Maximal Tori and the Classification of Rank One Compact Lie Groups
December 2023
The representation theory of non-commutative Lie groups is much more involved than that of commutative Lie groups. However, in the case of a compact and connected Lie group $G$, its maximal compact connected abelian subgroup (its maximal torus) contains a lot of information about $G$. Indeed, understanding the group of transformations of the torus under the action of internal symmetries (conjugations) of $G$ produces the Weyl group of $G$, which is a powerful invariant and simplifies much of the analysis of $G$. By rank of $G$ we mean the dimension of its maximal torus. While seemingly only a very crude invariant, it is enough to classify interesting examples of low dimension. One can show that any $G$ of rank $1$ is isomorphic to either $U(1)$, $SU(2)$ or $SO(3)$. As a consequence there are no compact connected Lie groups of dimension $2$, except for the torus of dimension $2$.
Cohomology and Characteristic Classes
November 2022
One of the most powerful techniques in modern geometry is cohomology. It allows to associate linear invariants (cohomology groups) to highly non-linear objects (here, smooth manifolds) in a systematic and homotopy invariant manner. In particular, de Rham cohomology allows to compute the cohomology groups by interpreting the geometry (or rather the homotopy type) of a manifold as an obstruction to the existence of global solutions to certain PDEs on that manifold. Together with homological algebra, the Mayer-Vietoris sequence allows to compute the cohomology of a manifold from the cohomology of a covering. Since having the same cohomology is a weaker notion than being homeomorphic, and since the dimension of the $0$-th order de Rham cohomology group $H^0_{dR}(M)$ equals the number of connected components of the manifold, de Rham cohomology can be applied to extensions of the Jordan curve theorem, i.p. the Jordan-Brouwer separation theorem, invariance of domain, and invariance of dimension.
Infinite Dimensional Analysis - Gaussian Random Variables
November 2021
Gaussian measures on infinite dimensional vector spaces tend to be singular with respect to each other. In particular, given a Gaussian measure $\mu$, there is only a very restricted subspace $H$ such that translation along elements of that subspace produces an equivalent measure (Cameron-Martin Theorem). This subspace is called the Cameron-Martin subspace, and despite the fact that the Gaussian measure assigns it $\mu(H) = 0$ mass, the Cameron-Martin subspace determines the measure uniquely. Given two Gaussian measures $\mu_1, \mu_2$, the Feldman-Hájek Theorem gives a characterization of mutual singularity or equivalence in terms of the eigenvalues of the covariance operator of the two measures. Given two sequences of equivalent Gaussian marginal distributions, the Kakutani theorem gives a criterion for the equivalence of the product measures on the product space.
Infinite Dimensional Analysis - Gaussian Measures on Hilbert Spaces
November 2021
In order to write down integral equations (for integral formulations of ODEs, mild solutions, action functionals, Picard iteration, etc.), one needs a reference measure. On a finite-dimensional Hilbert space $H$ there is a canonical choice for this: the Lebesgue measure. However, in the infinite-dimensional case, one can show that any measure that could reasonably be called 'Lebesgue measure' cannot exist. This makes it necessary to construct more subtle methods to deal with the measures encountered, with the most common and also most important case being Gaussian measures. The generalization of Gaussian measures from Euclidean space $\mathbb{R}^n$ to general Hilbert spaces brings some functional analytic subtleties with it e.g. the definition of Gaussianity without resorting to a density (which would necessarily have to involve the non-existent Lebesgue measure), the choice of $\sigma$-algebra on which the measure is defined, the covariance form/operator, generalizations to more general topological vector spaces, etc.
Pathwise Stochastic Integrals for Model Free Finance
June 2021
In order to price an option, one would usually proceed as follows: 1. Select a class of models suitable for the problem, 2. Calibrate the model using market data, 3. Compute the price as the discounted expectation with respect to the calibrated model. However, there is a problem with this approach: for complicated options, different models may produce different prices. Model-free finance seeks to solve this problem by not choosing a probabilistic model in the first place. However, this brings some substantial problems with it; for example, what do an SDE and its solution even mean if there is no underlying semi-martingale measure? and what are arbitrage and no-free-lunch, which are fundamentally stochastic notions? Solutions to these problems are given by the notions of model-free Itô isometry and rough paths, as well as typical price paths and model-free arbitrage opportunities.
Mathematical Quantum Mechanics - Complete Positivity & Stinespring Theorem
June 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 $U(t) = e^{-iHt/\hbar}$, which is itself induced by the Hamiltonian $H$ 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 $T \otimes \text{Id}$ for 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 Dilation Theorem gives a characterization of completely positive linear maps as $*$-representations followed by conjugation by some linear map.
Category Theory - Kan Extensions
December 2019
Left/Right Kan extensions concern the (universal) existence of extensions of a functor $F: \mathcal{C} \to \mathcal{E}$ to a functor on a larger domain $\mathcal{D}$ along a functor $K:\mathcal{C} \to \mathcal{D}$ i.e. a functor $L: \mathcal{D} \to \mathcal{E}$ 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 $[\mathcal{D},\mathcal{E}]$ and $[\mathcal{C},\mathcal{E}]$. 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 $\mathbb{Q}$ to $\mathbb{R}$, and induced group representations.
Category Theory - Commutativity of Limits
November 2019
Let $X: I \to \mathcal{C}$ be a diagram. A limit of this diagram is a representing object of the set valued functor $\text{Nat}(\Delta(-),X)$, which assigns to an object $A$ in $\mathcal{C}$ 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?'
Darstellungstheorie endlicher Gruppen - Tensor Produkte von FG-Moduln und der Darstellungsring
July 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 $M \oplus N$, eine andere das Tensorprodukt $M \otimes N$. 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.
Darstellungstheorie endlicher Gruppen - Charaktere
June 2019
Hat der für die Darstellung $\rho: G \to GL(V)$ verwendete Vektorraum $V$ endliche Dimension, so lässt sich die Spur $\text{tr}(\rho(g))$ der Darstellung eines bestimmten Elements $g \in G$ in der Gruppe bilden. Die Funktion $\chi: G \to \mathbb{C}$, welche einem Element der Gruppe diese Spur zuweist nennt man Charakter dieser Darstellung: $\chi(g) = \text{tr}(\rho(g))$. 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.
Topologie - Universelle Konstruktionen und topologische Basen
June 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.
Darstellungstheorie endlicher Gruppen - Gruppenalgebra & Satz von Maschke
May 2019
Eine ($F$-lineare) Darstellung einer Gruppe $G$ ist ein Homomorphismus $\rho: G \to GL(V)$ von $G$ in die Automorphismengruppe eines $F$-Vektorraumes $V$. Zu jeder solcher Darstellung lässt sich ein Modul über der sog. Gruppenalgebra $F[G]$ 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.
Workshop on Advances in Analysis
April 2019
We show that the set of discontinuities of a monotone function $f: \mathbb{R} \to \mathbb{R}$ is countable - so i.p. of Lebesgue measure $0$. We also give a constructive proof that for any countable set $S \subset \mathbb{R}$ there exists a monotone function having exactly those points of discontinuity - it is given as a limit of indicator functions $f = \lim_{n \to \infty} \sum_{s \in S} \mathbf{1}_{[s,\infty)}$. We also characterize (generalized) inverse functions of a monotone function.
Algebraic Geometry - Functor of Points on Schemes & Yoneda Lemma
November 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 $h_X$ assigning to a scheme $A$ the set of morphisms $\text{Hom}(A,X)$ from $A$ to $X$. Those should be interpreted as the $A$-points (think of a field $k$ and the associated affine scheme $\text{Spec}(k)$, 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.