Courses
Generalized Geometry, by Roberto Rubio
Low-Dimensional Topology, by Rafael Torres
Research talks
RT1) Rafael Torres (SISSA Trieste)
Knots, surgeries and smooth structures on R^4
There is a myriad of reasons why 4-manifolds are captivating objects. Two of them are as follows. 1. The study of 4-manifolds brings together several areas of research. 2. Dimension four hosts phenomena that does not occur in any other dimension. This talk will survey work of several mathematicians (e.g. Stallings, Donaldson, Freedman, DeMichelis, Gompf, Taubes) and describe a relation between classical knot theory and smooth structures on Euclidean 4-space.
RT2) Sebastián Camponovo (SISSA Trieste)
Exotic definite 4-manifolds with infinite fundamental group
We say that a 4-manifold X has an exotic smooth structure if X possesses more than one smooth structure, i.e., there exists a 4-manifold X′ that is homeomorphic but not diffeomorphic to X. Although examples of exotic smooth structures on closed 4-manifolds have been known since the 1980s, the definite case remained elusive for decades.
The first examples of exotic smooth structures on 4-manifolds with definite intersection form were produced by Levine-Lidman-Piccirillo in 2023. Their construction yielded manifolds with fundamental group Z/2. Subsequently, Baykur-Stipsicz-Szabó and Harris-Naylor-Park developed related techniques to obtain examples with various finite fundamental groups.
The aim of this talk is to discuss how a related approach can be used to obtain examples with new fundamental groups, starting from appropriate exotic building blocks, and construct infinitely many irreducible exotic smooth structures on some 4-manifolds with infinite fundamental groups.
The talk is based on joint work with Rafael Torres.
RT3) Matthieu Madera (UAB)
Complex foliations
Poisson structures elegantly describe symplectic foliations, with leaves of possibly different dimensions. Is there an analogue for complex foliations? The regular case is well understood via Levi-flat CR structures, and the holomorphic singular setting has a satisfying theory as well. However, a good definition in the smooth singular setting remains elusive. We discuss several natural candidates, examine why they prove difficult to handle in practice, and propose a definition that can be verified on a family of examples.
RT4) Jaime Pedregal (U Utrecht)
Generalized Riemannian Geometry
Apart from generalized complex structures, which encompass both complex and symplectic structures, one can also consider the notion of generalized Riemannian metrics. These are tightly related to pairs of a Riemannian metric and a 2-form. Going further, one can also consider generalized Levi-Civita connections for such generalized metrics, and these are related to Riemannian geometry “with torsion”. Contrary to classical Riemannian geometry, these generalized Levi-Civita connections are not unique, although in some cases there is a canonical choice. Depending on time, we will also touch upon the notion(s) of generalized curvature.
RT5) Tom Ariel (KU Leuven, UAB)
On the Dirac complement problem
When approaching the deformation theory of Dirac structures, the question of whether a given Dirac structure admits a Dirac complement — a complementary subbundle which is Dirac — pops up. This question is also related to the structure theory of Courant algebroids, since a Courant algebroid admitting a pair of complementary Dirac structures has the structure of the double of a Lie bialgebroid.
In my talk, I will present the Dirac complement problem, and propose geometric and algebraic obstructions to the existence of a Dirac complement for certain Dirac structures of interest.
RT6) Filip Moučka (CTU Prague, UAB)
Symmetry and skew-symmetry in generalized geometry
Generalized geometry is built around the canonical symmetric pairing on the direct sum of the tangent and cotangent bundles. However, there is also the equally natural skew-symmetric pairing. I will present the basic ideas of a theory analogous to generalized geometry obtained by replacing the symmetric pairing with the skew-symmetric one. I will describe connections with other areas of mathematics and hint at possible applications.
Lightning talks
LT1) Cristina Gómez Cirera
Lens space surgeries
The study and classification of 3-manifolds constitute one of the central areas of research in low-dimensional topology. One of the main tools in this field is Dehn surgery, which allows us to obtain any 3-manifold by performing an operation along a finite collection of disjoint knots in the 3-sphere (intuitively, a knot may be viewed as a piece of rope that we tie and then glue its endpoints together). Among the simplest 3-manifolds to construct in this way are lens spaces. A lens space can be defined as the 3-manifold that results from gluing together two solid tori along their boundaries by an orientation-reversing diffeomorphism. It is therefore natural to ask in which ways a lens space can be obtained via Dehn surgery on a knot.
In the 1990s, J. Berge introduced twelve families of knots in S^3 that yield lens spaces through Dehn surgery and conjectured that these are the only knots in S^3 admitting such surgeries. Although substantial progress has been made towards the resolution of this conjecture, it remains an open problem.
The aim of this talk is to introduce Berge’s construction and a generalisation to the setting in which the surgery is performed on a knot embedded in an arbitrary lens space.
LT2) Pablo Serrano
A new method for finding filling Dehn spheres that split knots on S^3
Filling Dehn surfaces are generically-immersed surfaces on a 3-manifold that induce a cellular decomposition on the manifold. This means that they can be used to study the topology of said manifold working only in two dimensions, using the information of the self-intersections of the surface, what is known as its Johansson diagram. Moreover, they can also be used in knot theory, as the knot can be fully reconstructed from its intersections with the surface, as long as they happen in non-singular points and the arcs left between them are unknotted: in that case, we say the surface splits the knot. In this talk, we will showcase a new combinatorial algorithm to find both a filling Dehn sphere that splits any knot in S^3 and its corresponding Johansson diagram, by following a series of simple steps on a given diagram of the knot.
LT3) Mariana Lisett Landín Munguía
Why Does the Homogeneous Dimension Appear in Singular Integrals?
In Euclidean harmonic analysis, many estimates depend on the dimension of the ambient space. On homogeneous groups, the relevant quantity is not the topological dimension but the homogeneous dimension (Q), which arises from the group’s dilation structure.
In this talk, I will explain the definition of homogeneous dimension and illustrate why it naturally appears in the study of singular integral kernels. Through an example, I will show how the geometry induced by dilations determines the order of singularities and the scaling properties of the associated operators.
The talk is based on material studied during my undergraduate thesis on singular integrals on homogeneous groups.
LT4) Florian Leander Aggias
Renormalizability of Yang-Mills-Dirac theory
The renormalizability of perturbative Yang–Mills theory coupled to Dirac spinors on Euclidean manifolds without boundary in the BV formalism is investigated, using the method of homotopic renormalization. The required mathematical background is recalled. A recollection of the BV formalism is given. Costello’s homotopic renormalization is introduced. The classical BV data of Yang–Mills theory coupled to Dirac spinors is derived, and the homological calculations necessary for the proof of renormalizability are demonstrated. The conclusion is that Yang–Mills–Dirac theory is renormalizable on R^4.
LT5) Luis Pizarro
Complex structures on nilmanifolds
A complex manifold is a topological manifold equipped with an holomorphic atlas or with a sheaf of holomorphic functions. It is known that every complex manifold of complex dimension n is a differential manifold of dimension 2n. However the other direction is false and an important problem in complex geometry is to determine when a 2n-dimensional manifold admit a complex structure. In this talk a special type of manifolds will be treated, those who are a quotient of a nilpotent Lie group and it is shown how this problem becomes an algebraic problem
LT6) Roberto Rubio
A gentle closure
I would like to look back at the last few days of the summer school, the last few months of preparation and the last (many) years of mathematical career to explain the why and how of GENTLE. Hopefully not a goodbye but a see you soon.