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
Teacher: Diyora Salimova
Assistant: Ilkhom Mukhammadiev
Language: Talk/participation possible in German and English
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
Mathematical Seminar
Elective
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
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Language: Talk/participation possible in German and English
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.
Mathematical Seminar
Elective
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
Teacher: Ernst Kuwert
Assistant: Florian Johne
Analysis I–III
Mathematical Seminar
Elective
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
Teacher: Nadine Große
Assistant: Maximilian Stegemeyer
Language: Talk/participation possible in German and English
Differential Geometry I
Mathematical Seminar
Elective
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
Teacher: Heike Mildenberger
Assistant: Maxwell Levine
Language: Talk/participation possible in German and English
Set Theory
Mathematical Seminar
Elective
Teacher: 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.
Mathematical Seminar
Elective
Seminar: Numerics of Partial Differential Equations
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.
Teacher: Sören Bartels
Assistant: Vera Jackisch, Tatjana Schreiber
Language: Talk/participation possible in German and English
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
Mathematical Seminar
Elective
Seminar on p-adic Geometry
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.
Teacher: Abhishek Oswal
Assistant: Ben Snodgrass
Language: in English
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.
Mathematical Seminar
Elective
Seminar: The Wiener Chaos Decomposition and (Non-)Central Limit Theorems
Teacher: Angelika Rohde
Assistant: Gabriele Bellerino
Language: Talk/participation possible in German and English
Mathematical Seminar
Elective