Seminar Notes

At TU Munich, students are required to attend one seminar during the course of their bachelor's degree. However, the organizers of some other seminars were so kind to let me join theirs as well. A seminar typically consists of 8-10 people, each of which holds one of the 90 min talks as a lecturer would. In some seminars, I was allowed to hold multiple lectures. Depending on whether it is a Bachelors or a Masters seminar, the talks are given in German or English.

Here are the lecture notes for the talks I held.

Seminar Notes

At TU Munich, students are required to attend one seminar during the course of their bachelors degree. However, the organizers of some other seminars were so kind to let me join theirs as well. A seminar typically consists of 8-10 people, each of which holds one of the 90 min talks as a lecturer would. In some seminars I was allowed to hold multiple lectures. Depending on whether it is a Bachelors or a Masters seminar, the talks are given in german or english.

Here are the lecture notes for the talks I held.

(M.Sc.) Mathematical Quantum Mechanics - Complete Positivity & Stinespring Theorem

June 4, 2021

For a closed quantum system, time evolution in the Heisenberg picture is given by a unitary transformation of the observable by the time evolution operator, which is itself induced by the Hamiltonian of the system and the Schrödinger equation. However, in Quantum Information Theory, when considering open systems with possible entanglement, a more general notion of time evolution is needed. In the Schrödinger picture, this is the notion of a quantum channel, in the Heisenberg picture it is that of completely positive maps between operator algebras. Essentially, a time evolution should not only map states to states of an isolated system but also states to states when the system is paired with an auxiliary system on which no action is performed. Mathematically speaking, not only T should be a positive map, but also T tensored with any identity operator.

For finite dimensional state spaces, the Choi–Jamiołkowski isomorphism gives the channel-state duality, relating complete positivity to positivity of a certain state: the Choi matrix. However, in infinite dimension this fails. The Stinespring Dialation Theorem gives a characterization of completely positive linear maps as *-representations followed by conjugation by some linear map.

(M.Sc.) Category Theory - Kan Extensions

December 12, 2019

Left/Right Kan extensions concern the (universal) existence of extensions of a functor F: C -> E to a functor on a larger domain D along a functor K:C -> D i.e. a functor L: D -> E s.t. the resulting diagram commutes. We first introduce left/right Kan extensions as initial/terminal objects in appropriate comma categories and then give another characterization in terms of left/right adjoint to a precomposition-functor K* between functor categories [D,E] and [C,E]. Furthermore, we give a concrete construction for the Kan Extensions for sufficiently complete categories. We also consider concrete examples of extending inclusions of posets, extending exponential functions from Q to R, and induced group representations.

(M.Sc.) Category Theory - Commutativity of Limits

November 28, 2019

Let X: I -> C be a diagram. A limit of this diagram is a representing object of the set valued functor Nat(Delta(-),X), which assigns to an object A in C the cone associated to A and the diagram. We consider sufficient and necessary conditions under which limits and colimits with small index category commute. We also consider some examples. A concrete example is the question "Is the pushout of cokernels uniquely isomorphic to the cokernel of pushouts?"

(B.Sc.) Darstellungstheorie endl. Gruppen - Tensor Produkte von FG-Moduln und der Darstellungsring

July 17, 2019

Unter den Voraussetzungen des Satzes von Maschke ist jeder FG-Modul eine direkte Summe aus irreduziblen Moduln. Es ist also natürlich nach Methoden zu suchen, mit welchen man aus zwei FG-Moduln einen neuen produziert. Eine dieser ist die direkte Summe, eine andere das Tensorprodukt. Wir führen das Tensorprodukt mittels seiner universellen Eigenschaft, als Faktorraum, sowie auch mit einer expliziten Basis ein. Die Charaktere der so gewonnen Darstellungen lassen sich ebenfalls schön charakterisieren.

 

Die Menge der Darstellungen einer Gruppe bildet mit den Operationen der direkten Summe und des Tensorproduktes einen Halbring. Der Gruppenring ist der Ring welchen man erhält, wenn man den unterliegenden (additiven) Monoid zu einer Gruppe vervollständigt. Das geschieht mittels Grothendieck-Konstruktion.

(B.Sc.) Darstellungstheorie endl. Gruppen - Charaktere

June 19, 2019

Hat der für die Darstellung verwendete Vektorraum endliche Dimension, so lässt sich die Spur der Darstellung eines bestimmten Elements in der Gruppe bilden. Die Funktion, welche einem Element der Gruppe diese Spur zuweist nennt man Charakter dieser Darstellung. Durch Übergang von Darstellung auf Charakter der Darstellung geht zunächst scheinbar sehr viel Information verloren, aber es stellt sich heraus dass viele Eigenschaften der Darstellung wiederhergestellt werden kann.

Wir führen den Charakter einer linearen Darstellung ein und beweisen erste Eigenschaften. Außerdem betrachten wir die Beispiele der regulären und Permutationscharaktere.

(M.Sc.) Topologie - Universelle Konstruktionen und topologische Basen

June 13, 2019

Wir führen Einbettungen und Quotientenraum als universelle Konstruktionen ein, betrachten Beispiele sowie Anwendungen. Außerdem führen wir den Begriff der Basen und Subbasen ein, beweisen erste Eigenschaften und geben Beispiele.

Schlussendlich geben wir einen Beweis für die Aussage dass ein zweit-abzählbarer Raum separabel ist. Die Umkehrrichtung gilt wenn der Raum metrisierbar ist, was auch bewiesen wird.

(B.Sc.) Darstellungstheorie endl. Gruppen - Gruppenalgebra & Satz von Maschke

May 22, 2019

Eine (F-lineare) Darstellung einer Gruppe G ist ein Homomorphismus von G in die Automorphismengruppe eines F-Vektorraumes. Zu jeder solcher Darstellung lässt sich ein Modul über der sog. Gruppenalgebra von G über F (den FG-Modul) zuordnen und vice versa. Diese Zuordnung ist funktoriell und induziert sogar einen Isomorphismus der entsprechenden Kategorien.

Der Satz von Maschke gibt ein hinreichendes Kriterium dafür wann dieser FG-Modul halbeinfach ist. Wir führen die Gruppenalgebra ein und formulieren und beweisen den Satz von Maschke für endliche Gruppen.

(B.Sc.) Workshop on Advances in Analysis

April 26, 2019

We show that the set of discontinuities of a monotone function is countable - so i.p. of Lebesgue measure 0. We also give a constructive proof that for any countable set in R there exists a monotone function having exactly those points of discontinuity - it is given as a limit of indicator functions.

We also characterize (generalized) inverse functions of a monotone function.

(M.Sc.) Algebraic Geometry - Functor of Points on Schemes & Yoneda Lemma

November 30, 2018

For a general topological space X, a point can be characterized by a morphism from the singleton into X which maps to said point. We want to take a similar approach in the study of schemes. However, of course, not any two one-pointed schemes are uniquely isomorphic - any two non-isomorphic fields give an example. Thus we need a more sophisticated notion of points: a functor assigning to a scheme A the set of morphisms from A to X. Those should be interpreted as the A-points (think of a field and the associated affine scheme, which is an actual point).

It turns out that the entire scheme can be recovered from this functor. This is a stronger version of the Yoneda Lemma, which is introduced in more generality.