AG-C: Forschungsseminar “Komplexe Analysis”
| Fachbereich Mathematik/Informatik | Institut für Mathematik | KVV | Impressum |
Termine
Dienstag,
15.12.2020, 16-18Uhr
Johannes
Rau
(Bogotá)
Tropical
geometry and some applications
Abstract
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
Abstract
A
holomorphic triple (E1,
E2,
φ)
on a smooth projective curve C
with
g(C)≥1
consists of a pair of coherent sheaves E1,
E2∈Coh(C)
and a morphism φ:
E1→E2.
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
ℂ.
Dienstag,
2.2.2021, 16-18Uhr
Johannes
Horn
(Frankfurt)
Spectral
data for singular Hitchin fibers and asymptotic solutions to the
Hitchin equation
Abstract
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
Abstract
Let
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
Abstract
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
Abstract
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.