Analysis Seminar in Crete (2016-17) |
Σεμιναριο Αναλυσης
Analysis Seminars in the World / Analysis seminars in Greece
DATE | TIME | ROOM | SPEAKER | FROM | TITLE | COMMENT |
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Tue, 27 Sep 2016 | 13:15 | A303 | Athanasios Pheidas | University of Crete | An analogue of Hilbert's Tenth Problem for the ring of holomorphic functions | |
Abstract: Consider polynomials $f(x)$ of an array $x$ of variables, with coefficients in $\mathbb{Z}[z]$, where $\mathbb{Z}$: integers, $z$: a variable. We ask: Question: Is there an algorithm (in the Caves of Plato) which, given any $f(x)$ as above, decides (with certainty) whether the (functional) equation $f(x)=0$ has solutions $x$, each of which is a holomorphic function of the variable $z$? Another way to ask the question is ''can one solve algorithmically algebraic differential equations of order zero over the ring of global analytic functions?'' Put that way the implications of the answer for the physical sciences should be obvious. The answer to the question is unknown. But we have evidence that it might be ''NO''. This evidence will be presented. It involves meromorphic parameterisations of elliptic curves and ''existentially definable subsets'' of the ring of holomorphic functions (i.e. properties of elements of that ring which can be defined by using only existential quantifiers and equations - no negations). Connections with other areas - especially Diophantine Geometry - will be mentioned. The talk will be dedicated to the memory of Lee Rubel, who was the first to ask these and other similar questions. | ||||||
Tue, 11 Oct 2016 | 13:15 | A303 | Themis Mitsis | University of Crete | Carleman estimates with convex weights | |
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Tue, 25 Oct 2016 | 13:15 | A303 | Ilya M. Spitkovsky | NYU -- Abu Dhabi | Factorization of almost periodic matrix functions: some recent results and open problems | |
Abstract: The set $AP$ of (Bohr) almost periodic functions is the closed sub-algebra of $L_\infty(\mathbb{R})$ generated by all the exponents $e_\lambda(x) := e^{i\lambda x}$ , $\lambda\in \mathbb{R}$. An $AP$ factorization of an $n$-by-$n$ matrix function $G$ is its representation as a product $ G = G_+ \text{diag}[e_{\lambda_1} ,...,e_{\lambda_n} ]G_-$, where $G_+^{\pm 1}$ and $G_{-}^{\pm 1}$ have all entries in $AP$ with non-negative (resp., non-positive) Bohr-Fourier coefficients. This is a natural generalization of the classical Wiener-Hopf factorization of continuous matrix-functions on the unit circle, arising in particular when considering convolution type equations on finite intervals. The talk will be devoted to the current state of $AP$ factorization theory. Time permitting, problems still open will also be described. | ||||||
Thu, 24 Nov 2016 | 13:15 | A303 | Antonis Manoussakis | Technical Univ. of Crete | Banach spaces with few operators | |
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Tue, 29 Nov 2016 | 13:15 | A303 | Kostas Tsaprounis | University of Crete | Large cardinal axioms | |
Abstract: The axiomatic system of ZFC set theory is nowadays generally accepted as the current foundation of mathematics. However, and despite the expressive and deductive power of this system, it is widely known that many problems and questions, coming from diverse mathematical areas, are (provably) independent from the ZFC axioms. In the direction of reinforcing this basic theory with additional assumptions, one dominant family of candidates for new axioms consists of the so-called large cardinal axioms. These postulates, which have been intensively studied during the last decades, assert, roughly speaking, the existence of stronger and stronger forms of infinity, thus creating a hierarchy of very potent assumptions beyond ZFC. In this talk, we will present (some of) the large cardinal axioms, while underlining their (very useful and) powerful reflection properties. Moreover, we will mention some connections and applications of these axioms in the context of other mathematical areas. | ||||||
Tue, 6 Dec 2016 | 13:15 | A303 | Mihalis Kolountzakis | University of Crete | Infinitely many electrons on a line, at equilibrium | |
Abstract: We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles
on the circle, such that the (repelling) force they exert on each other depends only on their distance.
The main question is whether each equilibrium configuration needs to be an arithmetic progression.
Under very broad assumptions on the force we show this for the particles on the circle. In the case of infinitely
many particles on the line we show the same result under the assumption that the maximal (or the minimal) gap between successive points is
finite (positive) and assumed at some pair of successive points.
Under the assumption of analyticity for the force field (e.g., the Coulomb force) we deduce some extra rigidity for the configuration: knowing
an equilibrium configuration of points in a half-line determines it throughout.
Various properties of the equlibrium configuration are proved.
Joint work with Agelos Georgakopoulos (Warwick) |
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Tue, 14 Feb 2017 | 13:15 | A303 | Nikos Frantzikinakis | University of Crete | Ergodicity of the Liouville system implies the Chowla conjecture | |
Abstract: The Liouville function assigns the value one to integers with an even number of prime factors and minus one elsewhere. Its importance stems from the fact that several well known conjectures in number theory can be rephrased as conjectural properties of the Liouville function. A well known conjecture of Chowla asserts that the signs of the Liouville function are distributed randomly on the integers, that is, they form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory this conjecture asserts that the "Liouville dynamical system" is a Bernoulli system. We prove that a much weaker property, namely, ergodicity of the "Liouville dynamical system", implies the Chowla conjecture. Our argument combines techniques from ergodic theory, analytic number theory, and higher order Fourier analysis. | ||||||
Tue, 21 Feb 2017 | 13:15 | A303 | Nikos Frantzikinakis | University of Crete | Ergodicity of the Liouville system implies the Chowla conjecture (part II) | |
Abstract: The Liouville function assigns the value one to integers with an even number of prime factors and minus one elsewhere. Its importance stems from the fact that several well known conjectures in number theory can be rephrased as conjectural properties of the Liouville function. A well known conjecture of Chowla asserts that the signs of the Liouville function are distributed randomly on the integers, that is, they form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory this conjecture asserts that the "Liouville dynamical system" is a Bernoulli system. We prove that a much weaker property, namely, ergodicity of the "Liouville dynamical system", implies the Chowla conjecture. Our argument combines techniques from ergodic theory, analytic number theory, and higher order Fourier analysis. | ||||||
Tue, 28 Mar 2017 | 13:15 | A303 | George Costakis | University of Crete | Invariant sets for operators | |
Abstract: The existence of invariant (closed) sets for linear operators is established for various classes of operators. Operator theorists are interested to such sets because of the still open invariant subspace (and subset) problem on Hilbert spaces. For instance, we show that a Hilbert space operator of the form non-unitary isometry plus compact is far from being weakly supercyclic; therefore every closed projective orbit is invariant. This is in sharp contrast to the case: unitary plus compact. | ||||||
Mon, 15 May 2017 | 13:15 | A303 | Bernard Host | Universite Paris-Est Marne-la-Vallee | Correlation sequences are nilsequences | |
Abstract: Many sequences defined as correlations appear to be nilsequences, up to the addition of a small error term. Results of this type hold both in the ergodic and in the finite setting. The proofs follow the same general strategy, although the context and the tools are completely different. I'll try to explain the ideas behind these results. This is a joint work with Nikos Frantzikinakis. | ||||||
Tue, 23 May 2017 | 12:30 | A303 | Dirk Pauly | Universität Duisburg-Essen | Maxwell equations, and their solutions. Some characteristic examples. | |
Abstract: We will give a simple introduction to Maxwell equations. Concentrating on the static case, we will present a proper $L^2$-based solution theory for bounded weak Lipschitz domains in three dimensions. The main ingredients are a functional analysis toolbox and a sound investigation of the underlying operators gradient, rotation, and divergence. This FA-toolbox is useful for all kinds of partial differential equations as well. | ||||||
Tue, 13 Jun 2017 | 15:15 | A303 | Ilias Tergiakidis | Univ. Göttingen | A local invariant for a four-dimensional Riemannian manifold | |
Abstract: The introduction of the Ricci flow equation by Hamilton in the late 80’s and the introduction of new techniques including Ricci flow with surgery by Perelman guided to very important results towards the understanding of the geometry and topology of $3$-dimensional manifolds. A crucial step was to understand the formation of singularities under the Ricci flow and the limits of their parabolic dilations. A natural question to ask is what happens in the four dimensional case. In this talk we will describe a geometric construction, which associates to every point of a $4$-dimensional Riemannian manifold an algebrogeometric object, called the branching curve, We hope that this construction will guide to new techniques for dealing with the formation of singularities in the $4$-dimensional Ricci flow. For any four-dimensional Riemannian manifold $(M,g)$, the exterior square of the tangent space at the point $x \in M$ denoted by $\Lambda^2 T_x M$, has three intrisically defined quadratic forms, $v_x$, $\Lambda^2 g_x$ and $R_x$. The first one is given by the exterior product evaluated at a volume form, the second by the second exterior square of the Riemannian metric $g_x$ and the third one by the Riemann curvature tensor. After complexifying $\Lambda^2 T_x M$ their projectivization defines three quadrics in $\mathbb{P} (\Lambda^2 T_x M \otimes \mathbb{C})$. Specifically, $\mathbb{P}(v_x)$ determines the Grassmann manifold of lines in $\mathbb{P}(T_x M \otimes \mathbb{C})$. For every $x \in M$ , the complete intersection of these three quadrics, corresponds to a singular $K3$ surface. Its resolution of singularities is as branched double cover for the quadric $\mathbb{P} (g_x)$ coming from the metric and living in $\mathbb{P}(T_x M \otimes \mathbb{C})$. |
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