Geometry and Topology

Higgs moduli spaces and O'Grady examples


Emilio Franco

Kiem proved that the moduli space of SL(2,C)-Higgs bundles over a smooth projective curve X admits a symplectic desingularization if and only if g(X) = 2. Based on work of Schedler and Bellamy, Tirelli reproduced this result to the case of GL(2, C)-Higgs bundles, proving as well that no symplectic desingularization of SL(n, C) and GL(n, C)-Higgs bundles exists besides these two cases where (n, g) = (2, 2).

Real analogues of Plücker formulas


José Basto-Gonçalves

Plücker formulas for complex plane algebraic curves, relate the genus, number of double points, cusps, inflections and bitangencies. These formulas are almost two hundred years old, but their real analogues for plane curves or surfaces in R4 are from the second half of the last century and later. I will briefly describe the theorem of Fabricius-Bjerre for plane curves and present some of the results and problems in generalising them for surfaces.

Mixed Hodge structures on symmetric products


Jaime Silva

In this talk, we will cover some results on the nature of mixed Hodge structures on the cohomology of complex quasi-projective varieties. Specifically, we will be interested in varieties whose cohomology behaves as an exterior algebra. For those, we will manage to deduce the mixed Hodge structure on their symmetric products. These results will allows us to obtain a general formula for the mixed Hodge structure of free abelian character varieties for some reductive groups. We will study this formula in some detail for G=Sp(n,C), a case handled by me and C.

On K^2 for stable surfaces


Giancarlo Urzúa

V. Alexeev proved in 1994 that the set S of self-intersections of the canonical class of stable surfaces satisfies the descending chain condition, this is, any monotone sequence is increasing. (This set S is a subset of the positive rational numbers.) In particular S has a minimum, and it may have accumulation points. I will discuss what is known about S, certain new theorems on accumulation points, and open questions. This is a joint work with José Ignacio Yáñez.

Non-existence of orthogonal complex structures on the round 6-sphere


Ana Cristina Ferreira

In this talk, we will review the famous Hopf problem, which dates back to 1948, of whether there is a complex structure on the round 6-sphere. Although some attempted answers have been advanced, both in the negative and in the positive directions, this problem is still open. We will review some of the work and partial results that appear in the literature and take a closer look at almost complex structures compatible with the metrics of constant sectional curvature.

Quadratic complexes, singular varieties and moduli


Dan Avritzer

Let G be the Grassmannian of lines in P3 embedded in P5 as the Plücker quadric Q. The intersection of Q with a second hypersurface of degree d is what is called a complex of lines of degree d. When we consider the intersection of Q with a second quadratic hypersurface in P5, P, we have a quadratic complex. Let X = Q ∩ P be a quadratic complex that, in this talk, we assume to be non-singular, meaning X is non-singular.

EPRL/FK Asymptotics and the Flatness Problem: a concrete example


José Ricardo Oliveira

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics - the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

Topological complexity of the Klein bottle


Lucile Vandembroucq

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2.

Mirror symmetry for Nahm branes


Emilo Franco

Using the Dirac–Higgs bundle, we consider a new class of space-filling (BBB)-branes on moduli spaces of Higgs bundles, given by a generalized Nahm transform of a stable Higgs bundle. We then use the Fourier–Mukai–Nahm transform to describe its dual brane, which is checked to be a (BAA)-brane supported on a complex Lagrangian multisection of the Hitchin fibration.


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