Mathematik - WiSe 2025/26

Mathematik I (Physik)

Einschreibeschlüssel: 25WSM1

Course Description

In this seminar, we will explore machine learning methods built on (strictly) positive definite kernels - a powerful and mathematically well-understood class of models that form the foundation of many learning algorithms.
We begin with a review of strictly positive definite kernels and their key properties, including an introduction to reproducing kernel Hilbert spaces (RKHS) and related theoretical concepts.
Building on this foundation, we will study classical kernel-based algorithms such as support vector machines and kernel ridge regression, and discuss techniques for scaling them to large data sets.
The seminar will then turn to the intersection of kernel methods and deep learning, including topics such as the neural tangent kernel (NTK) and modern kernel-based approaches in deep learning like Recursive Feature Machines (RFMs).

Throughout the seminar, we will cover topics ranging from theoretical and mathematical foundations to practical and algorithmic aspects, giving students a broad view how kernel methods appear in both classical and modern machine learning.

Target Participants

    Master students of Mathematics and Financial and Insurance Mathematics

Prerequisites

    Some familiarity with machine learning and/or functional analysis is beneficial.

Registration key

    ML

Einschreibeschlüssel: Tautologisch

Dieser Kurs richtet sich an angehende Studierende der Mathematik/Wirtschaftsmathematik sowohl im Bachelor als auch für das Lehramt Gymnasium. Ziel ist es, mathematische Grundfertigkeiten zu vermitteln und einzuüben, die für das Studium wichtig sind. 

Wir behandeln das Pauli-Fierz Modell für nichtrelativistisch beschriebene Materie, die an ein quantisiertes Strahlungsfeld gekoppelt ist. Wir untersuchen weitere Eigenschaften des bosonischen Fockraums, der bosonischen Erzeugungs- und Vernichtungsoperatoren und der Feldoperatoren, der zweiten Quantisierung von selbstadjungierten Operatoren -- die freie Energie der Photonen ist ein Beispiel dazu. Dies erlaubt den Hamiltonoperator des minimal gekoppelten Systems überhaupt genau hinschreiben zu können und dann die Selbstadjungiertheit für das UV- regularisierte Pauli-Fierz Modell zu zeigen.

Selbsteinschreibung mit Einschreibeschlüssel mqed

Analysis und Lineare Algebra I

Einschreibeschlüssel: LinAlg1W25

Einschreibeschlüssel:  MiQWS25

About this Lecture

This lecture discusses the interplay of Theory, Modelling, Numerical Methods and Implementation in Mathematical Finance.
All aspects learn from each other: one needs to understand the theory to build models and good implementations. Studying numerical experiments gives deeper theoretical insight. Using the computer to understand math can be fun!
We discuss how to build an industry-grade implementation of our models and allow future extensions while being efficient.
We discuss practical applications in the financial industry.

The lecture tries to be as self-contained as possible, but we will use some numerical methods developed in a previous course. We will start with a short recapitulation of the numerical methods needed for those who did not follow the last lecture. It is possible to consider most of these parts as “given” (“black box”). Don’t panic: We will assist you.

The lecture will discuss the theory and application of some prominent methods and models from mathematical finance. We focus on interest rate and hybrid models with high relevance for the financial industry. In an excursion, we consider a climate model (DICE), extend it and combine it with our interest rate models.
We will then use our implication to gain a deeper understanding of the theoretical properties of the model.

If time permits, we conclude the lecture by discussing running our models in a cloud.

Tentative Agenda

• Recapitulation of Numerical Methods (Monte-Carlo Method, SDEs, etc.)
• Interest Rates, Linear Interest Rates Products
• Multiple Interes Rate Curve Modelling and its Implementation
• Interest Rate Options, Convexity Adjustment
• Discrete Term-Structure Models (formerly known as LIBOR Market Models) and their Implementation
  • Model Calibration
  • Valuation of Complex Derivatives
  • Object Oriented Implementation, Definition of Model Interfaces
• Short Rate Models and their Implementation
• Climate Models
• Numerical Methods in the Context of Mathematical Finance
  • Model Calibration: Optimization and Root-Finding
  • Correlation Modelling: Prinzipal Component Analysis and Factor Reduction, Regression Methods
• IT Implementation of Models, e.g. in the Cloud

The lecture covers the object oriented implementation of the algorithms and using modern software development tools.
As part of the implementation of the models and the valuation algorithms, the lecture will discuss some of the latest standards in software development.

• revision control systems (git)
• unit-testing (junit)
• build management (maven, gradle)
• continuous integration (TravisCI, Jenkins)

Implementation will be performed in Java (Eclipse, IntelliJ)

Einschreibeschlüssel: Num2526

Course Description

In this reading course, we will study stochastic processes in continuous time that happen to be sub- or supermartingales. In particular, this includes martingales. After proving Doob's maximal inequalities and stopping theorem, we will derive the quadratic variation of a local martingale. As a result, we will be able to understand the notion of a semimartingale.

Target Participants

Die Problemlabs bieten einen Raum zum gemeinsamen Arbeiten und Lernen. Ob ihr Hausaufgaben bearbeitet oder nochmal das Skript nacharbeitet ist dabei ganz euch überlassen.
An mehreren Terminen wird der Raum von ehrenamtlichen Tutor:Innen aus höheren Semestern betreut, so dass ihr bei Fragen oder Unklarheiten die nötige Unterstützung bekommt.

Anmeldeschlüssel: NoProblem

The module is a continuation of the classical part of Mathematical Statistical Physics from summer. We'll cover some advanced topics, starting with

• Reflection positivity & proof of orientational symmetry breaking in dimensions 3 and higher
• Cluster expansions - a perturbative technique, related to but easier than Feynman diagrams
• Pirogov-Sinai theory: a technique for proving first-order phase transitions at low temperature. The technique builds on ideas of the Peierls argument but is considerably more involved.

Inhalt der Vorlesung ist eine Einführung in die Optimierung in - vornehmlich - endlicher Dimension. Wichtige Themen und Inhalte sind unter anderem:

Konvexität, Lineare Programme und ihre Standardformen, Existenz von Lösungen für lineare Programme, Dualitätstheorie für lineare Programme, das Simplexverfahren, Formulierung und Existenz von Lösungen konvexer Optimierungsprobleme, duale Darstellung konvexer Funktionen und die Kuhn-Tucker-Theorie.

Einschreibeschlüssel: Optimierung2526

In dieser Vorlesung behandeln wir gemeinsam die Grundlagen der (Hochschul-)Mathematik.
Thematisch beginnen wir bei fundamentalen Begriffen wie Mengen und Abbildungen, sprechen dann über zentrale Begriffe der Analysis wie Folgen und Stetigkeit und widmen uns am Ende Ableitungen und Integrale.

Die Veranstaltung richtet sich an Studierende verschiedener Bachelorstudiengänge, wie z.B. Geowissenschaften und Informatik mit großem Nebenfach.

Anmeldeschlüssel: MfNW