


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

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, FMIUB



