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Seminari de Geometria Algebraica 2023/2024 |
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Conferenciant
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Títol |
Data i hora |
Josep Àlvarez Montaner
UPC
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Differential operators over rings of invariants of finite groups
In this talk we will see that some facets of the theory of D-modules over polynomial rings can be extended to the case of rings of invariants of finite groups. A blend of different techniques allow us to define a notion of holonomicity in this setting, we can develop a theory of Bernstein-Sato polynomials, V-filtrations, Hodge ideals and we can study the de Rham cohomology of holonomic modules.
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Divendres 6 d'octubre, 15h10, Aula B1, FMI-UB.
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Souvik Goswami
Universitat de Barcelona
Contact:
ispelta at crm.cat
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Periods of mixed Hodge structures associated to algebraic cycles
Given a pair of algebraic cycles whose cohomology classes are zero, and are in complimentary codimensions in an appropriate sense, R.Hain, S.Bloch,
et.al., in the 1990s established a mixed Hodge structure associated to the pair. A period of this mixed Hodge structure is a real number, which is
called the 'height' pairing. In this talk I will explain the construction by Hain, and also motivate a similar construction attached to a pair of Bloch’s
higher algebraic cycles. This is a joint work in progress with Dr. Greg Pearlstein and Dr. José Ignacio Burgos Gil.
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Divendres 20 d'octubre, 15h10, Aula B1, FMI-UB.
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J. Jelisiejew
M. Lahoz.
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Jornada de Jóvenes Doctores en Geometría Algebraica II
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30 i 31 d'octubre, Aula B1, FMI-UB
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C. Voisin.
A-M. Castravet
Z. Patakfalvi
E. Elduque
V. González-Alonso
R. Pardini
J.C. Naranjo
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Barcelona Mathematical Days
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1 i 2 de novembre, IEC
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Fabio Gironella
Université de Nantes
Contact:
robert.cardona@ub.edu
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Vanishing cycles for symplectic foliations
The main objects of the talk will be symplectic foliations,
and more precisely a subclass of these called "strong". Strong
symplectic foliations are meant to be one of the possible rigid
generalizations of taut foliations to high dimensions, and indeed have
quite a rigid nature, with techniques such as pseudo-holomorphic curves
à la Gromov and asymptotically holomorphic sequences of sections à la
Donaldson working well in this setting. I will present a joint work (in
progress) with Klaus Niederkrüger and Lauran Toussaint that aims at
giving a new obstruction for a symplectic foliation to be strong, that
comes in the form of a symplectic high-dimensional version of vanishing
cycles for smooth codimension 1 foliations on 3-manifolds. The proof
relies on pseudo-holomorphic curve techniques, in a way which is
parallel to the case of Plastikstufe introduced by Niederkrüger '06 in
the contact case. Time permitting, I will also talk about a new
construction of symplectic foliations, and give an example of a
symplectic foliation which is not strong due to the presence of a
symplectic vanishing cycle (and to which other previously known
obstructions to strongness don't apply).
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Dilluns 6 de Novembre, 15h10, Aula T2, FMI-UB.
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V. Benedetti.
J. Roé
M. Bolognesi
N. Bouchareb
T. Dedieu
G.P. Pirola
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Séminaire Méditerranéen de Géométrie Algébrique
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16 i 17 de novembre, FMI-UB
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Federico Caucci
Università di Ferrara
Contact:
marti.lahoz at ub.edu
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On syzygies of abelian and Kummer varieties
Equations defining projective varieties have been classically studied by several authors.
In this talk, I will give an overview on some (recent) results about syzygies of projective varieties, especially focusing on the case of abelian and Kummer varieties.
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Divendres 24 de novembre, 15h10, Aula B1, FMI-UB.
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Andrés Fernández Herrero
Columbia University
Contact:
ignasi.mundet at ub.edu
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Curve counting on the classifying stack BGL_n
In this talk I will describe a version of stable maps into a quotient stack [Z/GLN], where Z is a projective variety with an action of the general linear group GLN. If time allows, I will also update on the ongoing piece of the story with marked points, which involves some surprises such as the inclusion of a notion of "orientation" for the markings in order to compactify the evaluation morphisms and recover reasonable gluing morphisms. This talk is based on joint work with Daniel Halpern-Leistner.
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Divendres 15 de desembre, 15h10, Aula B1, FMI-UB.
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Guillermo Sánchez Arellano
Universidad Complutense de Madrid
Contact:
robert.cardona@ub.edu
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Local h-principles for Holomorphic Partial Differential Relations
Whenever we face a geometric problem involving differential operators such as a partial differential equation or inequality of some order r we can try first to solve it just formally, i.e. try to find a section of the correspondent r-jet bundle that satisfies the prescribed conditions. If we can find a homotopy between a formal solution to an actual solution we say that the problem satisfies an h-principle. H-principles in the smooth category are satisfied under some well-known general conditions. In this talk we will see a way to translate these general conditions to the holomorphic category in a neighbourhood of a totally real submanifold of a complex manifold, or more generally in a neighbourhood of the Lagrangian skeleton of any Stein manifold.
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Divendres 19 de gener, 12h10, Aula T2, FMI-UB.
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Robert Cardona
Universitat de Barcelona
Contact:
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Stability of hypersurfaces in symplectic four-manifolds
On a symplectic manifold, a generalization of contact hypersurfaces are "stable" hypersurfaces, introduced by Hofer and Zehnder in 1994. A hypersurface is called stable if it admits a distinguished symplectic tubular neighborhood foliated by hypersurfaces with conjugate characteristic foliations. This neighborhood makes them suitable as boundaries of symplectic cobordisms in symplectic field theory, and convenient for a study of their induced dynamics from a Hamiltonian point of view. In this talk, we will introduce these objects, motivate their study, and give a complete answer to the following natural question, commonly attributed to G. Paternain: is stability an open or a generic condition for embedded hypersurfaces in symplectic four-manifolds?
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Divendres 26 de enero, 15h10, Aula T2, FMI-UB.
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Martin Sombra
ICREA - UB
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Equidistribution of small points in projective varieties
Let $X$ be a projective variety over a number field equipped with a metrized line bundle $\overline{L}$. A generic sequence of algebraic points of $X$ is small if their heights with respect to $\overline{L}$ converge to the smallest possible value, namely the essential minimum of the height function. Yuan's equidistribution theorem (2008) describes the asymptotic distribution of the Galois orbits of the points in a small generic sequence, under the assumption that the essential minimum coincides with the normalized height of $X$. This hypothesis holds in important cases such as dynamical heights on projective varieties, but it fails for most choices of $(X,\overline{L})$. In this talk I will present a generalization of this theorem, extending to the general projective setting a result by Burgos, Philippon, Rivera Letelier and the speaker for the toric case. In particular, it applies to the canonical height on a semiabelian variety, and thus permits to recover Kühne's equidistribution theorem (2019).
Joint work with François Ballaÿ (Caen).
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Divendres 9 de febrer, 15h10, Aula T2, FMI-UB.
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Carolina Tamborini
Universität Duisburg-Essen
Contact: ispelta@crm.cat
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Moduli spaces of curves: tautological rings and non-tautological double cover cycles
While the full cohomology ring H*(\bar{Mg,n}) of the moduli space of genus g, n-pointed stable curves is generally intractable to study, a distinguished subring RH*(\bar{Mg,n}), called tautological ring, stands out. This subring admits an explicit set of generators and is rich enough to contain most cohomology classes arising from natural algebraic cycles.
In the seminar, we aim to address the natural question of when the equality RH*=H* occurs.
After an introduction on moduli spaces of curves and their tautological rings, I will discuss joint work together with Arena, Canning, Clader, Haburcak, Li, and Mok on the construction of many new non-tautological algebraic cohomology classes arising from double cover-cycles, generalizing previous work of Graber-Pandharipande and van Zelm.
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Divendres 16 de febrer, 15h10, Aula T2, FMI-UB.
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Roberto Gualdi
UPC
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On the arithmetic Kähler package
The Kähler package, which includes the hard Lefschetz property and the Hodge–Riemann relations, plays a relevant role in several areas of mathematics, from differential and algebraic geometry to combinatorics. In the context of Arakelov geometry, an arithmetic version of these properties can be formulated for hermitian metrized line bundles; strongly related to them, the arithmetic analogue of Grothendieck’s standard conjectures has been proposed by Gillet and Soulé.
In this talk, based on a joint ongoing work with Paolo Dolce and Riccardo
Pengo, we will show how the validity of the arithmetic Kähler package is
linked to certain positivity conditions of the involved hermitian line bundle,
and we will mention some implications of these observations in the case of
projective spaces.
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Divendres 23 de febrer, 15h10, Aula T2, FMI-UB.
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Ciro Ciliberto
Università di Roma Tor Vergata
Contact: joaquim.roe@uab.cat
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A rationality criterion for varieties and applications to Fano threefolds
In 1938 U. Morin, improving on earlier results by G. Fano (1918),
stated a projective classification theorem for varieties of dimension
$n\geq 3$ whose general surface sections are rational. Although Morin's
result is correct, his proof is wrong. In the first part of this talk I
will explain how to fix Morin's argument by using ideas from Mori's
theory already exploited by F. Campana and H. Flenner to attack a quite
similar problem. This part is joint work with C. Fontanari. In the
second part of the talk I will make some application to rationality of
Fano threefolds.
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Divendres 15 de marzo, 15h10, Aula T2, FMI-UB.
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Rick Miranda
Colorado State University
Contact: joaquim.roe@uab.cat
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Moduli spaces for rational elliptic surfaces (of index 1 and 2)
Elliptic surfaces form an important class of surfaces both from the
theoretical perspective (appearing in the classification of surfaces)
and the practical perspective (they are fascinating to study,
individually and as a class, and are amenable to many particular
computations). Elliptic surfaces that are also rational are a special
sub-class. The first example is to take a general pencil of plane
cubics (with 9 base points) and blow up the base points to obtain an
elliptic fibration; these are so-called Jacobian surfaces, since they
have a section (the final exceptional curve of the sequence of blowups).
Moduli spaces for rational elliptic surfaces with a section were
constructed by the speaker, and further studied by Heckman and
Looijenga. In general, there may not be a section, but a similar
description is possible: all rational elliptic surfaces are obtained by
taking a pencil of curves of degree 3k with 9 base points, each of
multiplicity k. There will always be the k-fold cubic curve through the
9 points as a member, and the resulting blowup produces a rational
elliptic surface with a multiple fiber of multiplicity m (called the
index of the fibration). A. Zanardini has recently computed the GIT
stability of such pencils for m=2; in joint work with her we have
constructed a moduli space for them via toric constructions. I will try
to tell this story in this lecture.
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Divendres 15 de marzo, 16h10, Aula T2, FMI-UB.
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Alessio Caminata
Università di Genova
Contact: josep.alvarez@upc.edu
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F-signature Functions of Diagonal Hypersurfaces
We extend the theory of p-fractals introduced by Monsky and Teixeira to the setting of F-signature. We use this to prove that when p goes to infinity the F-signature function of diagonal hypersurfaces converges uniformly to a piecewise polynomial function. Moreover, we compute the F-signature of Fermat hypersurfaces. In particular, for the Fermat cubic in four variables we prove that the F-signature is strictly less than 1/8. This provides a negative answer to a question by Watanabe and Yoshida. This is a joint work with S. Schideler, K. Tucker, and F. Zerman.
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Divendres 5 de abril, 15h10, Aula T2, FMI-UB.
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Gregory Pearlstein
Università di Pisa
Contact: gossouvik@gmail.com
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Infinitesimal Torelli and rigidity results for a remarkable class of elliptic surfaces
I will discuss joint work with Chris Peters which extends rigidity results of Arakelov, Faltings and Peters to period maps arising from families of complex algebraic varieties which are non-necessarily proper or smooth. Inspired by recent work with P. Gallardo, L. Schaffler, Z. Zhang, I will discuss two classes of elliptic surfaces which can be presented as hypersurfaces in weighted projective spaces which have a unique canonical curve. In each case, we will show that infinitesimal Torelli fails for \(H^2\) of the compact surface, but is restored when one considers the period map for the complement of the canonical curve.
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Divendres 12 de abril, 15h10, Aula T2, FMI-UB.
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Janusz Gwoździewicz
Pedagogical University of Cracow
Contact: ana.belen.de.felipe@upc.edu
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Newton diagram of the discriminant
While studying local analytic mappings \((f, g) : (C^2, 0) \rightarrow (C^2, 0)\) two objects play an important rˆole.
These are the jacobian curve describing the set of critical values and the discriminant curve which describes
the set of critical values. During my talk I will present results about the Newton diagram and the Newton
polynomial of the discriminant curve.
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Divendres 19 de abril, 15h10, Aula T2, FMI-UB.
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Tong Zhang
East China Normal University (ECNU)
Contact: miguel.angel.barja@upc.edu
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Noether inequality for irregular threefolds of general type
The Noether inequality, named after M. Noether, describes the lower bound of the canonical volume for varieties of general type in terms of the geometric genus. For irregular varieties of general type, O. Debarre established in dimension two the optimal Noether inequality. In this talk, I will introduce an optimal Noether inequality for irregular threefolds of general type. This is a joint work with Y. Hu.
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Divendres 26 de abril, 15h10, Aula T2, FMI-UB.
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Leonid Ryvkin
Université Lyon 1
Contact: garmendia.alfonso@gmail.com
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Normal forms of singular foliations
The talk concerns itself with normal forms of singular foliations in the vicinity of singular leaves.
We will start by introducing the notion of singular foliation, a generalization of regular foliation inspired by Frobenius' theorem.
We will then proceed to discussing their normal forms using methods inspired by Poisson geometry.
Based on joint work with Camille Laurent-Gengoux.
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Divendres 10 de maig, 15h10, Aula T2, FMI-UB.
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Santiago Barbieri
UB
Contact: ispelta@crm.cat
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Semi-algebraic geometry and generic Hamiltonian stability
Many physical models (especially those used in Celestial Mechanics) can be described by smooth Hamiltonian dynamical systems whose Hamiltonian functions are the sum of an integrable part and of a small perturbation. In the 1970s, N. N. Nekhoroshev proved that if the integrable part verifies a local geometric transversality condition on its gradient - known as "steepness" - then its orbits remain stable under the effect of any sufficiently small perturbation over a very long time (exponential or polynomial in the inverse of the size of the perturbation, depending on the regularity of the functions at hand). In this seminar, by using tools of real-algebraic geometry, I will give a modern proof of a Theorem by Nekhoroshev (1973) stating that smooth functions are generically steep, both in measure and in topological sense. Namely, by combining an analytic version of Yomdin-Gromov Theorem on the reparametrization of semi-algebraic sets together with non-trivial estimates on the codimension of suitable algebraic varieties, I will show that - for a given high enough positive integer r - the r-jets of non-steep functions of class \(C^{2r-1}\) are contained in a semi-algebraic set S having positive codimension in the space of polynomials of degree r. Moreover, by applying suitable quantifier elimination algorithms to the "bad" semi-algebraic set S, I will state new explicit algebraic criteria ensuring that a given function is steep: this constitutes a very important result in view of applications, e.g. to celestial mechanics.
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Divendres 17 de maig, 15h10, Aula T2, FMI-UB.
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Luca Vai
Università di Pavia
Contact: ispelta@crm.cat
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Theta functions and projective structures
Given s compact Riemann surface \(X\), we consider the line, in the space of sections of \(2\Theta\) on \(J^0(X)\), orthogonal to all the sections that vanish at the origin. This line produces a natural meromorphic bidifferential on \(X\times X\) with s pole of order two on the diagonal. In this talk, I will describe this bidifferential and in particular show that it produces a projective structure on \(X\) which is different from the standard ones.
This is a joint work with I.Biswas. and A.Ghigi.
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Divendres 24 de maig, 15h10, Aula T2, FMI-UB.
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Alicia Dickenstein
Universidad de Buenos Aires
Contact: ispelta@crm.cat
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Sparse systems with high local multiplicity
Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots in the torus of the system is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, which is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. In joint work with Frédéric Bihan and Jens Forsgård, we give an upper bound that is always sharp for circuits and, under a generic technical hypothesis, it is considerably smaller for any codimension m. We also present a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.
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Divendres 31 de maig, 15h10, Aula T2, FMI-UB.
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Daniel Álvarez-Gavela
Massachusetts Institute of Technology
Contact: robert.cardona@ub.edu
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The Arborealization Program
Finite-type open Riemann surfaces can be understood combinatorially via the theory of ribbon graphs. The arborealization program aims to extend this theory to arbitrary dimensions, where the object of study consists of Stein manifolds (complex submanifolds of C^n) or more precisely their symplectic underpinning: Weinstein manifolds. The ultimate goal is to reduce the symplectic topology of (polarized) Weinstein manifolds to the differential topology of a certain class of combinatorial spaces, called (positive) arboreal spaces, up to certain combinatorial moves. Joint work in progress with Y. Eliashberg and D. Nadler.
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Dimercres 12 de juny, 12, Aula IMUB, FMI-UB.
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Eamon Quinlan-Gallego
University of Utah
Contact: josep.alvarez@upc.edu
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Towards Bernstein-Sato polynomials in mixed characteristic.
The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. In this talk I will survey some results on a characteristic-p analogue of this invariant, as well as recent progress towards a mixed characteristic version. Said recent progress is joint work with Thomas Bitoun.
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Divendres 14 de juny, 15h10, Aula T1, FMI-UB.
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Eduardo Fernández Fuertes
University of Georgia
Contact: robert.cardona@ub.edu
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Strongly overtwisted contact 3-manifolds and overtwisted disks
In 1989, Eliashberg established the celebrated dichotomy between tight and overtwisted contact structures in dimension 3. This dichotomy was extended to all dimensions by Borman, Eliashberg, and Murphy in 2015. While tight contact structures are geometrically rich, the overtwisted ones are flexible and can be classified by the underlying homotopical data. It is said that they are governed by an h-principle. For parametric families of overtwisted contact structures, this flexibility does not hold unless all of them are assumed to coincide with a specified germ near an embedded disk of the manifold, the so-called overtwisted disk. In fact, as shown independently by Chekanov and Vogel, this condition is essential in general. In this talk, I will introduce a subclass of overtwisted contact structures in dimension 3, called strongly overtwisted, for which the "fixed overtwisted disk" condition can be removed. That is, they satisfy a parametric h-principle without prescribing the germ of the contact structure over any subset of the manifold. In particular, two overtwisted disks in a strongly overtwisted contact 3-manifold are contact isotopic.
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Divendres 14 de juny, 16h10, Aula T1, FMI-UB.
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László Pyber
Rényi Institute of Mathematics de Budapest
Contact: ignasi.mundet@ub.edu
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Growth in linear groups
We prove a conjecture of Helfgott on the structure of sets of
bounded tripling in bounded rank, which states the following. Let A be a
finite symmetric subset of GL(n,F) for any field F such that \(\vert A3\vert \leq K \vert A\vert \).
Then there are subgroups \(H \leqslant Γ \leqslant \langle A \rangle\) such that \(A\) is covered by \(K^O_n(1)\)
cosets of \(Γ, Γ/H\) is nilpotent of step at most \(n−1\), and \(H\) is contained in
\(A^O_n(1)\). This theorem includes the Product Theorem for finite simple
groups of bounded rank as a special case. As an application of our
methods we also show that the diameter of sufficiently quasirandom
finite linear groups is poly-logarithmic. (This is work by Sean Eberhard,
Brendan Murphy, László Pyber, Endre Szabó.)
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Divendres 28 de juny, 15h10, Aula iA, FMI-UB.
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Eduard Casas-Alvero
UB
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Analytic classification of irreducible plane curve singularities: an approach to the many characteristic exponents case
We describe a rational procedure for coordinates in infinitesimal first neighbourhoods, extending the coordinates given by the coefficients of an equation in the case of a single characteristic exponent. As an application, we extend Zariski's 1st and 2nd elimination criteria.
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Dimecres 10 de julliol, 16, Aula T2, FMI-UB.
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Maria Alberich Carramiñana
UPC
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Stratifications of the Moduli Space of Equisingular Plane Branches
In this talk, we will consider two discrete analytic invariants of plane branches: the Bernstein-Sato polynomial and the semigroup of values of the Jacobian ideal. We will focus on the question of how the topology of the branch determines these invariants.
A comprehensive way to address this question is through the stratifications that these invariants define on the moduli space of equisingular (topologically equivalent) branches. We will present an overview of recent results in this direction and discuss some open problems.
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Dimecres 10 de julliol, 17.30, Aula T2, FMI-UB.
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Kei-ichi Watanabe
Nihon University
Contact: elias@ub.edu
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Theory of Integrally closed Ideals in 2-dimensional normal rings via resolution of singularities; almost Gorestein rings and Gorestein normal tangent cones
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Dimars 16 de julliol, 15h10, Aula , FMI-UB.
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Franco Giovenzana
Université Paris-Saclay
Contact: ispelta@crm.cat
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Projective duality of some generalized Kummer fourfolds
Motivated by finding locally complete projective families of generalized Kummer fourfolds, we study projective duality for some projective models of generalized Kummer fourfolds. This is based on joint work in progress with D. Agostini, P. Beri, and A. D. Rios-Ortiz.
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Dimercres 24 de julliol, 15h10, Aula T2, FMI-UB.
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