Seminari de Geometria Algebraica de Barcelona
 UB UPC UAB
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 Seminari de Geometria Algebraica 2018/2019 Conferenciant Títol Data i hora ARCADES Doctoral School II and ESR Days 3 a 7 de setembre, IMUB Constantin Shramov Steklov Math Inst & NRU HSE Moscou, Rússia Contact: ignasi.mundet at ub.edu Automorphisms of Kaehler manifolds I will survey various results about finite groups acting by automorphisms and birational automorphisms of Kaehler manifolds. I will show that in many cases such groups enjoy the Jordan property, similar to subgroups of general linear groups. The talk is based on joint works with Yu. Prokhorov. Divendres 7 de setembre, 15h, Aula T2, FMI-UB Martín Sombra ICREA - UB The zero set of the independence polynomial of a graph In statistical mechanics, the independence polynomial of a graph G arises as the partition function of the hardcore lattice gas model on G. The distribution of the zeros of these polynomials when G→∞ is relevant for the study of this model and, in particular, to the determination of its phase transitions. In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth k ≥ 0. Our main result states that for these graphs, the zero sets of their independence polynomials converge as k→∞ to the bifurcation measure, in the sense of DeMarco, of a certain family of dynamical systems on the Riemann sphere. This is ongoing work with Juan Rivera-Letelier (Rochester) Divendres 28 de setembre, 15h, Aula T2, FMI-UB Alberto F. Boix Ben-Gurion U. of the Negev Beer-Sheva, Israel Contact: szarzuela at ub.edu A Characteristic Free Approach to Finite Determinacy Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an “infinitesimal determinacy”, i.e. to the tangent spaces at the orbits of the group action. The goal of this talk is to formulate a universal approach to finite determinacy in arbitrary characteristic, not necessarily over a field, for a large class of group actions; along the way, we introduce the notion of “pairs of (weak) Lie type”, which are groups together with a substitute for the tangent space at the unit element, such that the group is locally approximated by its tangent space, in a precise sense. This construction may be regarded as a sort of replacement of the exponential/logarithmic maps and is of independent interest. In this generality we establish the “determinacy versus infinitesimal determinacy” criteria, a far reaching generalization of numerous classical and recent results, together with some new applications. The content of this talk is based on joint work with Gert–Martin Greuel (Universität Kaiserslautern, Germany) and Dmitry Kerner (Ben–Gurion University of the Negev, Israel) Divendres 5 d'octubre, 15h, Aula T2, FMI-UB Roberto Gualdi U. Bordeaux - UB - CRM Contact: sombra at ub.edu Height of cycles in toric varieties We present in this talk some relations between suitable heights of cycles in toric varieties and the combinatorics of the defining Laurent polynomials. To do this, we associate to any Laurent polynomial f with coefficients in an adelic field two families of concave functions on a certain real vector space: the upper functions and the Ronkin functions of f. For the choice of an adelic semipositive toric metrized divisor D, we give upper bounds for the D-height of a complete intersection in a toric variety in terms of the upper functions of the defining Laurent polynomials. In the one-codimensional case, we prove an exact formula relating the D-height of a hypersurface to the Ronkin function of the associated Laurent polynomial, generalizing the well-known equality for the canonical case. Our approach involves mixed integrals, Legendre-Fenchel duality and other notions from convex geometry. Divendres 19 d'octubre, 15h, Aula T2, FMI-UB Francisco Presas ICMAT, Madrid Contact: ignasi.mundet at ub.edu Homotopy type of the space of smooth embeddings of $$\Large{\mathbb{S}}^1$$ in $$\Large{\mathbb{R}}^4$$ via Engel geometry. We introduce the space of horizontal embeddings for the standard Engel distribution in the Euclidean 4-space. We prove that the space of smooth embeddings of the circle into R⁴ is simply connected (classical result), by checking that the space of horizontal embeddings has homotopy type very related to the space of smooth embeddings (they are related by an h-principle). We extend the method to sketch the computation of the $$\pi_2$$ of that space showing that is Z$$\oplus$$Z (more modern result). We finally comment on work in progress further generalizing the techniques by using Manifold calculus to try to compute the whole homotopy type of this space. This is joint work with E. Fernández and X. Martínez-Aguinaga. Divendres 26 d'octubre, 15h, Aula T2, FMI-UB Pedro D. González Pérez ICMAT - UCM, Madrid Contact: adefelipe at ub.edu The valuative tree is a projective limit of Eggers-Wall trees Consider a germ C of reduced curve on a smooth germ S of complex analytic surface. Assume that C contains a smooth branch L. Using the Newton-Puiseux series of C relative to any coordinate system (x,y) on S such that L is the y-axis, one may define the Eggers-Wall tree ΘL(C) of C relative to L. Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically ΘL(C) into Favre and Jonsson's valuative tree of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on ΘL(C) as pullbacks of other naturally defined functions on the valuative tree. As a consequence, we prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning one branch: if L' is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees ΘL'(C) and ΘL(C) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the valuative tree is the projective limit of Eggers-Wall trees over all choices of curves C. Joint work with Evelia García Barroso and Patrick Popescu-Pampu. Divendres 9 de novembre, 15h, Aula T2, FMI-UB Joana Cirici UB A Dolbeault cohomology theory for almost complex manifolds In this talk I will survey recent joint work with Scott Wilson which extends Dolbeault cohomology to all almost complex manifolds, and generalizes some foundational results for compact Kähler manifolds to the non-integrable setting. I will also explain Lie-algebra analogues of the theory which provide useful computational tools for compact Lie groups and nilmanifolds. Divendres 16 de novembre, 15h, Aula T2, FMI-UB Diletta Martinelli School of Mathematics Edinburgh, Escòcia (UK) Contact: marti.lahoz at ub.edu On the geometry of contractions of the Moduli Space of sheaves of a K3 surface I will describe how recent advances have made possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland. In particular Bayer and Macrì relate birational transformations of the moduli space M of sheaves of a K3 surface X to wall-crossing in the space of Bridgeland stability conditions Stab(X). I will explain how it is possible to refine their analysis to give a precise description of the geometry of the exceptional locus of any birational contractions of M. Divendres 23 de novembre, 15h, Aula T2, FMI-UB Bernd Sturmfels Max Plank Institut Leipzig, Alemanya Contact: cdandrea at ub.edu Moment Varieties of Polytopes The moments of the uniform distribution on a convex polytope are rational functions in its vertex coordinates. We study the projective varieties parametrized by these moments. This is work with Kathlen Kohn and Boris Shapiro. On our journey, we encounter Hankel determinantal ideals, splines, cumulants, multisymmetric functions, and invariants of nonreductive groups. While moment varieties are complicated, they offer many nice open problems. Article Dilluns 26 de novembre, 15h, aula T1, FMI-UB Vincenzo Antonelli Politecnico di Torino, Itàlia Contact: miro at ub.edu Ulrich bundles on Hirzebruch surfaces Ulrich bundles on a projective variety are vector bundles that admit a completely linear resolution as sheaves on the projective space. They carry many interesting properties and they are the simplest one from the cohomological point of view. In this talk we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over P¹. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. Finally we discuss some particular cases and we construct examples of indecomposable Ulrich bundles. Divendres 30 de novembre, 15h, Aula T2, FMI-UB Enrico Carlini Politecnico di Torino, Itàlia Contact: alessandro.oneto at upc.edu Hilbert function of double points Hilbert functions of zero dimensional schemes, reduced or not, play a crucial role in many areas of mathematics: from Waring ranks of forms to identifiability of tensors. However, while we have a very good understanding of the reduced case, we know very little in the not reduced case. In this talk we will explore the situation with a special focus to double points in the plane. Divendres 30 de novembre, 16h, Aula T2, FMI-UB FACARD 2019 Workshop 16 a 18 de gener, IMUB Laura Brustenga UAB Relative clusters for smooth families In the talk, we will discuss a generalisation of clusters of points to the relative setting. When the family is smooth, we are able to show that relative clusters form a representable functor. We will recall the construction of Kleiman's iterated blow-ups, which are the representing schemes for the absolute case. Thereafter we will focus on and work out an explicit example of length two relative clusters. The example is geometric and interesting in its own, but hopefully, it will also share some insight about the general situation. Divendres 1 de febrer, 15h, Aula T2, FMI-UB Elba Garcia-Failde IPHT (CNRS) París-Saclay, França Contact: carles.casacuberta at ub.edu Simple maps, topological recursion and a new ELSV formula In this talk, we call ordinary maps a certain type of graphs embedded on surfaces, in contrast to fully simple maps, which we introduce as maps with non-intersecting disjoint boundaries. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). We propose a combinatorial interpretation of the important and still mysterious symplectic transformation which exchanges x and y in the initial data of the TR (the spectral curve). We give elegant formulas for the disk and cylinder topologies which recover relations already known in the context of free probability. For genus zero we provide an enumerative geometric interpretation of the so-called higher order free cumulants, which suggests the possibility of a general theory of approximate higher order free cumulants taking into account the higher genus amplitudes. We also give a universal relation between fully simple and ordinary maps through double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics. In particular, we obtain an ELSV-like formula for double strictly monotone Hurwitz numbers with ramification profile (2,...,2) over 0 and arbitrary one over ∞. Divendres 8 de febrer, 15h, Aula T2, FMI-UB Marco Gualtieri University of Toronto, Canadà Contact: eva.miranda at upc.edu The potential of generalized Kahler geometry Since the introduction of generalized Kähler geometry in 1984 by Gates, Hull, and Roček in the context of two-dimensional supersymmetric sigma models, we have lacked a general understanding of the degrees of freedom inherent in the geometry. In particular, the description of a usual Kähler structure in terms of a complex manifold together with a local Kähler potential function is not available for generalized Kähler structures, despite many positive indications in the literature over the last decade. I will explain how holomorphic Poisson geometry may be used to solve this problem and to obtain new constructions of generalized Kähler metrics. Divendres 15 de febrer, 15h, Aula T2, FMI-UB Paula Escorcielo Universidad de Buenos Aires Argentina Contact: cdandrea at ub.edu A version of Putinar's Positivstellensatz for cylinders Let f be a polynomial in n variables with real coefficients. Assume f is positive (non-negative) in a basic closed semialgebraic set S, a certificate of the positivity (non-negativity) of f in S is an expression that makes evident this fact. For example, Hilbert's 17-th problem states that if a polynomial is non-negative in Rn, it can be written as a sum of squares of rational functions, which is a certificate of the non-negativity of f in Rn. It is well-known that Krivine's Positivstellensatz (which states necessary and sufficient conditions for a polynomial system of equations and inequations to have no solution in Rn) implies Hilbert's 17-th problem. There are also other versions of Positivstellensatz, which hold on particular situations. For instance, Putinar's Positivstellensatz states that given g1, ..., gs polynomials in n variables with real coefficients such that the quadratic module M(g1, ..., gs) generated by g1, ..., gs is archimedean, every polynomial f which is positive on the basic closed semialgebraic subset S of Rn where g1, ..., gs are non-negative, belongs to M(g1, ..., gs). The archimedeanity assumption on M(g1, ..., gs) implies that the set S is compact. In this talk, we will present a version of Putinar's Positivstellensatz in the case that the underlying basic closed semialgebraic set is not compact but a cylinder of type SxR. This is a joint work with Daniel Perrucci. Dilluns 18 de febrer, 15h, aula T2, FMI-UB Thomas Strobl Université Claude Bernard, Institut Camille Jordan, Lyon 1, França Contact: eva.miranda at upc.edu The universal Lie ∞-algebroid of a singular foliation We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated 𝒪-submodule of vector fields on the underlying manifold closed under Lie bracket, where 𝒪 is the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie ∞-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie ∞-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie ∞-algebroid of the singular foliation. For real analytic or holomorphic singular foliations, it can be chosen, locally, to be a Lie n-algebroid for some finite n. If time permits we mention how to apply this construction to the realm of geometrical invariants and/or the construction of gauge theories. This is joint work with Camille Laurent-Gengoux and Sylvain Lavau. Divendres 22 de febrer, 15h, Aula T2, FMI-UB Yairon Cid Ruiz UB Saturated special fiber ring and rational maps The idea of studying rational maps by looking at the syzygies of the base ideal is a relatively new idea that has now become an important research topic. In this talk, we will discuss some recent results that lead to birationality criteria and formulas for the degree of rational maps that depend on the algebraic properties of the syzygies of the base ideal. Mainly, we will introduce a new algebra called «saturated special fiber ring» and we will discuss its relations with the degree and birationality of rational maps between irreducible projective varieties. Time permitting, we will also discuss some results in the problem of specializing the coefficients of a rational map. This talk is based on joint works with Laurent Busé and Carlos D’Andrea and with Aron Simis. Divendres 1 de març, 15h, Aula T2, FMI-UB Ferran Dachs-Cadefau Martin-Luther-Universität Halle, Alemanya TBA Divendres 8 de març, 15h, Aula T2, FMI-UB Nick Vannieuwenhoven Katholieke Universiteit Leuven, Bèlgica Contact: marta.casanellas at upc.edu TBA Divendres 22 de març, 15h, Aula T2, FMI-UB

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