**Speaker:** Joana Cirici (UB)

**Title:** Hodge-de Rham numbers of almost complex 4-manifolds

**Date:** 22/3/2022

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** I will introduce a Frölicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures. As an application, I will focus on the case of almost complex 4-manifolds, showing how the Frölicher-type spectral sequence gives rise to Hodge-de Rham numbers with very special properties. This is joint work with Scott Wilson.

**Speaker:** Arturo Fernández (ICEX - Universidade Federal de Minas Gerais - Brasil)

**Title:** Number of Milnor and Tjurina of Foliations

**Date:** 21/10/2021

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this lecture, I will show the relationship between the Milnor and Tjurina numbers of a foliation in the complex plane. Such numbers are similar to the classic Milnor and Tjurina numbers for singular curves. This work is in collaboration with Evelia García Barroso (Universidad de la Laguna - Spain) and Nancy Saravia Molina (PUCP-Peru).

**Speaker:** Daniel Räde (Augsburg)

**Title:** Macroscopic band width inequalities

**Date:** 2/4/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Speaker:** Dan Agüero (IMPA)

**Title:** New invariants and a splitting theorem for complex Dirac structures

**Date:** 26/3/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk, we start by recalling the relationship between Poisson and Dirac geometry. We use this as motivation for studying complex Dirac structures with constant real index. Then we introduce a new invariant, the order and redefine the previously known invariant: the type. Finally we give a local description for complex Dirac structures with constant real index and order via a splitting theorem.

**Speaker:** Jérôme Los (Aix-Marseille)

**Title:** Volume entropy for surface groups via dynamics.

**Date:** 12/3/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk, I will explain how we can use the dynamics of Bowen-Series Like maps to compute explicitly the volume entropy for a class of presentations of surface groups. In particular the « classical » presentations are covered by this class and it gives a way to make the computation explicite for all surfaces.

**Speaker:** Jérôme Los (Aix Marseille)

**Title:** Geometrization of some piecewise homeomorphisms of the circle.

**Date:** 20 /2/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk I will describe a class of piecewise homeomorphisms of the circle from which we can construct a subgroup of the group Home^+ (S^1).
In this particular class we show that the group in question is a surface group and the map is a Bowen-Series-Like map for that group.

**Speaker:** Dmitry Novikov (Weizmann Institute)

**Title:** Complex Cellular Structures (joint with Gal Binyamini)

**Date:** 13 /2/2020

**Time:** 15:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes.
The Gromov-Yomdin Lemma (latter generalized by Pila and Wilkie) proves that the maps could be chosen of $C^r$-norm at most one, and the number of such maps is uniformly bounded for finite-dimensional families.
This number was not bounded by Yomdin or Gromov, but it necessarily grows as $r\to\infty$.
We explained the obstruction to complexification of this result in terms of the inner hyperbolic metric properties of complex holomorphic sets.
Further, we accidentally proved a new simple lemma about holomorphic functions in annulii, a quantitative version of great Picard theorem. This lemma allowed us to construct a proper holomorphic version of the above results in all dimensions, effective and with explicit polynomial bounds on complexity for families of complex (sub)analytic and semialgebraic sets, combining best properties of both aforementioned results.
As first corollaries we effectively bound the Yomdin-Gromov number (thus proving a long-standing Yomdin conjecture about tail entropy) and prove a bound on the number of rational points in $\log$-sets in the spirit of Wilkie conjecture.

**Speaker:** Lucas Ambrozio (Warwick)

**Title:** Min-max width and volume of Riemannian three-dimensional spheres

**Date:** 30/1/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** By the work of Simon and Smith, every Riemannian three-dimensional sphere contains an embedded minimal two-dimensional sphere. Their method of construction is a min-max method for the area functional and the area of this minimal sphere is bounded from above by a number depending only on the ambient geometry, known as the width. In this talk, we will discuss upper bounds for the width among certain classes of metrics with the same volume. This is joint work with Rafael Montezuma (UMass-Amhrest).

**Speaker:** Florent Balacheff (UAB)

**Title:** Entropia de volum i longituds de corbas homotòpicament independents

**Date:** 16/1/2020

**Time:** 12:00

**Web:** http:// mat.uab.cat/web/ligat/

**Abstract:** Presentaré una desigualtat per les varietats riemannianes tancades que impliqua l'entropía del volum i el conjunt de longituds de qualsevol família de corbes homotòpicament independents basats en un mateix punt. Aquesta desigualitat implica un teorema del collaret universal, és a dir sense restricció de curvatura. És un treball en col.laboració amb el Louis Merlin.

**Speaker:** Andrea Sambusetti (Roma)

**Title:** Topological rigidity and finiteness for non-geometric 3-manifolds

**Date:** 19/12/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** The Riemannian geometry of non-geometric 3-manifolds (that is, those which do not admit any of the eight complete maximal homogeneous 3-dimensional geometries) deserved considerably less attention than their geometric counterparts, with a few remarkable exceptions. In this seminar, we will explain some peculiar topological rigidity and finiteness properties of the class of non-geometric Riemannian 3-manifolds with bounded entropy and diameter, with respect to the Gromov-Hausdorff distance. The talk is based on the papers https://arxiv.org/abs/1705.06213 and https://arxiv.org/abs/1711.06210 in collaboration with F. Cerocchi"

**Speaker:** Sorin Dumitrescu (Université Côte d'Azur, Nice)

**Title:** Rational parallelisms and generalized Cartan geometries on complex manifolds

**Date:** 5/12/2019

**Time:** 12:00

**Web:** http:// mat.uab.cat/web/ligat/

**Abstract:** This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic conformal structures, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms.
We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries. This is joint work with Indranil Biswas (TIFR, Mumbai).

**Speaker:** Pallavi Panda (Lille)

**Title:** Parametrisation of the deformation spaces of ideal polygons and punctured polygon

**Date:** 21/11/ 2019

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** To every hyperbolic surface, one can associate a simplicial complex called the arc complex whose 0-th skeleton is the set of isotopy classes of non-trivial embedded arcs and there is a k-simplex for every (k+1)-tuple of pairwise disjoint and distinct isotopy classes. The arc complexes of ideal polygons and punctured polygons are finite and are homeomorphic to spheres. The deformation spaces of these two hyperbolic surfaces can be completely parametrised via their arc complexes using strip deformations along the arcs. This is a partial generalisation of a result by Dancinger-Guéritaud-Kassel.

**Speaker:** Marcos Cossarini (UPEM)

**Title:** Discrete surfaces with length and area and minimal fillings of the circle.

**Date:** 7/11/ 2019

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of crossings with the walls, and the area of the surface is the number of crossings between the walls themselves. We show how to approximate a Riemannian or self-reverse Finsler metric by a wallsystem.
This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere has minimum area among orientable Riemannian surfaces that fill isometrically a closed curve of given length. (A surface fills its boundary curve isometrically if the distance between each pair of boundary points measured along the surface is not less than the distance measured along the boundary.) We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n (n-1)/2 squares. This conjecture is equivalent to the FAC extended to surfaces with self-reverse Finsler metric.
If the surface is a disk, the discrete FAC follows from Steinitz's algorithm for transforming curves into pseudolines. This gives a new, combinatorial proof that the FAC holds for disks with Riemannian or self-reverse Finsler metric.
If time allows, we also discuss how to discretize a directed metric on a surface using a triangulation with directed edges. The length of each edge is 1 in one way and 0 in the other way, and the area of the surface is the number of triangles. These discrete surfaces are dual to Postnikov's plabic graphs.

**Speaker:** Adrien Boulanger (Aix-Marseille Université)

**Title:** Counting problems in infinite measure

**Date:** 31/10/2019

**Time:** 12:00

**Speaker:** Ignasi Mundet Riera (UB)

**Title:** Subgrups compactes de grups de difeomorfismes i propietat Jordan

**Date:** 24/10/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Speaker:** Gil Solanes (UAB)

**Title:** Valoracions de Lipschitz-Killing en varietats pseudo-riemannianes

**Date:** 10/10/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Les curvatures de Lipschitz-Killing són invariants riemannians que apareixen en situacions tan diverses com la fórmula dels tubs de Weyl i l'espectre del laplacià de les formes diferencials. Les valoracions de Lipschitz-Killing són l'extensió d'aquests invariants als subconjunts compactes suficientment regulars d'una varietat riemanniana.
Presentarem un treball conjunt amb A. Bernig i D. Faifman on estenem les valoracions de Lipschitz-Killing al context de les varietats pseudo-riemannianes.

**Speaker:** Samir Bedrouni (USTHB, Alger)

**Title:** Convex foliations of degree four on the complex projective plane

**Date:** 27/6/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk, I will present the main results of a recent paper in collaboration with D. Marín, cf. arXiv:1811.07735. First, I will explain the outline of the proof of the result which states that up to automorphism there are 5 homogeneous convex foliations of degree four on the complex projective plane. Second, we will see how to use this result to obtain a partial answer to a question posed in 2013 by D. Marín and J. Pereira about the classification of reduced convex foliations on the complex projective plane.

**Speaker:** Nabil Kahouadji (Northeastern Illinois University)

**Title:** Isometric Immersions of Pseudo-Spherical Surfaces via PDEs.

**Date:** 27/6/2019

**Time:** 10:45

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Pseudo-spherical surfaces are surfaces of constant negative Gaussian curvature. A way of realizing such a surface in 3d space as a surface of revolution is obtained by rotating the graph of a curve called tractrix around the z-axis (infinite funnel). There is a remarkable connection between the so- lutions of the sine-Gordon equation uxt = sin u and pseudo-spherical surfaces, in the sense that every generic solution of this equation can be shown to give rise to a pseudo-spherical surface. Furthermore, the sine-Gordon equation has the property that the way in which the pseudo-spherical surfaces corresponding to its solutions are realized geometrically in 3d space is given in closed form through some remarkable explicit formulas. The sine-Gordon equation is but one member of a very large class of differential equations whose solutions likewise define pseudo-spherical surfaces. These were defined and classified by Chern, Tenenblat [1] and others, and include almost all the known examples of "integrable" partial differential equations. This raises the question of whether the other equations enjoy the same remarkable property as the sine-Gordon equation when it comes to the realization of the corresponding surfaces in 3d space. We will see that the answer is no, and will provide a full classification of hyper- bolic and evolution equations [2, 3, 4]. The classification results will show, among other things, that the sine-Gordon equation is quite unique in this regard amongst all integrable equations.

**Speaker:** Robert Cardona (UPC)

**Title:** A contact topology approach to Euler flows universality

**Date:** 9/5/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** There has been several steps towards establishing universality of Euler flows in the last years, especially in two papers by Terence Tao in 2017 and 2019. In this talk, we propose a new approach to this question. After presenting a correspondence theorem between some steady Euler flows and Reeb vector fields in contact geometry, we provide new results in the direction of Tao’s programme. By means of high dimensional contact topology, we prove some realization theorems on Reeb dynamics. We deduce some new universality properties for steady Euler flows, for instance “Turing universality” which was one of the suggested open problems in Tao’s papers. This is a joint work in progress with Eva Miranda, Daniel Peralta-Salas and Francisco Presas.

**Speaker:** Martin Henk (Technische Universität Berlin)

**Title:** The dual Minkowski problem

**Date:** 25/4/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Speaker:** Cédric Oms (UPC)

**Title:** Hamiltonian Dynamics on Singular Symplectic Manifolds

**Date:** 4/4/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** The study of singular symplectic manifolds was initiated by the work of Radko, who classified stable Poisson structures on surfaces. It was observed by Guillemin—Miranda—Pires that stable Poisson structures can be treated as a generalization of symplectic geometry by extending the de Rham complex. Since then, a lot has been done to understand the geometry, dynamics and topology of those manifolds.
We will explore the odd-dimensional case of those manifolds in this talk by extending the notion of contact manifolds to the singular setting. We prove existence of singular contact structures. We will prove that singularities allow do construct Reeb plugs and thereby disproving Weinstein conjecture in this setting. In particular, this yields the existence of proper Hamiltonians on singular symplectic manifolds without periodic Hamiltonian orbits.
This is joint work with Eva Miranda.

**Speaker:** Stefan Suhr (University of Bochum)

**Title:** A Morse theoretic Characterization of Zoll metrics

**Date:** 28/3/2019

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** From the Morse theoretic point of view Zoll metrics are rather peculiar. All critical sets of the energy on the loop space are nondegenerate critical manifolds diffeomorphic to the unit tangent bundle. This especially implies that min-max values associated to certain homology classes coincide. In my talk I will explain that the coincidence of these min-max values characterises Zoll metrics in any dimension. A specially focus will lie on the case of the 2-sphere. This is work in collaboration with Marco Mazzucchelli (ENS Lyon).

**Speaker:** Clemens Huemer (UPC)

**Title:** Carathéodory's theorem in depth

**Date:** 21/3/2019

**Time:** 12:00

**Web:** http:// mat.uab.cat/web/ligat/

**Abstract:** Let $S$ be a finite set of points in $\mathbb{R}^d$. Carathéodory's theorem states that if a point $q$ is inside the convex hull of $S$, then there exist points $p_1, \dots, p_{d+1}$ of $S$ such that $q$ is contained in the convex hull of $\{p_1,\dots,p_{d+1}\}$; that is $q$ is contained in the simplex defined by $\{p_1,\dots,p_{d+1}\}$.
We present a depth version of this theorem. Informally, we prove that if $q$ is "deep inside" the convex hull of $S$ then there exist "large" pairwise disjoint subsets $S_1,\dots,S_{d+1}$ of $S$, such that every simplex having a vertex
from each $S_i$ contains $q$.
We also prove depth versions of Helly’s and Kirchberger’s theorems.
This is a joint work with Ruy Fabila-Monroy.

**Speaker:** Gilles Courtois (Institut de Mathématiques de Jussieu - Paris)

**Title:** Minimal entropy of manifolds and their fundamental group

**Date:** 14/3/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** The Milnor-Svarc theorem says that the entropy of a closed Riemannian manifold is non zero if and only if its fundamental group has exponential growth but does not give an explicit relation between the minimal entropy of a manifold and the minimal entropy of its fundamental group. The goal of this talk is to explain this through examples and state that such relations hold for manifolds with Gromov-hyperbolic fundamental group.

**Speaker:** Erika Pieroni (Università La Sapienza - Roma)

**Title:** Minimal Entropy of 3-manifolds

**Date:** 7/3/2019

**Time:** 12:00

**Web:** http:// mat.uab.cat/web/ligat/

**Abstract:** We compute the minimal entropy of every closed, orientable 3-manifold, proving that the cube of this invariant behaves additively both with respect to the prime decomposition and the JSJ decomposition.
As a consequence, we deduce that the cube of the minimal entropy, in restriction of closed, orientable 3-manifolds, is proportional to the simplicial volume, where the proportionality constant only depends on the dimension n=3.

**Speaker:** Laurent Meersseman (Université d'Angers)

**Title:** Kuranishi and Teichmüller

**Date:** 21/2/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Let $X$ be a compact complex manifold. The Kuranishi space of $X$ is an analytic space which encodes every small deformation of $X$. The Teichmüller space is a topological space formed by the classes of compact complex manifolds diffeomorphic to $X$ up to biholomorphisms smoothly isotopic to the identity. F. Catanese asked when these two spaces are locally homeomorphic. Unfortunatly, this almost never occurs. I will reformulate this question replacing these two spaces with stacks. I will then show that, if $X$ is Kähler, this new question has always a positive answer. Finally, I will discuss the non-kähler case.

**Speaker:** Matias del Hoyo (Universidade Federal Fluminense, i visitante del CRM dentro de la Convocatoria Lluís Santaló financiada por l'Institut d'Estudis Catalans)

**Title:** Deformations of compact foliations

**Date:** 14/2/2019

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** In a recent work with R. Fernandes we show that a compact Hausdorff foliation over a compact connected manifold is rigid, in the sense that every one-parameter deformation of it must be trivial. We study the foliation by using the Lie theory of Lie groupoids and Lie algebroids. A foliation is an example of a Lie algebroid and it can always be integrated to its so-called holonomy groupoid. In this talk I will present the theorem and illustrate with examples, provide a glimpse on Lie groupoids and Lie algebroids, and discuss a sketch of our proof.

**Speaker:** Alfredo Hubard (Paris Est-Marne La vallée)

**Title:** The branch-waist of Riemannian two-spheres.

**Date:** 31/1/2019

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** I will explain an equality between a min max quantity related to the diastole (or waist) of a map from a two sphere to a tree, and the largest antipodality of the sphere. The insights come from the theory of graph minors which we import to the Riemannian world. From these insights we improve some fundamental inequalities in curvature free Riemannian geometry. Joint work with Arnaud de Mesmay and Francis Lazarus.

**Speaker:** Marco Mazzucchelli (ENS Lyon)

**Title:** On the boundary rigidity problem for surfaces

**Date:** 24/1/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** The classical boundary rigidity problem asks whether, or to what extent, the inner geometry of a compact Riemannian manifold with boundary can be determined by means of boundary measurements, such as the distance function among boundary points, or the geodesic scattering map. In my talk I will review this problem and some of the known results that are valid for "simple" Riemannian manifolds. I will then sketch the proof of some recent boundary rigidity results for non-simple Riemannian surfaces, including surfaces with trapped geodesics or with non-convex boundary. The talk is based on joint work with Colin Guillarmou and Leo Tzou.

**Speaker:** Carles Sáez (UAB)

**Title:** Which finite groups act smoothly on a given 4-manifold ?

**Date:** 17/1/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Speaker:** Jean Raimbault (Universitat de Toulouse)

**Title:** The topology of arithmetic hyperbolic three-manifolds

**Date:** 10/1/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** I will discuss various recent results that demonstrate the "particular beauty" of arithmetic congruence manifolds among the family of all hyperbolic manifolds of finite volume. In particular I will talk about when cusped arithmetic manifolds can be link complements.

**Speaker:** Cyril Lecuire (Toulouse)

**Title:** Quasi-isometric rigidity of 3-manifold groups

**Date:** 13/12/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** I will discuss the quasi-isometric rigidity of 3-manifold groups: A finitely generated group that roughly (when viewed from far away) looks like the fundamental group of a compact 3-manifold is in fact the fundamental group of a compact 3-manifold (up to taking a finite index subgroup). I will first introduce some definitions to turn the previous sentence into a precise statement. Then I will explain how we use characteristic decompositions, the Perelman-Thurston's Geometrization Theorem and previous works on quasi-isometric rigidity to prove it. This is a joint work with Peter Haissinsky.

**Speaker:** Bram Petri (University of Bonn)

**Title:** Short geodesics on random hyperbolic surfaces

**Date:** 29/11/2018

**Time:** 10:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Random Surfaces can be used to study the geometric properties of typical (hyperbolic) surfaces of large genus. Moreover, they can sometimes be used in existence proofs. That is, sometimes the easiest way of proving that surfaces with certain properties exist is to prove that the probability that a random surface has these properties is non-zero. Of course there are multiple possible models of random surfaces. In this talk, a random surface will be a surface that is picked at random using the Weil-Petersson volume form on the moduli space of hyperbolic surfaces of a given genus. I will speak about the length spectrum of these random surfaces. This is joint work with Maryam Mirzakhani.

**Speaker:** Roberto Rubio

**Title:** From classical structures to Dirac structures and generalized complex geometry

**Date:** 22/11/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** This is an introductory talk to motivate the definition of Dirac structures, which encompass presymplectic and Poisson geometry, and generalized complex structures, which encompass complex and symplectic geometry. We will start by reviewing all these classical structures on a smooth manifold $M$, and then propose alternative ways to look at them. For instance, the graph of both a presymplectic or a Poisson structure can be seen as a subbundle of $T \, M+T^ *M$. This subbundle, by skew-symmetry, is maximally isotropic for the canonical symmetric pairing in $T \, M+T^*M$. Such a maximally isotropic subbundle is an almost Dirac structure. In order to talk about Dirac structures, we will need to introduce the suitable analogue to the Lie bracket on vector fields: the Dorfman bracket on sections of $T \, M+T^*M$. No previous knowledge about Dirac structures or generalized geometry will be assumed for this talk.

**Speaker:** Joan Porti (UAB)

**Title:** Holomorphic forms on the $SL(N,{\ mathbb C})$ moduli spaces of surfaces with boundary

**Date:** 15/11/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** For an oriented surface of finite type, we consider the moduli space of representations in a simply connected reductive Lie group (eg $SL(N,{\mathbb C})$), and also the moduli space relative to the boundary. Find a relationship between complex valued volume forms in those moduli spaces, the relative and the absolute one. This is joint work with M. Heusener.

**Speaker:** Constantin Vernicos (Montpellier)

**Title:** Volume growth in Hilbert Geometry

**Date:** 8/11/2018

**Time:** 12:00

**Web:** http:/ /mat.uab.cat/web/ligat/

**Abstract:** Hilbert geometries are metric spaces defined in the interior of a convex set thanks to the cross-ratio as in the so called projectif model of the hyperbolic geometry (also referred as Beltrami, Cayley or Klein model). While becoming acquainted with these geometries, we will survey what is nowadays known about the volume growth of metric balls with respect to they radius and in particular the volume entropy.

**Speaker:** Simon Allais (ENS Lyon)

**Title:** Application of generating functions to symplectic and contact rigidity

**Date:** 18/10/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In 1992, Viterbo introduced new means to study the Hamiltonian dynamics of ${\mathbb R}^{2n}$ by applying Morse-theoretical methods to generating functions. Among his results, he gave a new proof of Gromov's non-squeezing theorem (1985) and sketched a proof of the more subtle symplectic camel theorem. A part of this work was generalized to the contact case by Sandon (2011) who provided a new way to derive the contact non-squeezing theorem of Eliashberg, Kim and Polterovich (2006).
We will recall the main points of this theory and show how it allows us to derive a proof of the symplectic camel theorem which can easily be extended to the contact case.

**Speaker:** Carlos Florentino (Universitat de Lisboa)

**Title:** Geometry, Topology and Arithmetic of Character Varieties

**Date:** 11/10/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Character varieties are spaces of representations of finitely presented groups F into Lie groups G. For some choices of F and G, these play important roles in theories such as Chern-Simons or 2d conformal field theory, as well as in hyperbolic geometry and knot theory. Some character varieties are also interpreted as moduli spaces of G-Higgs bundles over Kähler manifolds. When G is a complex algebraic group, character varieties are algebraic and have interesting geometry and topology. We can also consider more refined invariants such as Deligne's mixed Hodge structures, which are typically very difficult to compute, but also provide relevant arithmetic information.
In this seminar, we present some explicit computations of polynomial invariants of character varieties, concentrating on a generalization of the Euler-Poincaré characteristic - the so-called E-polynomial - and when G is a group such as SL (n,C), (P)GL(n,C)or Sp(n,C). We also show a remarkable relation between these invariants and those for the corresponding irreducible loci inside the character varieties. All concepts will be motivated with several examples, and we will give an overview of known calculations of E-polynomials, as well as some conjectures and open problems.
This is joint work with A. Nozad, J. Silva and A. Zamora

**Speaker:** Teresa García (UAB)

**Title:** Actions on products of CAT(-1) spaces

**Date:** 4/10/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk I will discuss the main result of my PhD thesis: for X a proper CAT(-1) space there is a maximal open subset of the horofunction compactification of $X \times X$ with respect to the maximum metric that compacti fies the diagonal action of an infinite quasi-convex group of the isometries of $X$.
I will also discuss briefly the case of a product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper CAT(-1) spaces.

**Speaker:** Dmitry Faifman (University of Toronto)

**Title:** Curvature in contact manifolds and integral geometry

**Date:** 12/6/2018

**Time:** 12:00

**Abstract:** Valuations are finitely additive measures on nice subsets, for example the Euler characteristic, volume and surface area are valuations. During the 20th century, valuations have been studied predominantly on convex bodies and polytopes, in linear spaces and lattices. Valuations on manifolds were introduced about 15 years ago by S. Alesker, with contributions by A. Bernig, J. Fu and others, and immediately brought under one umbrella a range of classical results in Riemannian geometry, notably Weyl's tube formula and the Chern-Gauss-Bonnet theorem. These results circle around the real orthogonal group.
In the talk, the real symplectic group will be the central player. Drawing inspiration from the Lipschitz-Killing curvatures in the Riemannian setting, we will construct some natural valuations on contact and dual Heisenberg manifolds, which generalize the Gaussian curvature. We will also construct symplectic-invariant distributions on the grassmannian, leading to Crofton-type formulas on the contact sphere and symplectic space.

**Speaker:** Florent Balacheff (Univ. Lille/UAB)

**Title:** Producte de longituds de geodèsiques tancades homòlogicament independents

**Date:** 5/6/2018

**Time:** 12:00

**Abstract:** En aquesta xerrada considerarem generalitzacions del segon teorema de Minkowski per a varietats Riemannianes. Per exemple, explicarem per què els tors, les superfícies i la suma connexa de dos espais projectius reals de dimensió 3, amb una mètrica Riemanniana de volum normalitzat, sempre admeten una base d’homologia mòdul 2 induïda per geodèsiques tancades, per a les quals el producte de les longituds està acotat per sobre. Basat en un treball conjunt amb S. Karam i H. Parlier.

**Speaker:** José Andrés Rodríguez Migueles (Université de Rennes I)

**Title:** Geodésicas en superficies hiperbólicas y complementos de nudos

**Date:** 8/5/2018

**Time:** 12:00

**Abstract:** Toda geodésica cerrada en una superficie hiperbólica tiene un levantamiento canónico en su haz tangente unitario, y lo podemos ver como un nudo dentro de una variedad de dimensión 3. Por ejemplo, Ghys demostró que los nudos de Lorentz son precisamente esos que se encuentran de dicha manera si usamos la superficie modular. Genéricamente el complemento de los nudos así construidos admiten una estructura hiperbólica, única por el teorema de rigidez de Mostow. En esta plática voy a relacionar algunas propiedades de la geodésica
inicial con la geometría de la 3-variedad.

**Speaker:** Antonin Guilloux (Univ. Pierre et Marie Curie)

**Title:** Volume function and Mahler measure of polynomials

**Date:** 17/4/2018

**Time:** 12:00

**Abstract:** The Mahler measure of a polynomial is a kind of size of this polynomial, introduced around 1930 for questions related to prime numbers. It has since proven useful and intriguing in a variety of contexts of number theory. It is very hard to compute in general.
I will present this Mahler measure, and its computation for a class of 2-variables polynomials through elementary considerations about a function called volume function. Indeed, for this class, strong and surprising links exist between Mahler measure and volume of hyperbolic 3-manifolds.

**Speaker:** Philippe Castillon (Université de Montpellier)

**Title:** Prescribing the Gauss curvature of hyperbolic convex bodies

**Date:** 10/4/2018

**Time:** 12:00

**Abstract:** The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure $\ mu$, Alexandrov problem consists in proving the existence of a convex body whose curvature measure is $\mu$. In the Euclidean space, this problem is equivalent to an optimal transport problem on the sphere.
In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain how to relate this problem to a non linear Kantorovich problem on the sphere and how to solve it.
Joint work with Jérôme Bertrand.

**Speaker:** Laurent Meersseman (Université d'Angers)

**Title:** Local Structure of the Teichmüller Stack

**Date:** 6/3 /2018

**Time:** 12:00

**Abstract:** In this talk, I will discuss the local structure of the Teichmüller stack of a fixed smooth compact manifold M, which encodes the set of complex structures on M up to biholomorphisms isotopic to the identity. I will focus on the differences between Kähler components (that is components corresponding to Kähler structures) and non-Kähler ones.

**Speaker:** Agustí Reventós (UAB)

**Title:** Algunes aplicacions geomètriques de les sèries de Fourier

**Date:** 29/1/2018

**Time:** 12:00

**Speaker:** Jean Gutt (Univ. Köln)

**Title:** Knotted symplectic embeddings

**Date:** 22/1/2018

**Time:** 12:00

**Abstract:** I will discuss a joint result with Mike Usher, showing that many toric domains $X$ in the 4-dimensional euclidean space admit symplectic embeddings $f$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $f (X)$ to $X$.

**Speaker:** Michael Heusener (Univ. Clermont Auvergne)

**Title:** Deformations of abelian representations of knot groups into $\mathrm{SL}(n,\mathbb C)$

**Date:** 8/1/2018

**Time:** 12:00