Resúmenes
Alberto Verjovsky Solá
"Some geometry, arithmetic, and dynamics on arithmetic quaternionic manifolds."
We will present some results about modular orbifolds and Hilbert-Blumental varieties over the quaternions.
This is joint work with Helena Lizarraga Collí, Otto Romero, and Adrián Zenteno
Dennis Sullivan
"Building general three manifolds with 3D projective constructions."
TBA
Ernesto Lupercio
"Singularity theory, zeta functions and representation theory."
In this talk I will tour the intersection of the subjects in the title for some interesting conclusions. Joint work with Enrique Becerra and Ludmil Katzarkov.
John Parker
"New non-arithmetic complex hyperbolic lattices."
I will give an overview of a joint project with Martin Deraux and Julien Paupert, where we constructed the first new examples of non-arithmetic complex hyperbolic lattices since the work of Deligne and Mostow in the 1980s. I will give all the necessary background, including a definition of (non-)arithmeticity of lattices.
Laurent Meersseman
"The geography of Teichmüller stacks."
Given a smooth compact manifold M, the set of complex structures on M modulo isomorphisms isotopic to the identity can be turned into an analytic stack, the so-called Teichmüller stack, under a mild hypothesis. This being done, it is important to understand where and why there exist some bad points, that is points that prevent the Teichmüller stack from being a manifold/analytic space, and to analyse their geography and the local models around them. In this talk, we will discuss these questions and associate to any Teichmüller stack a simpler object - an orbifold in the case of Kähler complex structures - that captures the geometry of the locus of good points.
Mathias Zach
"Singularities in the search for automorphisms of Enriques surfaces."
Oguiso and Yu proved the existence of an Enriques surface $X$ with a specific automorphism $f \colon X \to X$ with maximal entropy. The universal covering of such a surface is a $2:1$ mapping $Y \to X$ from K3 surface $Y$. By virtue of the Torelli theorems, the automorphisms of the latter allow for a purely combinatorial description via the induced action on their integer lattices. This combinatorial fingerprint of the sought for automorphism $f$ is known. But as the Torelli theorems are highly non-constructive, it yet remains to construct an explicit K3 surface $X$, its Enriques involution, and the automorphism $f$. In this talk I will explain how we use singularity theory, intersection theory, and computer algebra systems to take on this task. This is a work in progress with Simon Brandhorst.
Thais Maria Dalbelo
"Local Euler obstruction of Isolated Determinantal Singularities."
An important invariant studied in singularity theory is the local Euler obstruction, which was defined by
MacPherson in [3] as one of the main ingredients of its proof for the Deligne and Grothendieck’s conjecture.
This conjecture concerns the existence and uniqueness of the Chern classes in the singular case.
Although it is very important, the local Euler obstruction is not easily computed using its definition,
which motivated many authors to find formulas to facilitate its computation. In [1], Brasselet, Lˆe and Seade
provided a Lefschetz type formula, therefore a topological formula, for the local Euler obstruction.
Using the formula presented by Brasselet, Lˆe and Seade and Newton polyhedra we compute the local Euler obstruction of isolated determinantal singularities.
Joint work with Anne Fruhbis-Kruger, Luiz Hartmann and Maicom Varella
Universidade Federal de Sao Carlos
thaisdalbelo@ufscar.br
References
[1] J.-P. Brasselet, D. T. Lˆe, and J. Seade. Euler obstruction and indices of vector fields. Topology, 39(6):1193–1208, 2000.
[2] R. D. MacPherson. Chern classes for singular algebraic varieties. Ann. of Math. (2), 100:423– 432, 1974.
Yenni Cherick
"Lipschitz Geometry of Complex Surface Germs."
It has been known since the work of Tadeusz Mostowski in 1985 that the set of complex surface germs up to bilipschitz equivalence is countable. By leveraging the work of Walter Neumann and Anne Pichon on the bilipschitz classification of complex surface germs, we describe how to construct an infinity of complex surface germs that have the same topological type and whose normalization to a fixed analytic type but are not pairwise bilipschitz equivalent.
Yuri Tschinkel
"New invariants in equivariant geometry."
The classification of actions of finite groups on algebraic varieties has a long history. I will discuss new invariants in equivariant birational geometry, introduced in joint work with A. Kresch, as well as their applications to concrete problems in small dimensions.
Waldemar Barrera
"Counting the connected components of the discontinuity region in certain families of Complex Kleinian Groups."
Given Γ a finitely generated, discrete, and non-elementary subgroup of PSL(2, C) whose region of discontinuity is Ω, it is well known that Ω can have one, two, or infinitely many connected components. When attem- pting to make an analogous version of this result for discrete subgroups of PSL(3, C), we face the difficulty that many classical techniques cannot be applied; furthermore, it is challenging to exhibit interesting examples of discrete groups of PSL(3,C) that do not come from the geometry of complex hyperbolic space H2 . The aim of this conference is to provide an C overview of the work that has been done to study the counting of compo- nents for discrete subgroups of PSL(3, C), and we will show that there are examples with one, two, three, and four components, as well as a poten- tial candidate to be an example of a discrete subgroup of PSL(3, C) with infinitely many components.
Patrick Popescu
"Combinatorics of real analytic morsifications."
I will present a work done in collaboration with A. Bodin, E. García Barroso and M.S. Sorea. We study a large class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the associated Morse functions in terms of a contact tree built from the real Newton-Puiseux roots of the polar curve of the morsification.
Michelle Morgado
"Integrated Equivariant Milnor Number."
This work embarks on an exploration within the realm of equivariant theory, delving into their well-established utility as demonstrated by seminal works of Ohmoto and Roberts. Drawing inspiration from the theory of singular characteristic classes, we establish interconnections between equivariant characteristic classes and a novel variation of equivariant Milnor number. Our principal objective is to establish a category of equivariant classes reminiscent of Milnor and Fulton-Johnson classes of singular hypersurfaces, while also deriving consequential outcomes pertaining to these classes.
In this talk, we introduce definitions for the integrated equivariant Milnor number $\mu_I^G$ and the equivariant Milnor class $\mathcal{M}_G(Z)$, applicable to singular hypersurfaces. Noteworthy outcomes encompass the constancy of $\mu_I^G$ across strata in a Whitney stratification of $Z$, along with the correlation $\mathcal{M}_G(Z)=\mathcal{M}_{G,0}(Z)=\displaystyle\frac{1}{|G|}\sum_{i=1}^k \mu_I^G(x_i)$ applicable to hypersurfaces hosting isolated singularities $x_1,\cdots, x_k$, where $\mathcal{M}_{G,0}(Z)$ denotes the $0$-th equivariant Milnor class of $Z$.
Furthermore, we introduce the equivariant Fulton-Johnson class of singular hypersurfaces, suggesting equivariant adaptations of specialization homomorphisms in constructible functions and homology. An equivariant rendition of the Verdier specialization property emerges, resulting in a nexus between equivariant Fulton-Johnson and Schwartz-MacPherson classes.
Joint work with Amanda Monteiro and Nivaldo de Goes Grulha Junior.
Laura Ortiz
"Local models in the understanding of analytic invariants of foliations by curves."
In the talk we will address the problem of the geometrical understanding of the analytical classification of foliations in neighborhoods of the origin in the complex plane. For this sake, we will have to consider pairs of differential equations. Namely, in the domains to consider, there will be two foliations of different types in order to give coordinate systems in certain regions. In the places where such foliations are tangent, such a coordinate system cannot be defined in the same way, however the places of tangencies (the curves of tangencies) will provide fundamental information for the required classification.
In the talk we will focus on this question using simple local analytical representatives (models) and normalizing transformations.
The talk is based on a work in collaboration with Jessica Jaurez-Rosas and Sergei Voronin.
Jorge Vitorio Pereira
"On the reduction of foliations to positive characteristic."
Foliations on projective manifolds defined over fields of positive characteristic exhibit rich and intriguing geometric features without a parallel in characteristic zero. I will explain how to explore this geometry to obtain results about the irreducible components of the space of codimension one holomorphic foliations. Joint work with Wodson Mendson.
Lev Birbrair
"Lipschitz geometry of semialgebraic surface germs."
TBA
Richard Canary
"Patterson-Sullivan theory and relatively Amosov groups."
It is classical that if X is a closed hyperbolic surface, then its geodesic flow is ergodic and one may obtain asymptotic counting results for the number of closed geodesics on X less than a given number. Patterson and Sullivan developed the theory of Patterson-Sullivan measures which allow generalizations of these results to infinite volume hyperbolic manifolds. We develop a similar theory in the presence of a convergence group action and an associated cocycle. We give applications of this approach to establish counting results for relatively Anosov groups. Joint work with Pierre-Louis Blayac, Feng Zhu and Andrew Zim
Daniel Duarte
"Higher Jacobian matrix of weighted homogeneous polynomials and derivation algebras."
In this talk we discuss some homogeneity properties of the higher Jacobian matrix of weighted homogeneous polynomials. As an application, we present a partial answer to a recent conjecture by Hussain-Ma-Yau-Zuo about the non-existence of negative weight derivations on the higher Nash blowup local algebra. This is a joint work with Wágner Badilla-Céspedes and Abel Castorena.
Aurelio Menegon
"Milnor fibration and deformations of maps on analytic and subanalytic sets."
We discuss Milnor fibration and Thom property for analytic maps defined on either a subanalytic set or a complex analytic space. We also discuss deformations of such maps.
Bernard Teissier
"Reimbedding singular germs: what is the use?."
I shall present examples where features of a singular germ which are significant for equisingularity of for resolution are encoded in the sections by coordinate hyperplanes in some very special reimbeddings.
Juanjo Nuño Ballesteros
"On the topology of complex map-germs and a general Lê-Greuel formula."
The Lê-Greuel formula is a classical result that give us an iterative method to compute the Milnor number of an isolated complete intersection singularity (ICIS). We will discuss about some extensions of this formula to the general case where we consider a pair of functions $f,g$ on a complex analytic variety $X$. The idea is to give a formula for the difference between the Euler characteristics of the general fibre of $f$ and the general fibre of the pair $(f,g)$ (provided it admits a fibration). This is a joint work in progress with Lê and Seade.
Agustin Romano
"Reflexive modules on quotient surface singularities."
Let (X, x) be a normal surface singularity and denote by L its link. The first complete classification of the finite dimensional representations of the fundamental group of L was done by McKay in the case of rational double point singularities. Later, Artin and Verdier, reformulate the McKay correspondence in a more geometrical setting. Their correspondence gives a complete classification of the indecomposable reflexive modules. In the case of quotient surface singularities, Esnault classified all the reflexive modules of rank one. Moreover, Esnault proved that quotient surface singularities are the only surface singularities with a finite number of indecomposable reflexive modules, such singularities are called Cohen–Macaulay finite representation type.
In this talk, we classify all the reflexive modules on quotient surface singularities. For this, we will use the Atiyah-Patodi-Singer theorem and the theory of secondary characteristic classes to construct our classification. As a consequence, the classification problem of reflexive modules over surface singularities of Cohen–Macaulay finite representation type is completely finished.
Joint work with
José Antonio Arciniega-Nevárez and José Luis Cisneros-Molina.
Ángel Cano
"Representation Variety of knots."
In this talk we will recover classical result on knot theory using quandle theory.
Hellen Santana
"The geometrical information encoded by the Euler obstruction map."
In this work we investigate the topological information captured by the Euler obstruction of a germ map $f(X,0)\to (\mathbb{C}^2,0)$ where $(X,0)$ denotes a germ of a complex $d$-equicimensional singular space, here $d>2$, and its relation with the local Euler obstruction of the cordinate function and, consequently, with Brasselet number. Moreover, under some technical conditions on the domain we relate the Chern number of a special collection to the map germ $f$ at the origin with the number of cusps of a generic perturbation of $f$ at a stabilization.
Patricia Dominguez
"Kleinian groups and its relations with holomorphic dynamics."
We will give some results related to the residual set in holomorphic dynamics and int analogue.
Moira Chas
"There May Be More Klein Bottles and Möbius Bands Than Are Dreamt of in Your Mind."
The Klein bottle and the Möbius band are among the most famous topological objects, yet they still have the power to make us ponder. We will discuss their history and reveal some of their singularities, many of which are not well known, even among seasoned mathematicians. If time permits, we will also explore sides of the projective plane.
Jean Paul Brasselet
"Poincaré-Hopf from the origins to the present day."
The Poincaré-Hopf theorem is an example of a result which over the years does not lose its relevance and gains in extension and new versions. Lately, the history of its first ingredient, the Euler-Poincaré characteristic, has yet again had new discoveries. Since its extension to singular varieties, by Marie-Hélène Schwartz, new results have appeared, both historical and purely mathematical. The presentation will present this evolution, from the genesis of the notions to today.
Irma Pallares
"Rational homology manifolds and perverse exact cubical hyperresolutions."
Rational homology manifolds are classical objects of study in topology, resembling manifolds from the perspective of homology. In this talk, we will introduce a characterization of rational homology manifolds in terms of cubical hyperresolutions and discuss various applications of this approach. This is a joint work with J. Fernández de Bobadilla.