Lecturer: Michael Růžička
Assistant: Maximilian Stegemeyer
Language: in German
Lecture: Mo, Mi, 8-10h, HS Rundbau, Albertstr. 21
Tutorial: 2 hours, various dates
Analysis II is the continuation of Analysis I from the winter semester and one of the basic lectures of the study programmes in Mathematics. Central concepts of Analysis I (limits and derivations) will be generalized to the case of higher dimension.
Central topics are the topology of \(\mathbb R^n\), metrics and norms, differential calculs in several variables, ordinary differential equations and in particular linear differential equations.
Analysis I, Linear Algebra I (or bridge course linear algebra)
Analysis
Linear Algebra II
Lecturer: Stefan Kebekus
Assistant: Christoph Brackenhofer
Language: in German
Lecture: Di, Do, 8-10h, HS Rundbau, Albertstr. 21
Tutorial: 2 hours, various dates
Linear algebra II is the continuation of the lecture linear algebra I from the winter semester and one of the basic courses of math studies. Central topics are: Jordan’s normal form of endomorphisms, symmetrical bilinear forms with especially the Sylvester’s theorem, Euclidian and Hermitian vector spaces, skalar products, orthonormal bases, orthogonal and (self-) adjugated , spectral theorem, principal axis theorem.
Linear Algebra I
Linear Algebra
Lecturer: Nadine Große
Assistant: Jonah Reuß
Language: in German
Lecture: Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
29.07., 08:30-11:30, HS Rundbau, Albertstr. 21
Sit-in exam (resit) 08.10., 08:30-11:30, HS Weismann-Haus, Albertstr. 21a
The lecture gives an introduction to elementary geometry in Euclidian and non-Euclidian space and its mathematical foundations. We get to know Euclidean, hyperbolic, and projective geometry as examples of incidence geometries, and study their symmetry groups.
The next main topic is the axiomatic characterization of the Euclidean plane. The focus is on the story of the fifth Euclidian axiom (and the attempts to get rid of it).
Linear Algebra I
Compulsory Elective in Mathematics
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in German
Lecture: Mi, 14-16h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
31.07., 10:00-12:00
Sit-in exam (resit) 17.10., 10:00-13:00
Numerics is a discipline of mathematics that deals with the practical solution of mathematical problems. As a rule, problems are not precisely solved but approximated, for which a sensible compromise of accuracy and computing effort has to be found. In the second part of the two -semester course, questions of the analysis such as the approximation of functions by polynomials, the approximately solution of non -linear equations and the practical calculation of integrals are treated. Attendance at the accompanying computer exercise sessions is recommended. These take place fortnightly, alternating with the tutorial for the lecture.
necessary: Linear Algebra I and Analysis I
useful: Linear Algebra II, Analysis II
Numerics
Lecturer: Johannes Brutsche
Assistant: Dario Kieffer
Language: in German
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
05.09.
After gaining an insight into the basics and various methods and questions of stochastics and probability theory in the Stochastics I lecture, this lecture will mainly focus on statistical topics, especially those that are relevant for students studying to become secondary school teachers. However, the lecture can also be a (hopefully) useful supplement and a good basis for later attendance of the course lecture ‘Mathematical Statistics’ for students in the B.Sc. in Mathematics with an interest in stochastics.
After clarifying the term ‘statistical model’, methods for constructing estimators (e.g. maximum likelihood principle, method of moments) and quality criteria for these (reliability of expectations, consistency) are discussed. Confidence intervals and hypothesis tests are also introduced. Linear models are considered as further applications and, if time permits, other statistical methods. The properties of exponential families and multivariate normal distributions, which are useful for many test and estimation methods, are also introduced.
Linear Algebra I+II and Analysis I+II
Compulsory Elective in Mathematics
Differential Geometry II – Geometry of Submanifolds
Lecturer: Guofang Wang
Assistant: Xuwen Zhang
Language: in German
Lecture: Mo, Mi, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
In the lecture, we discuss the geometry of the sub-manifolds of Euclidian spaces. Examples of such sub-manifolds are curves in the plane and surfaces in the 3-dimensional space. In the 1st part we introduce the external geometry of the sub-manifold, e.g. the second fundamental form, the mean curvature, the first variation of the area, the equations of Gauss, Codazzi and Ricci. In the 2nd part we examine tminimal hypersurfaces (minimal surfaces), the hypersurface with constant mean curvature and the geometric inequalities, the isoperimetric inequality and its generalisations.
Analysis III and "Differential Geometry" or "Curves and Surfaces"
Compulsory Elective in Mathematics
Lecturer: Patrick Dondl
Assistant: Luciano Sciaraffia
Language: in German
Lecture: Mo, 12-14h, HS II, Albertstr. 23b, Mi, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
06.08., 14:00-16:00, HS Rundbau, Albertstr. 21
Attention: Change of time and room!
Linear functional analysis, which is the subject of the lecture, uses concepts of linear algebra such as vector space, linear operator, dual space, scalar product, adjoint map, eigenvalue, spectrum to solve equations in infinite-dimensional function spaces, especially linear differential equations. The algebraic concepts have to be extended by topological concepts such as convergence, completeness and compactness.
This approach was developed at the beginning of the 20th century by Hilbert, among others, and is now part of the methodological foundation of analysis, numerics and mathematical physics, in particular quantum mechanics, and is also indispensable in other mathematical areas.
Linear Algebra I+II, Analysis I–III
Compulsory Elective in Mathematics
Lecturer: Wolfgang Soergel
Assistant: Xier Ren
Language: in German
Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
In linear algebra you studied linear systems of equations. In commutative algebra, we study polynomial equation systems such as \(x^2+y^2 = \) 1 and their solution sets, the algebraic varieties. It will turn out that such a variety is closely related to the ring of the restrictions of polynomial functions on that variety, and that we can extrapolate this relationship to a geometric understanding of any commutative rings, in particular the ring of the integers. Commutative algebra, algebraic geometry, and number theory grow together in this conceptual building. The lecture aims to introduce into this conceptual world. We will especially focus on the dimension of algebraic varieties and their cutting behavior, which generalizes the phenomena known from the linear algebra on the case of polynomial equation systems.
necessary: Linear Algebra I+II
useful: Algebra and Number Theory
Compulsory Elective in Mathematics
Lecturer: Amador Martín Pizarro
Assistant: Stefan Ludwig
Language: in German
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
28.07., 14:00-17:00
This introductory course in mathematical logic consists of several parts. It the basics of predicate logic and a brief introduction to model theory and the axiom system as well as the axiom system of set theory. The aim of the lecture is to explain the recursion-theoretical content of the predicate calculus, in particular the so-called Peano-arithmetic and Gödel's incompleteness theorems.
Basic knowledge of mathematics from first semester lectures
Compulsory Elective in Mathematics
Lecturer: Heike Mildenberger
Assistant: Hannes Jakob
Language: in German
Lecture: Di, Do, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
22.07.
Sit-in exam (resit) 13.10.
A topological space consists of a basic set \(X\) and a family of open subsets of the basic set, which is called topology on \(X\). Examples over the basic sets \(\mathbb R\) and \({\mathbb R}^n\) are given in the analysis lectures. The mathematical subject \glqq{}Topology\grqq\ is the study of topological spaces and the investigation of topological spaces. Our lecture is an introduction to set-theoretic and algebraic topology.
Analysis I and II, Linear Algebra I
Compulsory Elective in Mathematics
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: in English
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, date to be determined
22.09., 10:00-12:00
The problem of axiomatising probability theory was solved by Kolmogorov in 1933: a probability is a measure of the set of all possible outcomes of a random experiment. From this starting point, the entire modern theory of probability develops with numerous references to current applications.
The lecture is a systematic introduction to this area based on measure theory and includes, among other things, the central limit theorem in the Lindeberg-Feller version, conditional expectations and regular versions, martingales and martingale convergence theorems, the strong law of large numbers and the ergodic theorem as well as Brownian motion.
necessary: Analysis I+II, Linear Algebra I, Elementary Probability Theory I
useful: Analysis III
Compulsory Elective in Mathematics
Lecturer: David Criens
Assistant: Samuel Adeosun
Language: in English
Lecture: Mi, 14-16h, HS II, Albertstr. 23b, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
This lecture builds the foundation of one of the key areas of probability theory: stochastic analysis. We start with a rigorous construction of the It^o integral that integrates against a Brownian motion (or, more generally, a continuous local martingale). In this connection, we learn about It^o's celebrated formula, Girsanov’s theorem, representation theorems for continuous local martingales and about the exciting theory of local times. Then, we discuss the relation of Brownian motion and Dirichlet problems. In the final part of the lecture, we study stochastic differential equations, which provide a rich class of stochastic models that are of interest in many areas of applied probability theory, such as mathematical finance, physics or biology. We discuss the main existence and uniqueness results, the connection to the martingale problem of Stroock-Varadhan and the important Yamada-Watanabe theory.
Probability Theory I and II (Stochastic Processes)
Compulsory Elective in Mathematics
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in English
Lecture: Mi, 10-12h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The lecture addresses algorithmic aspects in the practical realization of mathematical methods in big data analytics and machine learning. The first part will be devoted to the development of recommendation systems, clustering methods and sparse recovery techniques. The architecture and approximation properties as well as the training of neural networks are the subject of the second part. Convergence results for accelerated gradient descent methods for nonsmooth problems will be analyzed in the third part of the course. The lecture is accompanied by weekly tutorials which will involve both, practical and theoretical exercises.
Lectures "Numerik I, II" or lecture "Basics in Applied Mathematics"
Compulsory Elective in Mathematics
Lecturer: Mikhail Tëmkin
Language: in English
Lecture: Mo, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
The notion of a manifold is fundamental importance. On one hand, it is a common ground for many branches of pure and applied mathematics, as well as mathematical physics. On the other hand, it itself is a lush source of elegant, unexpected and structural results. Next, algebraic topology is to mathematics what the periodic table is to chemistry: it offers order to what seems to be chaotic (more precisely, to topological spaces of which manifolds is an important example). Finally, differential topology studies smooth manifolds using topological tools. As it turns out, narrowing the scope to manifolds provides many new beautiful methods, structure and strong results, that are applicable elsewhere -- as we will see in the course. Necessary notions from algebraic topology will be covered in the beginning.
Point-set topology (e.g. "Topology" from summer semester of 2024)
Compulsory Elective in Mathematics
Lecturer: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in German
Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Groups without any non-trivial normal subgroup are called simple groups. Similar to prime numbers for the natural numbers, simple groups form the building blocks for finite groups. It is easy to see that Abelian finite simple groups are cyclic. Non-Abelian examples are alternating groups and Lie-type groups.
The classification of finite simple groups is far beyond the scope of this course. However, we will illustrate some of the recurring ideas of classification and, in particular, prove the following result of Brauer and Fowler:
Theorem: Let G be a finite group of even order such that the centre is of odd order. Then there is an element \(g \neq 1_G\) with \(|G| < |C_G (g)|^3\) .
This theorem had a particularly large impact on the classification of finite simple groups, as it suggests that these could be classified by examining the centralisers of elements of order 2.
Algebra and Number Theory
Compulsory Elective in Mathematics
Lecturer: Ernst August v. Hammerstein
Assistant: Sebastian Hahn
Language: in English
Lecture: Mo, 14-16h, HS II, Albertstr. 23b
Exercise session: Mi, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Lévy processes are the continuous-time analogues of random walks in discrete time as they possess, by definition, independent and stationary increments. They form a fundamental class of stochastic processes which has widespread applications in financial and insurance mathematics, queuing theory, physics and telecommunication. The Brownian motion and the Poisson process, which may already be known from other lectures, also belong to this class. Despite their richness and flexibility, Lévy processes are usually analytically and numerically very tractable because their distributions are generated by a single univariate distribution which has the property of infinite divisibility.
The lecture starts with an introduction into infinitely divisible distributions and the derivation of the famous Lévy-Khintchine formula. Then it will be explained how the Lévy processes emerge from these distributions and how the characteristics of the latter influence the path properties of the corresponding processes. Finally, after a short look at the method of subordination, option pricing in Lévy-driven financial models will be discussed.
necessary: Probability Theory I
useful: Probability Theory II (Stochastic Processes)
Compulsory Elective in Mathematics
Lecturer: Thorsten Schmidt
Assistant: Simone Pavarana
Language: in English
Lecture: Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
In this lecture we will study new and highly efficient tools from machine learning which are applied to stochastic problems. This includes neural SDEs as a generalisation of stochastic differential equations relying on neural networks, transformers as a versatile tool not only for languages but also for time series, transformers and GANs as generator of time series and a variety of applications in Finance and insurance such as (robust) deep hedging, signature methods and the application of reinforcement learning.
The prerequisites are stochastics, for some parts we will require a good understanding of stochastic processes. A (very) short introduction will be given in the lectures – so for fast learners it would be possible to follow the lectures even without the courses on stochastic processes.
Compulsory Elective in Mathematics
Lecturer: Patrick Dondl
Assistant: Eric Trébuchon
Language: in English
Lecture: Mi, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
This course provides a comprehensive introduction to mathematical modeling. We will learn the systematic process of translating real-world problems into mathematical frameworks, analyzing them using appropriate mathematical tools, and interpreting the results in practical contexts. The course covers both discrete and continuous modeling approaches, with emphasis on differential equations, variational problems, and optimization techniques. Through case studies in physics, biology, engineering, and economics, students will develop skills in model formulation, validation, and refinement. Special attention is given to dimensional analysis, stability theory, and numerical methods necessary for implementing solutions with a focus on numerical methods for ordinary differential equations. The course combines theoretical foundations with hands-on experience in computational tools for model simulation and analysis.
Analysis I, II, Linear Algebra I, II, Numerics I, II
Compulsory Elective in Mathematics
Numerical Optimization
Lecturer: Moritz Diehl
Assistant: Léo Simpson
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
20.08., 10:00-12:00
The aim of the course is to give an introduction into numerical methods for the solution of optimization problems in science and engineering. The focus is on continuous nonlinear optimization in finite dimensions, covering both convex and nonconvex problems. The course divided into four major parts:
The course is organized as inverted classroom based on lecture recordings and a lecture manuscript, with weekly alternating Q&A sessions and exercise sessions. The lecture is accompanied by intensive computer exercises offered in Python (6 ECTS) and an optional project (3 ECTS). The project consists in the formulation and implementation of a self-chosen optimization problem or numerical solution method, resulting in documented computer code, a project report, and a public presentation. Please check the website for further information.
necessary: Analysis I–II, Linear Algebra I–II
useful: Introduction to Numerics
Compulsory Elective in Mathematics
Advanced Course in Schemes
Lecturer: Andreas Demleitner
Language: in German
Lecture: Mo, 12-14h, SR 403, Ernst-Zermelo-Str. 1
Exercise session: Do, 14-16h, -, -
Compulsory Elective in Mathematics
Organisation: Susanne Knies
Language: in German
What characterizes a good tutorial? This question will be discussed in the first workshop and tips and suggestions will be given. Experiences will be shared in the second workshop.
Elective
Introduction to Programming for Science Students
Lecturer: Ludwig Striet
Language: in German
Lecture: Mo, 16-18h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
none
BOK course
Computer exercises in Numerics
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in German
In the practical exercises accompanying the Numerics II lecture, the algorithms developed and analysed in the lecture are implemented in practice and tested experimentally. The implementation is carried out in the programming languages Matlab, C++ and Python. Elementary programming skills are assumed.
See the lecture Numerics II.
In addition elementary programming knowledge.
Numerics
Undergraduate seminar: One-Dimensional Maximum Principle
Lecturer: Guofang Wang
Assistant: Xuwen Zhang
Language: in German
Seminar: Mi, 16-18h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 05.02., 16:00, SR 125, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Analysis I and II
Undergraduate Seminar
Lecturer: Susanne Knies
Assistant: Vivien Vogelmann
Language: in German
Seminar: Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Preregistration: until 30.01. to Vivien Vogelmann
Preliminary Meeting 04.02., 12:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
The seminar deals with statements that are known from finite-dimensional vector spaces but no longer apply in the infinite-dimensional case. What are the consequences of this? What applies instead? With which additional conditions can one possibly save oneself?
For more detailed information see the webpage!
Analysis I, II and Linear Algebra I, II
Undergraduate Seminar
Undergraduate seminar: Lattice Theory
Lecturer: Markus Junker
Language: in German
Seminar: Mo, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Preregistration: until 31.01. to Markus Junker
Preliminary Meeting 07.02., 11:15, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Lattices are similarly basic mathematical structures as orderings or groups. A lattice is a set with two associative and commutative operations \(\cap\) and \(\cup\) that satisfy the absorption laws \(a \cap (a \cup b) = a\) and \(a \cup (a \cap b) = a\). For example, the subsets of a fixed set form a lattice; or the sub vector spaces of a fixed vector space if \(\cup\) is interpreted as the sub vector space generated by the set-theoretic union . Lattices with special additional properties are, for example, Boolean algebras,
In this seminar, we will look at what can be said about arbitrary lattices and then at some results about more specialised lattices.
Linear Algebra I and II, Analysis I
Undergraduate Seminar
Undergraduate seminar: Counter-Examples in Probability Theory
Lecturer: David Criens
Assistant: Andreas Demleitner
It is not only theorems, proofs or illustrative examples, but also counterexamples that show the depth and beauty of a theory. Natural questions are: (a) are the requirements of a theorem necessary and not only sufficient; (b) are the requirements sufficient and not just necessary; (c) is an implication an equivalence, i.e. does the implication in the other direction also hold.
In this undergraduate seminar we deal with counter-examples from probability theory. Possible topics range from classic questions such as measurability, independence of random variables, expectations or conditional probabilities, to more advanced topics such as limit value rates, martingals or Markov processes. A suitable topic can be found for any interested student.
Elementary Probability Theory I (topics from probability theory I–III can also be assigned)
Undergraduate Seminar
Lecturer: Diyora Salimova
Assistant: Ilkhom Mukhammadiev
Language: Talk/participation possible in German and English
Seminar: Mo, 12-14h, online, -
Preregistration: by e-mail to Diyora Salimova
Preliminary Meeting 14.04., 15:00, via zoom (please write the lecturer in case the time slot does not fit you)
Preparation meetings for talks: Dates by arrangement
In recent years, deep learning have been successfully employed for a multitude of computational problems including object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of differential equations. Such simulations indicate that neural networks seem to admit the fundamental power to efficiently approximate high-dimensional functions appearing in these applications.
The seminar will review some classical and recent mathematical results on approximation properties of deep learning. We will focus on mathematical proof techniques to obtain approximation estimates on various classes of data including, in particular, certain types of PDE solutions.
Basics of functional analysis, numerics of differential equations, and probability theory
Compulsory Elective in Mathematics
Mathematical Seminar
Lecturer: Wolfgang Soergel
Assistant: Damian Sercombe
Language: Talk/participation possible in German and English
Seminar: Do, 10-12h, SR 403, Ernst-Zermelo-Str. 1
Preregistration: by e-mail to Wolfgang Soergel
Preliminary Meeting 28.01., 14:15, SR 127, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
This seminar is intended to introduce to the theory of linear algebraic groups. Linear algebraic groups are generalizations of the matrix groups known from linear algebra.
I imagine a format in which I or Sercombe lecture and in between the seminar participants give talks. The seminar is a reasonable addition to the commutative algebra lecture. Reference to that lecture increase in the course of the seminar.
Algebra and Number Theory (where the details of Galois theory and field theory are less relevant than the general theory of groups and rings) and Linear Algebra.
Compulsory Elective in Mathematics
Mathematical Seminar
Lecturer: Ernst Kuwert
Assistant: Florian Johne
Seminar: Di, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 04.02., 12:15, SR 218, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Analysis I–III
Compulsory Elective in Mathematics
Mathematical Seminar
Lecturer: Nadine Große
Assistant: Maximilian Stegemeyer
Language: Talk/participation possible in German and English
Seminar: Di, 12-14h, SR 226, Hermann-Herder-Str. 10
Preliminary Meeting 04.02., 10:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Differential Geometry I
Compulsory Elective in Mathematics
Mathematical Seminar
Lecturer: Heike Mildenberger
Assistant: Maxwell Levine
Language: Talk/participation possible in German and English
Seminar: Di, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Preliminary Meeting 29.01., 13:15, Raum 313, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Set Theory
Compulsory Elective in Mathematics
Mathematical Seminar
Lecturer: Harald Binder
Language: Talk/participation possible in German and English
To answer complex biomedical questions from large amounts of data, a wide range of analysis tools is often necessary, e.g. deep learning or general machine learning techniques, which is often summarized under the term ``Medical Data Science''. Statistical approaches play an important rôle as the basis for this. A selection of approaches is to be presented in the seminar lectures that are based on recent original work. The exact thematic orientation is still to be determined.
Good knowledge of probability theory and mathematical statistics.
Compulsory Elective in Mathematics
Mathematical Seminar
Seminar: Numerics of Partial Differential Equations
Lecturer: Sören Bartels
Assistant: Vera Jackisch, Tatjana Schreiber
Language: Talk/participation possible in German and English
Seminar: Mo, 12-14h, SR 226, Hermann-Herder-Str. 10
Preliminary Meeting 04.02., 12:00, Raum 209, Hermann-Herder-Str. 10, Or registration by e-mail to Sören Bartels.
The seminar will cover advanced topics in the theory and numerics of partial differential equations. This includes the iterative solution of the resulting linear systems of equations with multigrid and domain decomposition methods, the adaptive refinement of finite element grids, the derivation of an approximation theory with explicit constants, and the solution of nonlinear problems.
Introduction to Theory and Numerics of Partial Differential Equations
Compulsory Elective in Mathematics
Mathematical Seminar
Seminar on p-adic Geometry
Lecturer: Abhishek Oswal
Assistant: Ben Snodgrass
Language: in English
Seminar: Mo, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Preliminary Meeting 13.02., 14:30, SR 404, Ernst-Zermelo-Str. 1, Please email Abhisehk Oswal, and Ben Snodgrass if you are interested in the seminar but cannot make it to the preliminary meeting.
It has become clear over the last several decades that \(p\)-adic techniques play an indispensable role in arithmetic geometry. At an elementary level, \(p\)-adic numbers provide a compact and convenient language to talk about congruences between integers. Concretely, just as the field of real numbers \(\mathbb R\) arise as the completion of the field \(\mathbb Q\) of rational numbers with respect to the usual notion of distance on \(\mathbb Q\), the field \(\mathbb Q_p\) of \(p\)-adic numbers arise as the completion of \(\mathbb Q\) with respect to an equally natural \(p\)-adic metric. Roughly, in the \(p\)-adic metric, an integer \(n\) is closer to \(0\), the larger the power of the prime number \(p\) that divides it. A general philosophy in number theory is then to treat all these completions \(\mathbb R\), \(\mathbb Q_p\) of the field \(\mathbb Q\) on an equal footing. As we shall see in this course, familiar concepts from real analysis (i.e. notions like analytic functions, derivatives, measures, integrals, Fourier analysis, real and complex manifolds, Lie groups...), have completely parallel notions over the \(p\)-adic numbers.
While the Euclidean topology of \(\mathbb R^n\) is rather well-behaved (so one may talk meaningfully about paths, fundamental groups, analytic continuation, ...), the \(p\)-adic field \(\mathbb Q_p\) on the other hand is totally disconnected. This makes the task of developing a well-behaved notion of global \(p\)-adic analytic manifolds/spaces rather difficult. In the 1970s, John Tate’s introduction of the concept of rigid analytic spaces, solved these problemsand paved the way for several key future developments in \(p\)-adic geometry.
The broad goal of this course will be to introduce ourselves to this world of \(p\)-adic analysis and rigid analytic geometry (due to Tate). Along the way, we shall see a couple of surprising applications of this circle of ideas to geometry and arithmetic. Specifically, we plan to learn Dwork’s proof of the fact that the zeta function of an algebraic variety over a finite field is a rational function.
Field theory, Galois theory and Commutative algebra.
Some willingness to accept unfamiliar concepts as black boxes. Prior experience with algebraic number theory, or algebraic geometry will be beneficial but not necessary.
Compulsory Elective in Mathematics
Mathematical Seminar
Seminar: The Wiener Chaos Decomposition and (Non-)Central Limit Theorems
Lecturer: Angelika Rohde
Assistant: Gabriele Bellerino
Language: Talk/participation possible in German and English
Compulsory Elective in Mathematics
Mathematical Seminar