AG-C: Forschungsseminar “Komplexe Analysis”

| Fachbereich Mathematik/Informatik | Institut für Mathematik | KVV | Impressum |


Dienstag, 15.12.2020, 16-18Uhr
Johannes Rau (Bogotá)
Tropical geometry and some applications

In my talk, I would like to introduce the main ideas of tropical
geometry (from a geometric point of view) and then focus on some of the main applications, in particular, in the context of real algebraic geometry. In this context, tropical geometry can be understood as a generalization of Viro's patchworking technique to construct real algebraic varieties with specific topological properties. Tropical methods can be used here to extend the technique to singular varieties and higher codimensions.

Dienstag, 19.1.2021, 16-18Uhr
Alejandra Rincón Hidalgo (Trieste)
Moduli of Bridgeland semistable triples on curves

A holomorphic triple (
E1, E2, φ) on a smooth projective curve C with g(C)≥1 consists of a pair of coherent sheaves E1, E2Coh(C) and a morphism φ: E1E2. Let TCoh(C) be the abelian category of holomorphic triples. In this talk we study Bridgeland stability conditions on TC=Db(TCoh(C)) and the moduli spaces of Bridgeland semistable objects. In particular, we shall prove that the moduli stacks are algebraic of finite type over.

Dienstag, 2.2.2021, 16-18Uhr
Johannes Horn (Frankfurt)
Spectral data for singular Hitchin fibers and asymptotic solutions to the Hitchin equation

A recent breakthrough in the theory of Higgs bundle moduli spaces is the description of the asymptotics of the hyperkähler metric by Mazzeo-Swoboda-Weiss-Witt (MSWW) and Fredrickson (F). They show that (on the regular locus) its asymptotics are described by the semi-flat metric associated to the Hitchin system - a completely integrable system fibering a dense subset of the Higgs bundle moduli space by abelian torsors.
In this talk, we will explain how one can describe degenerate Hitchin fibers by abelianisation on the normalized spectral curve and Hecke modifications at the singularities of the spectral curve. This naturally leads to a construction of solutions to the decoupled Hitchin equation generalizing the limiting objects in the work of MSWW and F.

Dienstag, 9.2.2021, 16-18Uhr
George Hitching (Oslo Metropolitan University)
Nonemptiness and smoothness of twisted Brill-Noether loci of bundles over a curve

V be a vector bundle over a smooth curve C. The twisted Brill-Noether locus Bkn, e(V) parametrises stable bundles of rank n and degree e such that VE has at least k independent sections. This is a determinantal variety, whose tangent spaces are controlled by a Petri trace map. Generalising a construction of Mercat, we show for many values of the parameters that Bkn, e(V) is nonempty. When C and V are general, we show that under certain numerical conditions, Bkn, e(V) has a component which is generically smooth and of the expected dimension.

This is joint work with Michael Hoff (Saarbrücken) and Peter Newstead (Liverpool).

Dienstag, 16.2.2021, 16-18Uhr

Florent Schaffhauser (Bogotá)
On the Harder-Narasimhan-Shatz stratification for moduli spaces of real bundles

Atiyah and Bott have shown via gauge-theoretic methods that the moduli stack of holomorphic vector bundles of rank r and degree d has an equivariantly perfect stratification, which enabled them to compute the equivariant Poincaré series of the moduli stack of semistable such bundles. The stratification in question is indexed by the possible Harder-Narasimhan types of holomorphic vector bundles of rank r and degree d. In joint work with Melissa Liu, we studied the analogous situation over the field of real numbers, where a variety of difficulties arise. First, one needs to consider mod 2 coefficients to show equivariant perfection. Second, depending on the real topological invariants of the curve, the Harder-Narasimhan type of a real bundle will depend on more parameters than the Harder-Narasimhan type of the associated complex bundle. Nonetheless, in the vector bundle case, it is possible to sort out those difficulties, and arrive at a formula for the equivariant Poincaré series of moduli stacks of semistable real vector bundles. In this talk, after briefly reviewing the Atiyah-Bott approach, we will describe the Harder-Narasimhan-Shatz stratification in the real case, and explain how it leads to such a formula. Time permitting, we will show how to use the formula to compute Betti numbers of moduli stacks of semistable real vector bundles of small rank.

Dienstag, 23.2.2021, 16-18Uhr
Marco Antonio Armenta Armenta (Sherbrooke, Québec, Canada)
The moduli space of a neural network

In this talk I will introduce the notation required to model artificial neural networks in terms of quiver representations, and then show how the decisions of a neural network are completely determined by a space of orbits of quiver representations under the action of a reductive group, and therefore a moduli space.

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