François Labourie (Nice)
Positivity and Surface Groups / Ghost algebra, Poisson bracket and convexity abstract
Abstract:
“Positivity and surface groups”Positivity in a generalizsd flag manifold $M$ was introduced by Guichard and Wienhard. It generalizes both Lusztig poitivity and the geometry of the Shilov boundary. We will describe this notion which is based on a simple « nesting » property and show how it helps defining positivity maps of the circle in $M$, as well as representation of surface groups with properties mimicking those arising as monodromies of hyperbolic structures. Based on work with Guichard, Wienhard, and then Pozzetti and Beyrer.“Ghost algebra, Poisson bracket and convexity”The moduli space of Anosov representations of a surface group in a semisimple group – an open set in the character variety – admits many more natural functions than the regular functions. We will study in particular length functions and correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called ghost polygons and ghost bracket encoded in a formal algebra called the ghost algebra related in some cases to the swapping algebra. As a consequence of our main theorem, we show that the set of those functions – length and correlation – is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalizing a result of Kerckhoff in Teichmüller space. Based on work with Bridgeman.
10:00 • Université de Genève, Conseil Général 7-9, Room 1-07
Aitor Iribar López (ETH Zürich)
Complex abelian varieties and their moduli IX abstract
Abstract:
Intersection theory on A_g and on Mumford\'s partial compactification
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Gurvan Mével (UNIGE)
Floor diagrams and some tropical invariants in positive genus abstract
Abstract:
Göttche-Schroeter invariants are a rational tropical refined invariant, i.e. a polynomial counting genus 0 curves on toric surfaces, that can be computed with a floor diagrams approach. In this talk I will explain that this approach extends in any genus. This gives new invariants, related to ones simultaneously defined by Shustin and Sinichkin. I will then say few words on a quadratically enriched (and not refined !) version of this extension.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-07
Fabien Pazuki (University of Copenhagen)
Abstract:
Title: Parallelogram inequality for abelian varieties and applications Abstract: Let $A$ be an abelian variety defined over a number field. A theorem of Rémond states that for any two finite subgroup schemes $G, H$, the Faltings height of the four isogenous abelian varieties $A/G, A/H, A/(G+H), A/(G\\cap H)$ are linked by an elegant inequality. The goal of the talk is to present an analogous inequality for abelian varieties defined over function fields, and discuss some applications in diophantine geometry. This is joint work with Richard Griffon and Samuel Le Fourn. Please note the unusual time and location! Spiegelgasse 1, Seminarraum 00.003
14:15 • Universität Basel
Dr. Hassan Ou-azzou (Universität St.Gallen)
Equivalence of Classes of Polycyclic and Skew Polycyclic Codes over Finite Fields abstract
Abstract:
We study the equivalence of families of polycyclic codes associated with polynomials of the form \\(x^n - a_{n-1}x^{n-1} - \\ldots - a_1x - a_0\\) over a finite field, \\(\\mathbb{F}_q,\\ q=p^s\\). We begin with the specific case of polycyclic codes associated with a trinomial \\(x^n - a_{\\ell} x^{\\ell} - a_0\\) (for some \\(0< \\ell <n\\)), which we refer to as \\(\\ell\\)-trinomial codes, after which we generalize our results to general polycyclic codes. We introduce an equivalence relation called \\(n\\)-equivalence, which extends the known notion of \\(n\\)-equivalence for constacyclic codes. We compute the number of \\(n\\)-equivalence classes for this relation and provide conditions under which two families of polycyclic (or \\(\\ell\\)-trinomial) codes are equivalent. In particular, we prove that when \\(\\gcd(n, n-\\ell, q-1) = 1\\), any \\(\\ell\\)-trinomial code family is equivalent to a trinomial code family associated with the polynomial \\(x^n - x^{\\ell} - 1\\). For \\(n=p^r\\), we give a complete classification of \\(p^{r-1}\\)-trinomial codes, and \\(p^{r-s}\\)-trinomial codes. Secondly, we extend our study to the case of skew polycyclic codes. Finally, we provide some examples as an application of the theory developed in this paper.
15:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Bernoulli Instructor
Low temperature asymptotic expansions in spin O(N) models in dimensions 3 and higher abstract
Abstract:
I will review old and recent results on the problem of obtaining asymptotic series in powers of the temperature (T) around T=0 for local correlation functions of lattice spin models with continuous symmetries, using Reflection Positivity. After presenting the models and the problem, I will mostly discuss how to obtain the formal power series in a way that can be translated into a proof, and only briefly mention how to actually turn the formal procedure into a proof. If time permits, I will also mention some features of the d=2 case.Based on recent joint work with A. Giuliani, and on (less recent) work of Bricmont, Fontaine, Lebowitz, Lieb, and Spencer.
16:15 • EPF Lausanne, seminar room S. Ott