Geometry and Topology

The modular class of a singular foliation

Speaker: 

Sylvain Lavau

Date: 

Monday, 11 June, 2018 - 10:00

Venue: 

Room 0.06

The modular class is a generalization of the divergence of vector fields to other geometric structures such as Poisson manifolds and Lie algebroids. The modular class of a regular foliation involves a volume form on the conormal bundle and the associated Bott connection. It is a closed one form along the leaves. The vanishing of this modular class implies that there exists a volume form which is invariant along the leaves.  In the singular case, the above definition can not be summoned since the conormal bundle may not be even well defined.

Contact topology of Gorenstein toric isolated singularities

Speaker: 

Miguel Abreu

Date: 

Friday, 8 June, 2018 - 15:30

Venue: 

Room 1.09

Links of Gorenstein toric isolated singularities are good toric contact 
manifolds with zero first Chern class. In this talk I will present some 
results relating contact and singularity invariants in this particular 
toric context. Namely, 
(i) I will explain why the contact mean Euler characteristic is equal 
to the Euler characteristic of any crepant toric smooth resolution of 
the singularity (joint work with Leonardo Macarini). 
(ii) I will discuss applications of contact invariants of Lens spaces 

Complex Lagrangian subvarieties of the Higgs bundle moduli space

Speaker: 

Lucas Branco

Date: 

Thursday, 10 May, 2018 - 10:00

Venue: 

Room 0.06

Given a complex reductive group G, the moduli space M(G) of G-Higgs bundles on a curve has a natural hyperkähler structure and it comes equipped with an algebraically completely integrable system through the Hitchin fibration. These moduli spaces have played an important role in mirror symmetry and in the geometric Langlands program and thus it has become of particular interest the study of certain decorated special subvarieties (branes) of M(G).

Higgs moduli spaces and O'Grady examples

Speaker: 

Emilio Franco

Date: 

Friday, 16 March, 2018 (All day)

Venue: 

Room 1.09

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

Speaker: 

José Basto-Gonçalves

Date: 

Friday, 9 March, 2018 (All day)

Venue: 

Room 1.09

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

Speaker: 

Jaime Silva

Date: 

Friday, 2 March, 2018 (All day)

Venue: 

Room 1.09

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

Speaker: 

Giancarlo Urzúa

Date: 

Friday, 19 January, 2018 (All day)

Venue: 

Room 1.09

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

Speaker: 

Ana Cristina Ferreira

Date: 

Friday, 17 November, 2017 (All day)

Venue: 

Room 1.09, 16h00

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

Speaker: 

Dan Avritzer

Date: 

Friday, 27 October, 2017 (All day)

Venue: 

Room 1.09

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.

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