AG-C: Forschungsseminar “Komplexe Analysis”
| Fachbereich Mathematik/Informatik | Institut für Mathematik | KVV | Impressum |
Termine
Donnerstag,
den 15.7.2021, 16 bis 18Uhr
Prof.
Dr. Adel Khalfallah (King Fahd University of Petroleum and
Minerals, Dhahran, Saudi Arabia)
Linking
Complex Analytic to Nonstandard Algebraic Geometry
The
value of nonstandard mathematics is in serving as a "guiding
star" and often offering a conceptually simple and elegant
interpretation and generalization of classical theory and
sometimes leads to new concrete standard results. Not so much is
known about nonstandard complex analysis, unlike nonstandard real
analysis, topology and metric spaces theory. Only some very
specific applications of model theory are used to be known
as for instance the Lefschetz principle, the theorem of
Tarski-Seidenberg or some simple proofs of Hilbert’s
Nullstellensatz. Recently, in collaboration with S. Kosarew, we
started a program to develop a theory of analytic geometry using
nonstandard methods. One of our fundamental constructions is that
of a category of certain ringed spaces, called bounded schemes,
which contains the category of algebraic C-schemes and which
admits an essentially surjective functor, called the standard part
functor, to the category of complex spaces. The advantage of this
new more algebraic category is that it allows us to apply many
constructions of standard algebraic geometry which are not evident
in the analytic context. We obtain analytic results just by taking
the standard part functor.
Keywords:
Nonstandard analysis; Bounded schemes; internal polynomials;
convex subrings; Colombeau’s algebras
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Donnerstag,
den 15.4.2021, 16 bis 18Uhr
Zhen
Guan (Freie Universität Berlin)
Fast
Exponentially Convergent Solution of Electromagnetic Scattering
From Multilayer Concentric Magnetodielectric Cylinders by the
Spectral Integral Method
The
multilayer surface integral equations (SIEs) for electromagnetic
scattering by infinitely long magnetodielectric cylinders with an
arbitrary number of layers are derived and solved by the spectral
integral method (SIM). Singularity subtraction for the 2-D Green’s
function is used to enhance the computation accuracy and achieve
exponential convergence. The final matrix equation after
discretization is formed in the Fourier spectral domain rather
than the spatial domain, which greatly expedites the SIM solution
by accelerating the convolution via the fast fourier transform
(FFT) algorithm. A recursive method is proposed to solve the
spectral integral equations instead of using an iterative method
to lower the computation complexity. Numerical examples for
ordinary multilayer cylinders and invisibility cloak cylinders are
presented to validate the SIM results by comparing the total
fields, scattered fields, and radar cross section (RCS) to
analytical solutions or finite-element simulations. They verify
that the recursive solution has a complexity of O(MN log N) for an
M-layer cylinder with N discrete points on each interface.
Meanwhile, the SIM outperforms the analytical method because only
the 0th-order and 1st-order special functions (Bessel functions
and Hankel functions) are used in the SIM but higher-order
functions are necessary for the analytical method to maintain the
accuracy.
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Donnerstag,
den 20.5.2021, 16 bis 18Uhr
Prof.
Dr. Markus
Reineke
(Ruhr-Universität
Bochum)
Quiver
moduli from neural networks
Certain
aspects of neural networks can be modelled using so called
double-framed
representations of quivers. We define moduli spaces parametrizing
such objects and describe their geometric properties, using
Geometric Invariant Theory. We then derive explicit linear algebra
descriptions of these moduli spaces using classical invariant
theory and representation theory.
This is joint work in
progress with Marco Armenta, Thomas Bruestle and Souheila
Hassoun.
Handout
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Donnerstag,
den 27.5.2021, 16 bis 18Uhr
Prof.
Dr. Peter Bürgisser (Technische Universität
Berlin)
Optimization,
Complexity and Invariant Theory
Invariant
and representation theory studies symmetries by means of group
actions and is
a well established source of unifying principles in mathematics
and physics. Recent research suggests its relevance for complexity
and optimization through quantitative and algorithmic questions.
The goal of the talk is to give an introduction to new algorithmic
and analysis techniques that extend convex optimization from the
classical Euclidean setting to a general geodesic setting. We also
point out surprising connections to a diverse set of problems in
different areas of mathematics, statistics, computer science, and
physics.
The
talk is mainly based on this joint article with Cole Franks, Ankit
Garg, Rafael Oliveira, Michael Walter and Avi Wigderson:
http://arxiv.org/abs/1910.12375
---
Freitag,
den 28.5.2021, 14 bis 16Uhr
Marwan
Benyoussef (Freie Universität Berlin)
Computing
E-Polynomials for certain character varieties
We
report on arithmetic techniques introduced by Hausel and
Rodriguez-Villegas for computing the E-polynomial of certain
families of algebraic varieties, called character varieties of
surface groups. The construction of such varieties follows from
Geometric Invariant Theory. We will discuss these arithmetic
techniques based on a fundamental theorem of Katz, that reduces
the computation of E-polynomials to point counting over finite
fields. Then we give an idea on how to apply them to
more
general character varieties.
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Donnerstag,
den 3.6.2021, 16 bis 18Uhr
Prof.
Dr. Vikram Balaji
(Chennai
Mathematical Institute)
Parahoric
groups, Bruhat-Tits group schemes and representations of Fuchsian
groups
The
talk will introduce parahoric groups and Bruhat-Tits group schemes
and torsors and relate them to representations of Fuchsian groups.
The talk will be reasonably self-contained and will contain basic
examples.
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Donnerstag, den 24.6.2021, 16 bis 18Uhr
Juan
Martin Perez Bernal (Nice)
The
moduli stack of elliptic curves
By
a moduli space of Riemann surfaces of genus g,
we mean the set of isomorphism classes of complex analytic
structures on a closed oriented surface of genus g,
fixed once and for all. It is not clear a priori why this
definition makes sense, nor whether this set has an extra
structure, turning it into a "space". In order to
explicitly view the space-like properties of this set, we shall
fix g=1
and we also fix some extra data: a base point on the curve. The
moduli space we obtain by doing this is called the moduli space of
elliptic curves and the goal of the talk is to show that this
space is a complex analytic space in a coarse sense, and to
introduce the notions of a stack in order to have a finer
representation of the classification problem.
Donnerstag,
den 1.7.2021, 16 bis 18Uhr
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Prof.
Dr. Francesco Sala (Pisa)
Two-dimensional
cohomological Hall algebras
The
goal of the present talk is the introduction of two-dimensional
cohomological Hall algebras of quivers, curves, and surfaces, and
discuss their categorification. In the first part of the talk, I
will also provide some motivation to the theory of cohomological
Hall algebras arising from the study of Hilbert schemes of points
and moduli spaces of stable sheaves on smooth surfaces. In the
last part of the talk, I will discuss in detail an ongoing joint
work with Diaconescu, Schiffmann, and Vasserot, in which we
consider the cohomological Hall algebra of a Kleinian surface
singularity.