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Seminari de Geometria Algebraica 2024/2025 |
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Conferenciant
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Títol |
Data i hora |
Robert Auffarth
Universidad de Chile
Contact:
jcnaranjo@ub.edu
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Counting principal polarizations on abelian varieties with automorphisms
In this talk we will explore the problem of counting the number of principal polarizations (modulo the action of the automorphism group) on a given abelian variety. More specifically, given a positive-dimensional irreducible component of the singular locus of the moduli space of principally polarized abelian varieties, we will want to answer the question of how many principal polarizations a very general member can have. We will give some motivation to this problem, and show that the answer to the previous question is not always 1!
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Divendres 4 d'octubre, 15h10, Aula T2, FMI-UB.
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Enric Florit Zacarias
Universitat de Barcelona
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Abelian varieties that split modulo all but finitely many primes
Let k be a number field and let A be an abelian variety defined over k.
We say A splits if it is isogenous to a product of abelian varieties of smaller dimension. Otherwise, A is simple. When A is simple, it may well happen that A splits modulo some prime p of k. We then have the following problem: given a simple abelian variety A, describe the set of primes p such that A mod p splits.
A standard fact says that the splitting behaviour of A is given by the semisimple decomposition of its endomorphism algebra. On the other hand, the endomorphism algebra of A modulo p was completely described by Honda and Tate by looking at the Frobenius endomorphism. Since End(A) is a subalgebra of End(A mod p), we can study the relation of one algebra to another.
In this talk we will explain the characterisation of subalgebras of division algebras. As a corollary we can show that, when End(A) is noncommutative, A splits modulo all but finitely many primes p of k.
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Divendres 18 d'octubre, 15h10, Aula T2, FMI-UB.
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Anatoli Shatsila
Jagiellonian University
Contact:
jcnaranjo@ub.edu
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Hyperelliptic genus 3 curves with involutions and Prym map
We will call a Galois covering of smooth curves Klein hyperelliptic if both curves are hyperelliptic and the Galois group is isomorphic to the Klein four-group. Borowka and Ortega (2023) proved that the Prym map for such coverings is injective. However, if one allows the bottom curve to have genus less than 2 then there is one additional non-trivial case, namely, covers f: C --> E with g(C) = 3 and g(E) = 1. I will show that the Prym map in this case is generically 2-to-1. I will also give a description of the moduli space of such covers by tuples of points on the projective line and use it to construct an involution on the moduli space induced by the Prym map. This is a joint work with Pawel Borowka.
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Divendres 25 d'octubre, 15h10, Aula T2, FMI-UB.
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Bernard Teissier
IMJ-PRG
Contact:
ana.belen.de.felipe@upc.edu
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A path to resolution of singularities through toric geometry
Resolution of singularities of a toric variety by toric birational maps is blind to the characteristic. I will explain a conjecture and some encouraging results about the use of this to resolve singularities
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Divendres 8 de novembre, 15h10, Aula T2, FMI-UB.
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Marc Masdeu
Universitat Autònoma de Barcelona
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Efficiently computing Theta functions for p-adic Schottky groups
Let K / Qp be a finite extension of the p-adics. A subgroup G
in GL2(K) is Schottky if it is finitely generated by hyperbolic
matrices. These groups are always discrete and free, and act on the
projective line minus a set of “bad points”. The quotient by this action
is known as a Mumford curve, and the p-adic theta functions allow for
the computation of their Jacobians, among other applications. In this
talk we will explain join work with Xavier Xarles, whereby we devise a
practical polynomial-time algorithm to compute these p-adic Theta functions.
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Divendres 15 de novembre, 15h10, Aula T2, FMI-UB.
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Marta Casanellas
UPC
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Obtaining equations for equivariant evolutionary models
Phylogenetics studies the evolutionary relationships among species using their molecular sequences. These relationships are represented on a phylogenetic tree or network. Modeling nucleotide or amino acid substitution along a phylogenetic tree is one of the most common approaches in phylogenetic reconstruction. One can use a general Markov model or one of its submodels given by certain substitution symmetries. If these symmetries are governed by the action of a permutation group G on the rows and columns of a transition matrix, we speak of G-equivariant models. A Markov process on a phylogenetic tree or network parametrizes a dense subset of an algebraic variety, the so-called phylogenetic variety.
During the last decade algebraic geometry has been used in phylogenetics for reconstructing phylogenetic trees and for establishing the identifiability of parameters of complex evolutionary models (and thus guaranteeing model consistency). Since G-equivariant models have fewer parameters than a general Markov model, their phylogenetic varieties are defined by more equations and these are usually hard to find. We will see that we can easily derive equations for G-equivariant models from the equations of a phylogenetic variety evolving under a general Markov model. As a consequence, we will discuss the identifiability of networks evolving under G-equivariant models.
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Divendres 22 de novembre, 15h10, Aula T2, FMI-UB.
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Martí Lahoz
Universitat de Barcelona
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Prym semicanonical pencils
In the moduli space of double étale covers of curves of genus g > 1, the
locus of covers of curves with a semicanonical pencil (a
theta-characteristic with two sections) is formed by two irreducible
divisors distinguished by the parity of the dimension of a certain space
of sections. I will explain the behavior of the Prym map on each of
them, which is significantly different, and has a rich geometry in the
low genus cases. At the end I will focus on the odd divisor and on
genera 5 and 6, which are especially interesting. This is joint work in
collaboration with Joan Carles Naranjo, Andrés Rojas and Irene Spelta.
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Divendres 29 de novembre, 15h10, Aula T2, FMI-UB.
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Joana Cirici
Universitat de Barcelona
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Configuration spaces of algebraic varieties
I will use the theory of weights in étale cohomology to give a simple and conceptual proof of a theorem of Kriz stating that a rational model for the ordered configuration space of a smooth complex projective variety is given by the second page of the Leray spectral sequence for the obvious inclusion, along with its only non-trivial differential. Our proof builds on Totaro's study of this spectral sequence, combined with a basic observation related to the formality of filtered dg-algebras. An advantage of this proof is that it allows for a generalization to study the p-adic homotopy type of configuration spaces for certain algebraic varieties defined over finite fields. This is joint work in progress with Geoffroy Horel.
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Divendres 20 de desembre, 15h10, Aula T2, FMI-UB.
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Ricardo Menares
Pontificia Universidad Católica de Chile
Contact:
sombra@ub.edu
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On CM values of modular functions that are S-units
Two classical modular functions are the j invariant (which classifies elliptic curves) and the lambda invariant (which classifies elliptic curves plus a basis of the 2-torsion). We are interested in their CM values, that is the values they take on the set of imaginary quadratic numbers. When the functions are correctly normalized, these CM values are algebraic numbers.
We show that for any finite set S of prime numbers, only finitely many CM values of the j invariant can be algebraic S-units. In stark contrast, it is well known that all of the CM values of the lambda invariant are 2-units. In this talk I will present results and conjectures towards the characterization of modular functions having only finitely many CM values that are S-units. This is joint work with Sebastián Herrero and Juan Rivera-Letelier.
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Divendres 17 de gener, 15h10, Aula iA, FMI-UB.
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Pietro Speziali
Universidade Estadual de Campinas
Contact:
jcnaranjo@ub.edu
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The Hurwitz curve: a case study
Due to its many fascinating properties, the celebrated Klein quartic, defined by the plane homogeneous equation
H3:X3Y+Y3Z+Z3X=0
is arguably one of the most studied objects in algebraic and arithmetic geometry. For instance, it is the only curve of genus 3 with 168 automorphisms, making it the smallest example of a curve attaining the Hurwitz bound. This result holds over any algebraically closed field of characteristic different from 3; in characteristic 3, however, the number of automorphisms increases to 6048. Moreover, the Klein quartic, with its abundance of automorphisms, admits non-trivial twists over non-algebraically closed fields, some of which may possess numerous rational points even when $\mathcal{H}_3$ itself does not. These aspects underscore the significance of studying this curve and motivate the exploration of its generalization, the so-called $n$-th Hurwitz curve, defined by the plane homogeneous equation
Hn:XnY+YnZ+ZnX=0.
For $n \geq 4, \mathcal{H}_n$ does not generally attain the Hurwitz bound (except in cases where it exceeds it, as occurs over an algebraically closed field K of characteristic p > 0 whenever $n = p^h$ for some $h \geq 1$). Nonetheless, $\mathcal{H}_n$ exhibits many intriguing geometric and arithmetic properties.
In this talk, we will provide a (mostly) comprehensive presentation of results concerning the Hurwitz curve and its invariants, including its automorphism group, $p$-rank, $a$-number, Weierstrass points, bounds on its number of rational points, as well as its Jacobian and Jacobian decomposition in both characteristic $0$ and $p > 0$. This will serve as a case study of problems, techniques, and results in the theory of algebraic curves.
The talk is based on several joint works with Nazar Arakelian, Herivelto Borges, and Maria Montanucci, as well as ongoing research.
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Divendres 24 de gener, 15h10, Aula T2, FMI-UB.
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David Senovilla Sanz
Universidad de Cantabria
Contact:
maria.alberich@upc.edu
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Saito bases for branches with one Puiseux pair
Let $C$ be a plane curve in $(\mathbb C^2,\mathbf 0)$. According to K. Saito, the $\mathbb C\{x,y\}$-module $\Omega^1[C]$ of holomorphic 1-formas with $C$ invariant is free of rank 2. We call Saito basis of $C$ to any basis of $\Omega^1[C]$. In this talk we explain how, by means of Saito bases, we can obtain a series of analytic invariants of $C$. In particular, we see how to generalize the family of invariants defined by Y. Genzmer. Moreover, when $C$ is an irreducible branch with a Puiseux pair, we explain the underlying combinatorics of an algorithm to compute a Saito basis.
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Divendres 31 de gener, 15h10, Aula T2, FMI-UB.
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A. Bayer
S. Canning
P. Frediani
V.G. Alonso
F. Gounelas
B. Klinger
M. Lelli-Chiesa
E. Macrì
R. Pardini
L. Pertusi
G.P. Pirola
T. Zhang
X. Zhao
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Birational Geometry: from moduli to geography
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5-7 de febrer, FMI-UB
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Dumitru Stamate
University of Bucharest
Contact:
marchesi@ub.edu
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Trace ideals and classes of Gorenstein related rings
The Gorenstein rings are an important part of the class of Cohen-Macaulay (CM) rings, as the former enjoy many beautiful properties.
For instance, a local CM ring R is Gorenstein if and only if the ring itself is a canonical module for R, equivalently, the trace of the canonical module of R is the whole ring R.
In this talk we discuss two classes of CM rings that have been introduced in recent years ( nearly Gorenstein, and far flung Gorenstein) in terms of the size of the trace of the canonical module of the ring R, assuming the latter exists. We discuss some basic properties, provide examples and some questions around these topics.
This is based on joint works with J. Herzog, T. Hibi, and with J. Herzog and S. Kumashiro.
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Divendres 14 de febrer, 15h10, Aula T2, FMI-UB.
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Pip Goodman
Universitat de Barcelona
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Mod-ℓ variants of Faltings' isogeny theorem
For an abelian variety A defined over a number field K, Faltings' isogeny theorem tells us that End(A)⊗Zℓ is
determined by the action of Galois on the ℓ-adic Tate module. In general, the action of Galois on A[ℓ], the ℓ-torsion
of A, does not tell us anything about the endomorphisms of A. However, Zarhin has shown that in particular cases when the
image of Galois acting on A[ℓ] is a "large" non-soluble group, then one can deduce information on End(A). The advantage
of these results being that they can be easily applied to explicit examples.
In this talk we will see that one still obtains a lot of information about End(A) when the image of Galois on A[ℓ] merely
contains an element of large prime order. Furthermore, given favourable arithmetic conditions on K, we will see it is possible
to give a finite list of possibilities for End(A)⊗Q up to isomorphism.
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Divendres 21 de febrer, 15h10, Aula T2, FMI-UB.
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Ignasi Mundet
Universitat de Barcelona
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Finite automorphisms of compact Kaehler manifolds and deformations
Let X be a compact Kaehler manifold, and let Xi be a sequence
of compact Kaehler manifolds, each of which is deformation equivalent to X.
Suppose that there exists a sequence of prime numbers pi→∞ and,
for each i, an effective action of (Z/pi)k on Xi by biholomorphisms.
How is the existence of these actions reflected in the geometry of X?
We will see that, if k is the real dimension of X, then X is biholomorphic
to a complex torus. For smaller values of k, despite the fact that X can
fail to have any nontrivial holomorphic symmetry, we will see that (assuming
the actions are free) there are shadows of the actions on Xi in the geometry
of the Albanese map of X.
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Divendres 28 de febrer, 15h10, Aula T2, FMI-UB.
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Chenying Lin
Universität Regensburg
Contact:
roberto.gualdi@upc.edu
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Galois orbits of torsion points over polytopes near atoral sets
A celebrated equidistribution theorem states that the Galois orbit of a torsion point in an algebraic torus becomes equidistributed
with respect to the Haar measure of the compact torus as the strictness degree of the torsion points tends to infinity. In this talk,
we present a quantitative version of an extension of this theorem, where torsion points are considered over polytopes, and the
functions involved are of the form log∣P∣, with P being an essentially atoral polynomial—an important class of functions in height theory.
We will discuss an application of this result, providing an explicit rate of convergence of heights for a sequence of projective points
in a specific two-dimensional example. Additionally, we will introduce a classical approach to estimating the speed of convergence
in equidistribution theorems, namely Koksma's inequality, which is closely related to the discrepancy of points.
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Divendres 7 de març, 15h10, Aula T2, FMI-UB.
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Marta Mazzocco
ICREA and UPC
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Quantum character varieties and Calabi-Yau algebras
In this talk we will propose a quantization for del Pezzo surfaces belonging to a certain class. In particular
we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This
algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains
quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.
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Divendres 14 de març, 15h10, Aula T2, FMI-UB.
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Lidia Stoppino
Università di Pavia
Contact: miguel.angel.barja@upc.edu
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Fibered surfaces and their unitary rank
The geography of fibred surfaces is the study of the relations and of the possible range of variation of their relative invariants.
In particular given a fibred surface f:S→B, I will focus on the following invariants:
- the self-intersection of the relative canonical divisor Kf:=KS−f∗KB;
- the relative Euler characteristic χf
- the unitary rank uf, which is the rank of the unitary summand in the second Fujita decomposition of the Hodge bundle.
I will define these invariants and present the main known geographical results relating them.
Then I will prove some new lower bounds for the slope K2f/χf, depending increasingly on the unitary rank.
>From one of these inequalities we can derive a new Xiao-type bound on uf.
I will end by describing open problems and related questions.
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Divendres 21 de març, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 28 de març, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 4 d'abril, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 25 d'abril, 15h10, Aula T2, FMI-UB.
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Ilya Smirnov
BCAM
Contact:
cdandrea@ub.edu
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Divendres 9 de maig, 15h10, Aula T2, FMI-UB.
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Barbara Fantechi
SISSA
Contact:
miro@ub.edu
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Divendres 9 de maig, 16h10, Aula T2, FMI-UB.
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Florian Enescu
Georgia State University
Contact: cdandrea@ub.edu
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Divendres 16 de maig, 15h10, Aula T2, FMI-UB.
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Beatriz Pascual
Universidad Politécnica de Madrid
Contact: rhoms@crm.cat
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Divendres 23 de maig, 15h10, Aula T2, FMI-UB.
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De-Qi Zhang
National University of Singapore
Contact:
ignasi.mundet@ub.edu
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Divendres 30 de maig, 15h10, Aula T2, FMI-UB.
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Remy van Dobben de Bruyn
Universiteit Utrecht
Contact:
francesc.fite@gmail.com
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Divendres 6 de juni, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 13 de juni, 15h10, Aula T2, FMI-UB.
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Francesc Fité
Universitat de Barcelona
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Divendres 20 de juni, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 27 de juni, 15h10, Aula T2, FMI-UB.
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Divendres 4 de juliol, 15h10, Aula T2, FMI-UB.
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A. M. Botero
J. Giansiracusa
O. Lorscheid
M. Ulirsch
X. Xarles
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Geometry over semirings
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7-11 de juliol, CRM
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Contact:
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Divendres 18 de juliol, 15h10, Aula T2, FMI-UB.
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Divendres 25 de juliol, 15h10, Aula T2, FMI-UB.
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