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Contacta amb els organitzadors:
Joaquim Roé
Martí Lahoz

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Seminari de Geometria Algebraica 2018/2019 imatge de diagramació
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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

ignasi.mundet at
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.
Martín Sombra
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

szarzuela at
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

sombra at
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

ignasi.mundet at
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

adefelipe at
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
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)

marti.lahoz at
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

cdandrea at
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

miro at
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

alessandro.oneto at
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
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

carles.casacuberta at
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

eva.miranda at
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

cdandrea at
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

eva.miranda at
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
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
Multiplicities of jumping points for mixed multiplier ideals In this talk I want to present a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. If time allows, we would present some results about which topological information can be deduced from the jumping walls.
This is a joint work with Maria Alberich-Carramiñana, Josep Àlvarez Montaner and Víctor González Alonso.
Divendres 15 de març, 15h, Aula T2, FMI-UB
Nick Vannieuwenhoven
Katholieke Universiteit Leuven, Bèlgica

alessandro.oneto at
Geometry of the tensor rank decomposition The tensor rank decomposition or CPD expresses a tensor as a minimum-length linear combination of elementary rank-1 tensors. It has found application in fields as diverse as algebraic statistics, psychometrics, chemometrics, signal processing and machine learning, mainly for data analysis purposes. In these applications, the theoretical model is oftentimes a low-rank CPD and the elementary rank-1 tensors are usually the quantity of interest. However, in practice, this mathematical model is corrupted by measurement or sampling errors. In this talk, we will investigate the sensitivity of the CPD using techniques from algebraic and differential geometry.
Divendres 22 de març, 15h, Aula T2, FMI-UB
Jarosław Buczyński
Uniwersytet Warszawski - IMPAN, Polònia

jroe at
Strassen's additivity of tensor rank for small three-way tensors For a tensor \(T\in A \otimes B \otimes C\) (for vector spaces \(A\), \(B\) and \(C\)) the tensor rank of \(T\) is the minimal number of simple tensors such that \(T\) is the sum of those simple tensors. In this talk we address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. We show that for some small tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in \(\mathbb{C}^k \otimes \mathbb{C}^3 \otimes \mathbb{C}^3\) for any \(k\), then the additivity also holds.
Based on a joint work with Elisa Postinghel and Filip Rupniewski.
Divendres 22 de març, 16h, Aula T2, FMI-UB
Patricio Almirón
Universidad Complutense de Madrid

maria.alberich at
On the quotient of Milnor and Tjurina number Two of the main invariants of plane curves singularities are Milnor number μ, of topological nature, and Tjurina number τ, of analytical nature. In 2017 A. Dimca and G.M. Greuel posed the following question:
Is it true that for any isolated plane curve singularity we have μ/τ<4/3?
In this talk I will present a partial answer to this question in the case of semi-quasihomogeneus singularities (joint work with G. Blanco) which constitute a new evidence to believe in a positive answer to Dimca and Greuel's question in the general case.
Divendres 29 de març, 15h, Aula T2, FMI-UB
Joan Carles Naranjo
Universitat de Barcelona
Hyperelliptic Jacobians and Isogenies We mainly consider abelian varieties isogenous to hyperelliptic Jacobians. In the first part of the talk we will prove that a very general hyperelliptic Jacobian of genus \(g\geq 4\) is not isogenous to a non-hyperelliptic Jacobian. As a consequence we will obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary is that the Jacobian of a very general \(d\)-gonal curve of genus \(g\geq 4\) is not isogenous to a different Jacobian.

In the second part we will consider a closed subvariety \(\mathcal Y \subset \mathcal A_g\) of the moduli space of principally polarized varieties of dimension \(g\geq 4\). We will show that if a very general element of \(\mathcal Y\) is dominated by the Jacobian of a curve \(C\) and \(\dim \mathcal Y\geq 2g\), then \(C\) is not hyperelliptic. In particular, if the general element in \(\mathcal Y\) is simple, its Kummer variety does not contain rational curves. Finally, if time permits, we will show that a closed subvariety \(\mathcal Y\subset \mathcal M_g\) of dimension \(2g-1\) such that the Jacobian of a very general element of \(\mathcal Y\) is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus.

These results have been obtained in collaboration with G.P. Pirola and can be found in arXiv:1705.10154v2 or
Divendres 5 d'abril, 15h, Aula T2, FMI-UB
Federico Caucci
Università di Roma 1 La Sapienza, Itàlia

marti.lahoz at
Derived invariants arising from the Albanese map Given a smooth complex projective variety, it is natural to ask which geometric information are preserved under derived equivalence. Namely, if two varieties have equivalent derived categories, what can we say about their geometry? We prove a general result in this direction: the derived invariance of the cohomology ranks of pushforward under the Albanese map of the canonical line bundle (twisted with elements of the Picard variety). In the case of maximal Albanese dimension this settles conjectures of Popa and Lombardi-Popa, including the derived invariance of the Hodge numbers \(h^{0,j}\).
This is a joint work with G. Pareschi.
Divendres 26 d'abril, 15h, Aula T2, FMI-UB
Marcin Dumnicki
Uniwersytet Jagielloński w Krakowie, Polònia

jroe at
Divendres 10 de maig, 15h, Aula T2, FMI-UB
Sascha Timme
Technische Universität Berlin, Alemanya

piotr.zwiernik at
Divendres 17 de maig, 15h, Aula T2, FMI-UB
Juan Margalef

eva.miranda at
Divendres 24 de maig, 15h, Aula T2, FMI-UB
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