Vorträge

The trend towards massively parallel high-performance computers requires the development of parallel algorithms to employ their computational power.
The Parareal algorithm computes the solution of time-dependent problems parallel in time, meaning that approximations to the solution at different times are computed simultaneously. In this talk, we will focus on hyperbolic one-dimensional problems, where a combination of Parareal and a discontinuous Galerkin method will be used. The practical use and challenges of this method will be illustrated by means of a Python implementation for shallow water equations and corresponding numerical results.

How will the next generation of Artificial Intelligence (AI) look like? Comparing today's AI algorithms with biological intelligence, one of the most remarkable differences is the ability of the human brain to somehow understand the 'essence' of things: A small child can easily identify any type of object after having seen only a few examples or recognize a song even when played on different instruments or in a different key. In other words: Brains are able to create abstract concepts of real-world entities - and today's algorithms are not.

With today's AI largely being based on neuron models already invented by the mid of last century, I will argue that we should take a new look at the brain to find inspiration for the next generation of machine learning algorithms. Even though there is still only a very limited understanding of how the brain works computationally, I'll explain why there is hope that we can reverse-engineer some of its algorithmic principles and implement them in a computer. I'll explain why a highly interdisciplinary approach is required from neuroscience, computer science, mathematics and physics to make progress in this question.

The talk will be held on Zoom:
https://tuhh.zoom.us/j/86836210324?pwd=ajJURGY2T3pFNWMvUzVQTkduSTNCQT09
Meeting-ID: 868 3621 0324
Kenncode: 521014

For many processes in sciences, the coefficients of the partial differential equation describing a dynamical system as well as the boundary conditions of it may vary with time. In such cases one speaks of non-autonomous (or time-varying) evolution equations. From an operator theoretical point of view one considers families of Banach space operators which depend on the time parameter and studies the associated non-autonomous abstract Cauchy problem. We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on Lp -spaces. This is joint work with Christian Seifert (TUHH).

I will introduce myself and present the topic of my master thesis.

A fundamental principle in developing the Theory of Quantum Mechanics is to take
well-studied concepts from the Theory of Classical Mechanics and to define
analogues in the quantum mechanical setting.
One such important tool in Classical Mechanics is the theory of optimal
transport and in particular the Wasserstein distance.

In my thesis I studied the mathematical objects needed to translate
the concepts of the optimal transport problem to the realm of Quantum
Mechanics. In particular,
one wants to establish a relation between density matrices (trace-class operators
of trace one) and
probability measures. This can be done by the so-called
(generalized) Toeplitz operators and the (generalized) Husimi
transform.

After I give a brief introduction into both the Optimal Transport and Quantum
Mechanics I will introduce both
the Toeplitz operators and the Husimi transform and discuss some of their
properties.

Suppose that $A$ is a bounded linear operator on the Hardy space $H^p$ that satisfies
\[\langle Az^j,z^k \rangle = a_{k-j} \quad (j,k \in \mathbb{N}_0)\]
for some sequence of complex numbers $\{a_n\}_{n \in \mathbb{Z}}$. By the Brown--Halmos theorem, $A$ must be a Toeplitz operator with bounded symbol, that is, $\{a_n\}_{n \in \mathbb{Z}}$ is the Fourier sequence of a bounded function. Likewise, Nehari's theorem shows that if $A$ satisfies $\langle Az^j,z^k \rangle = a_{k+j+1}$ instead, then $A$ is equal to a Hankel operator with bounded symbol. These results were proven in the 50's and 60's and have become classical in the theory of Hardy spaces.

More recently, due to some applications in mathematical physics, there has been a lot of interest in so-called Toeplitz+Hankel operators. Quite simply put, a Toeplitz+Hankel operator is the sum of a Toeplitz operator $T(a)$ and a Hankel operator $H(b)$. Now clearly, if both $T(a)$ and $H(b)$ are bounded, then $A = T(a)+H(b)$ is necessarily bounded as well. It is therefore natural to ask whether the converse is also true or if the ``unboundedness'' of $T(a)$ and $H(b)$ can somehow cancel out. I will elaborate on this question and present a Brown--Halmos type result for Toeplitz+Hankel operators for both the Hardy spaces $H^p$ and the sequence spaces $\ell^p(\mathbb{N}_0)$. A similar characterization for compactness will be obtained as well.

Based on joint work with Torsten Ehrhardt and Jani Virtanen.

Hallo liebe Institutsmitarbeiter*innen,

anbei der offizielle Einladungstext zum Promotionsvortrag von Alexander Haupt:

_________________________________________________________________________

Sehr geehrte Damen und Herren,

im Rahmen seines Promotionsverfahrens wird

Herr M. Sc. Alexander Michael Haupt

einen kombinierten Live-Online-Vortrag mit dem Titel

„New Combinatorial Proofs for Enumeration Problems and Random Anchored Structures“

halten. Der Vortrag findet statt am

Dienstag, dem 21. September 2021 um 11:00 Uhr.

Zu diesem universitätsöffentlichen Vortrag lade ich Sie herzlich ein.

Aufgrund der aktuell geltenden Regelungen können Interessierte nur per Zoom am Vortrag teilnehmen. Bitte benutzen Sie hierzu die folgenden Zugangsdaten:

https://tuhh.zoom.us/j/88353220627?pwd=amZFYjRNOHl2TkdGb2c1Z29MVGNCUT09

Meeting-ID: 883 5322 0627
Kenncode: 747604

Mit freundlichen Grüßen
Prof. Dr. Matthias Schulte

(Vorsitzender des Prüfungsausschusses)