


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, FMIUB

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 RiveraLetelier (Rochester)

Divendres 28 de setembre, 15h, Aula T2, FMIUB

Alberto F. Boix
BenGurion U. of the Negev
BeerSheva, 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 complexanalytic category and in the differentiable
case. It means that the mapgerm 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, FMIUB

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 Dheight of a complete intersection in a toric variety
in terms of the upper functions of the defining Laurent polynomials.
In the onecodimensional case, we prove an exact formula relating the Dheight of a hypersurface to the Ronkin function of the associated Laurent
polynomial, generalizing the wellknown equality for the canonical case.
Our approach involves mixed integrals, LegendreFenchel duality and other
notions from convex geometry.

Divendres 19 d'octubre, 15h, Aula T2, FMIUB

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 4space. 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 hprinciple). 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. MartínezAguinaga.

Divendres 26 d'octubre, 15h, Aula T2, FMIUB

Pedro D. González Pérez
ICMAT  UCM, Madrid
Contact:
adefelipe at ub.edu

The valuative tree is a projective limit of EggersWall 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 NewtonPuiseux series of C relative to any coordinate system (x,y) on S such that L is the yaxis,
one may define the EggersWall tree Θ_{L}(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 NewtonPuiseux series, their denominators and contact orders.
The main objective of this paper is to embed canonically Θ_{L}(C) into Favre and
Jonsson's valuative tree of realvalued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural
functions on Θ_{L}(C) as pullbacks of other naturally defined functions on the valuative tree.
As a consequence, we prove an inversion theorem generalizing the wellknown AbhyankarZariski inversion theorem concerning one branch:
if L' is a second smooth branch of C, then the valuative embeddings of the EggersWall trees
Θ_{L'}(C) and Θ_{L}(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 EggersWall trees over all choices
of curves C.
Joint work with Evelia García Barroso and Patrick PopescuPampu.

Divendres 9 de novembre, 15h, Aula T2, FMIUB

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 nonintegrable setting.
I will also explain Liealgebra analogues of the theory which provide useful computational tools for compact Lie groups and nilmanifolds.

Divendres 16 de novembre, 15h, Aula T2, FMIUB

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 K3type using the machinery of wallcrossing 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 wallcrossing 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, FMIUB

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, FMIUB

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, FMIUB

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, FMIUB


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 blowups, 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, FMIUB

Elba GarciaFailde
IPHT (CNRS) ParísSaclay, 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 nonintersecting disjoint boundaries.
It is wellknown 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 socalled 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 ELSVlike 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, FMIUB

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 twodimensional 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, FMIUB

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 (nonnegative) in a basic closed semialgebraic set S, a certificate of the positivity (nonnegativity) of f in S is an expression that makes evident this fact.
For example, Hilbert's 17th problem states that if a polynomial is nonnegative in R^{n}, it can be written as a sum of squares of rational functions, which is a certificate of the nonnegativity of f in R^{n}.
It is wellknown that Krivine's Positivstellensatz (which states necessary and sufficient conditions for a polynomial system of equations and inequations to have no solution in R^{n}) implies Hilbert's 17th problem.
There are also other versions of Positivstellensatz, which hold on particular situations.
For instance, Putinar's Positivstellensatz states that given g_{1}, ..., g_{s} polynomials in n variables with real coefficients such that the quadratic module M(g_{1}, ..., g_{s}) generated by g_{1}, ..., g_{s} is archimedean, every polynomial f which is positive on the basic closed semialgebraic subset S of R^{n} where g_{1}, ..., g_{s} are nonnegative, belongs to M(g_{1}, ..., g_{s}). The archimedeanity assumption on M(g_{1}, ..., g_{s}) 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, FMIUB

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 𝒪submodule 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 subfoliations factorizes through it in an uptohomotopy 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 nalgebroid 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 LaurentGengoux and Sylvain Lavau.

Divendres 22 de febrer, 15h, Aula T2, FMIUB

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, FMIUB

Ferran DachsCadefau
MartinLutherUniversität Halle, Alemanya

TBA

Divendres 8 de març, 15h, Aula T2, FMIUB

Nick Vannieuwenhoven
Katholieke Universiteit Leuven, Bèlgica
Contact:
marta.casanellas at upc.edu

TBA

Divendres 22 de març, 15h, Aula T2, FMIUB



