discovered in 1915, constitutes the most famous family of solutions of the vacuum Einstein equations of general relativity. I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

Ζωντανή μετάδοση: https://video.ict.uoc.gr/live/show/457

Όλοι μαθαίνουμε ότι στην κβαντική μηχανική παύει να ισχύει ο ντετερμινισμός. Η «κλασική», μη κβαντική δηλαδή, φυσική υποτίθεται όμως ότι είναι ντετερμινιστική: Η πλήρης γνώση των αρχικών συνθηκών προσδιορίζει μονοσήμαντα το μέλλον. Αυτή η έννοια «ντετερμινισμού» σχετίζεται με το όνομα του Laplace και βρίσκει μαθηματική πραγμάτωση στα κλασικά θεωρήματα ύπαρξης και μοναδικότητας για τις Συνήθεις και Μερικές Διαφορικές Εξισώσεις. Τα πράγματα όμως τελικά δεν είναι τόσο απλά όσο φαίνονται, ακόμα και στην μη κβαντική φυσική: Η ομιλία θα ερευνήσει το θέμα του ντετερμινισμού στην κλασική φυσική ξεκινώντας από την Νευτώνεια μηχανική και φτάνοντας μέχρι την Γενική Σχετικότητα όπου θα συναντήσουμε πολύ απρόσμενους τρόπους που ο ντετερμινισμός μπορεί να καταρρεύσει ολοσχερώς.

The problem of timely warning about the danger of nearfield tsunami after the strong offshore earthquake is still unresolved, even if the number of publications is rather large. For the coast of Japan it takes nearly 20 minutes for tsunami wave to approach the nearest dry land after offshore seismic event. Robust evaluation of tsunami wave danger should be based on correct process simulation: wave generation, propagation, and inundation to a dry land. There are several software tools to calculate the wave propagation over the real digital bathymetry. However, it takes rather long in case of the fine calculation mesh. Results of code execution acceleration are discussed. With the use of modern Graphic Processing Units (GPUs) it is possible to achieve performance gain 100 (to calculate the wave propagation 100 times faster) and even more with Field Programmable Gates Arrays (FPGAs). Fast algorithms to determine initial sea surface disturbance at tsunami source are also considered.

Σε αυτήν την ομιλία εξετάζονται διδακτικές πρακτικές καθηγητών μαθηματικών σε πανεπιστήμια και πρακτικές που οι ερευνητές μαθηματικοί συζητούν ότι χρησιμοποιούν στην έρευνά τους στα μαθηματικά. Τα δεδομένα προέρχονται κυρίως από την παρατήρηση διδασκαλιών των μαθηματικών σε φοιτητές πρώτου έτους και συζητήσεις με τους διδάσκοντες σε πανεπιστήμια της Ελλάδας, της Αγγλίας και πολιτειών της Αμερικής. Στην ομιλία αναλύονται ορισμένες διδακτικές πρακτικές όπως η χρήση παραδειγμάτων από ερευνητές μαθηματικούς. Επίσης, συζητούνται θέματα σχετικά με τη χρήση αυτών των πρακτικών στη διδασκαλία και τις δυνατότητες που μπορούν να δοθούν στους φοιτητές-μελλοντικούς καθηγητές σχολείων ώστε να προετοιμαστούν για την σχολική εκπαίδευση.

Abstract: An undecidability result for the first-order theory of a ring structure leads to a "weakening" of the language in order to find at which point the theory changes from decidable to undecidable. The standard weakenings consist of replacing either or both operations by a fragment of it, i.e., an operation or relation that can be defined from it, but for which the converse is not (or at least not trivially) true. In this talk I will discuss two different fragments of multiplication: the set of images of a degree two polynomial and the relation of coprimeness. I will present my work on the undecidability of the first-order theories of certain polynomial rings for each language, and comment on possible further ideas in each case.

Θα δούμε μια σειρά από ομοιότητες ανάμεσα σε φαινόμενα των πολλαπλοτήτων διάστασης 3 και των σωμάτων αριθμών, όπως αυτά έχουν καθοριστεί στο λεξικό MKR. Θα δοθεί μια περιγραφή της απόλυτης ομάδας Galois ως mapping class group και θα δοθεί μια εξήγηση των η ομοιοτήτων ανάμεσα στις δύο θεωρίες βασισμένη στις αναπαραστάσεις ομάδων στους αυτομορφισμούς της ελεύθερης προπεπερασμένης ομάδας.

In statistical methodology, the techniques for assessing if a distribution is symmetrical or not are based on characteristic properties of symmetric distributions. These properties form the basis for developing measures and statistical tests through the skewness coefficient of the distribution. However the skewness coefficient is affected by the tail behavior of the distribution and naturally this affects the performance of the measures/tests. For this reason, here we follow a different path and based on a necessary and a necessary and sufficient condition for the symmetry of a probability density function we develop two measures of symmetry and a statistical test for symmetry. This is developed on the basis of calculus techniques. Apart from the properties of the measures, we study the distribution of the statistical test as well as the implementation details of the methods in practice.

This talk gives an overview on the use of non-local or fractional models for modelling the electrophysiology of the heart when taking into account the heterogeneous nature of cardiac tissue. Space fractional partial differential equations will be introduced along with some new computational approaches. We will then show how model parameters can be extracted from Diffusion Tensor Magnetic Resonance Images and using clinical data we will attempt to construct spatial biomarkers for understanding hypertrophic cardio-myopathy – disarray of the cardiac fibres.

Synchronization of a large collection of coupled, simple dynamical systems is a problem that has applications from neuroscience to traffic modeling to modeling of consensus formation.

Consider an array of identical linear oscillators, each coupled to its front and rear neighbor. The coupling may be asymmetric. If we kick the front oscillator (the leader), how does this signal propagate through the system? In some isolated cases, for certain values of the parameters, the answer is well-known, but until recently the only general results applicable to large were very qualitative.

We developed a theory that gives the correct quantitative description. The theory uses ideas from partial differential equations, but without taking a continuum limit. We will describe the theory and the conjectures it is based upon, as well as the quantitative results.

Abstract:

I will present some models where a density $\rho$ of particles move, following a velocity field, but subject to a density constraint $\rho\leq 1$. When needed, they adapt their "spontaneous" velocity in order to preserve this density constraint. The model was originally introduced for crowd motion but is adapted to many kind of particle flows. Mathematically, the PDE solved by the density is easy to write, but it lacks regularity, and is difficult to study. It has been possible to study it thanks to optimal transport techniques, which also suggested efficient numerical methods. I will briefly introduce the main ingredients from optimal transport that we need to understand the method, and the talk will be essentially self-contained.

Abstract:

In 2010, P. Sarnak formulated the following Moebius disjointness conjecture: $\frac1N\sum_{n\leq N} \mu(n)f(T^nx)\to 0$ as $N\to \infty$, for an arbitrary zero entropy homeomorphism $T$ of a compact metric space $X$, arbitrary $f\in C(X)$ and $x\in X$. Here, $\mu(\cdot):{\mathbb N}\to\{-1,0,1\}$ denotes the classical arithmetic Moebius function. I will show how this conjecture is connected on one hand with ergodic theory and on the other hand with other problems in number theory, concentrating on the Chowla conjecture from 1960 on the multiple correlations of the Moebius function.

For distinct points $p$ and $q$ in a two-dimensional connected Riemannian manifold $M$, we define their mediatrix $L_{pq}$ as the set of points equidistant to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curve. We show additional geometric regularity properties of mediatrices: at each point they have the radial linearizability property, which means that they are tangent to a finite collection of lines meeting in the origin. In the case of mediatrices on the sphere, where mediatrices are simple closed Lipschitz curves, we show these curves have at most countable singularities, and the total angular deficiency has a finite upper bound related to the total curvature of the metric on the sphere.

On the other hand mediatrices have the topological property that they separate the manifold $M$ into two parts and that non proper subset of them does. This allows for their topological classification. This classification is in some sense a generalization of the Jordan Brouwer Theorem. We will briefly discuss the classification of minimal separating sets in the orientable surfaces of genus 0, 1, and 2

Just last year, mediatrices found an application in a territorial conflict between Peru and Chile. We'll show the opinion of the International Court of Justice in The Hague.

Let $F$ be a family that consists of an odd number of parallel translates of a given (compact and with positive measure) set $F$ in the plane. We are interested in the area of those points in the plane that belong to an odd number of sets in $F$. What is the minimum possible such area in terms of the area of $F$? This depends a lot on the set $F$. We will resolve completely the cases of triangles, parallelograms, and trapezoids. We will also prove the existence and construct a set $F$ such that this area may be arbitrarily small. Many beautiful questions still remain open. This is an on-going project based on works with Assaf Oren and Igor Pak and with Uri Rabinovich.

Answering a question of Benjamini & Schramm, we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which circle arises from a discrete version of Riemann's mapping theorem. When the graph is hyperbolic, the aforementioned circle coincides with the usual boundary. http://arxiv.org/abs/1301.1506

Partition regularity is a classical notion in combinatorics: A collection of subsets of the integer is partition regular if for every finite partition of the integers at least one atom contains an element of the collection. On the other hand, multiplicative functions are a standard topic in number theory. In this talk we explain how a question of partition regularity led us to an unusual problem involving multiplicative functions, and we plan to say a few words about the methods needed to solve it.This is a joint work with Nikos Frantzikinakis.

Geometric questions for discrete objects are often easier to understand by first considering an appropriate continuous analogue, using additional structure that is available in the continuous setting to prove the desired theorem, and then passing back to a discrete object via an appropriate "discretization" procedure. This last discretization step often relies on ad hoc arguments that are quite involved but technical, yet sometimes it requires conceptually new ideas, and there are instances in which it remains an open problem. In this talk we will discuss results that systematize the discretization step in certain important situations. We will present both older and recent theorems along these lines, and describe several basic open questions.

Block copolymers are a classical example of self-assembling materials. By carefully selecting the monomers of each block and the composition of the copolymer, a rich collection of morphologies can be obtained. The theoretical approach to understand the phase separation of polymers is the well-established self-consistent field theory (SCFT), first proposed by Helfand and fully developed by Matsen and Fredrickson. However, this theory is not directly applicable to the rather stiff aromatic polymers used in photovoltaic applications. In this presentation, we investigate a recently proposed numerical approach for the solution of the SCFT for wormlike chains, and its potential application to the design of OPV cells.