Noetherianity and equationality
Tuesday, 2.5.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A theory is noetherian if there is a family of definable sets with the descending chain condition such that every definable set is a boolean combination of those in the family. Noetherianity captures some of the desired properties of algebraically closed fields in any characteristic or differentially closed fields in characteristic 0. Noetherian theories are in particular omega-stable and equational. In recent work with M. Ziegler, we have shown that the theory of proper pairs of algebraically closed fields in any characteristic is noetherian.
Shifted Lagrangian structures in Poisson geometry
Tuesday, 2.5.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
It is well known that BG carries a 2-shifted symplectic structure. In this talk, I will study the shifted lagrangian groupoids of BG. I will show how many constructions on Poisson geometry unify using the language of shifted symplectic groupoids. This is work in progress with Daniel Alvarez and Henrique Bursztyn.
Entwicklung algebraischen Denkens in der Schule
Tuesday, 2.5.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Das Verstehen und der flexible Gebrauch von Variablen, Termen und Gleichungen ist ein wichtiges Ziel im Mathematikunterricht. Der souveräne Umgang mit diesen algebraischen Objekten ist ein wichtiges Instrumentarium bei inner- und außermathematischen Problemstellungen, insbesondere im Rahmen der Funktionenlehre. Im Vortrag wird erörtert, welche Aspekte verstanden werden müssen und welche Fehlvorstellungen hierbei hinderlich sind. Es werden Erkenntnisse aus verschiedenen Forschungsprojekten vorgestellt, die den Blick für Lehrkräfte schärfen können, algebraisches Denken erfolgreich zu entwickeln.
Heights on commutative algebraic groups
Friday, 5.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
In diophantine geometry a height function is measure of the complexity of an algebraic number. This talk is a presentation of my master thesis and will look at height estimates on connected commutative algebraic groups for taking integral multiples of points. Such estimates are used in the Analytic Subgroup Theorem by Gisbert Wüstholz.
Workshop: Numerik in BaWü
Monday, 8.5.23, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
Dreitägiger Workshop mit Sprecher:innen verschiedener Universitäten in Baden-Württemberg
Paths towards Open World Generalization
Monday, 8.5.23, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
ODD Riemannian metrics
Monday, 8.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We describe a generalization of Riemannian metrics motivated from\nKähler geometry of singular complex varieties. These generalizations\nare semipositive symmetric 2-tensors, but degenerate in such way, that\ne.g. they still induce a metric space structure on the underlying\nmanifold.\n\nIn this talk, we will mostly use instructive examples to sketch how far\nRiemannian Geometry can (hopefully) be pursued for these ODD metrics.\n
Acceleration of quantum mechanical systems by exploiting similarity
Tuesday, 9.5.23, 09:00-10:00, Raum 226, Hermann-Herder-Str. 10
Total (generalized) variation for images and shapes
Tuesday, 9.5.23, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
A categorical perspective on scattering amplitudes
Friday, 12.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
Scattering amplitudes are physical observables which play a central role in interpreting scattering experiments at particle colliders. In recent years, a new perspective on scattering amplitudes, known as amplituhedron programme, has revealed a fascinating link to various mathematical structures of a combinatorial nature, such as positive Grassmannians and cluster algebras. In this talk I will explain this connection from the point of view of derived and cluster categories of type A quivers, from which the formulae for scattering amplitudes can be obtained from projectives of hearts of intermediate t-structures. This talk is based on arXiv:2101.02884 joint with K. Ray and on arXiv:2112.14288 joint with P. Oak, A. Pal, K. Ray and H. Treffinger.
Shifted convolution sums
Monday, 15.5.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
In talk I will evaluate shifted convolution sums of divisor functions of the form \(\bdisplaystyle\bsum_{n_1,n_2\bin\bmathbb{Z}\bsetminus\b{0\b}, n_1+n_2=n}\nQ_{d}^{(r_1,r_2)}\bBig(\bfrac{n_2-n_1}{n_1+n_2}\bBig)\bsigma_{r_1}(n_1)\bsigma_{r_2}(n_2)\) where \(\bsigma_{r}(n) = \bsum_{d \bmid n} d^ r\) and \(Q_{d}^{(r_1,r_2)}(x)\) is the Jacobi function of the second kind. These sums can be considered as a shifted version of the Ramanujan sum \(\bsum_{n_1 \bin \bmathbb{Z}} \bsigma_{r_1}(n_1) \bsigma_{r_2}(n_1) n_1^s\). \n\nKey words that appear in the proof and the final result: non-holomorphic Eisenstein series, cusp forms, values of \(L\)-functions, Mellin transform and Whittaker's functions.
Nonlinear bending-torsion theory for inextensible tapered rods by \(\bGamma\)-convergence
Tuesday, 16.5.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The upcoming presentation will focus on a modified version of the main theorem initially stated and proven by Mora and Mueller in their\nwork titled "Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-Convergence", published in 2003. Specifically, we will direct our attention towards studying a rod with a non-constant cross-section, known as a taper, and answer the question of how the \(\bGamma\)-limit of the elastic energy changes in this case. Additionally, we will combine this new result with previous findings, using banana plants (Musa sp.) as an example.\n
Statistical Learning for Structured Models: Tree Based Methods and Neural Networks
Friday, 19.5.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Estimation in regression and classification problems which include low dimensional structures are considered. The underlying question is the following. How well do statistical learning methods perform for models with low dimensional structures? We approach this question using various algorithms in various settings. First, we introduce a new tree based algorithm we named random planted forest. It adapts particularly well to models which consist of low dimensional structures. We examine its performance in simulation studies and include some theoretical backing by proving optimal convergence rates in certain settings for a modification of the algorithm. A generalized version of the algorithm is included, which can be used in classification settings. Furthermore, we prove optimal convergence rates in a classification setting using neural networks. While non-optimal rates existed for this problem, we are the first to prove optimal ones.
The exterior Dirichlet problem for the homogeneous k-Hessian equation
Friday, 19.5.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
We studied the exterior Dirichlet problem for the homogeneous k-Hessian equation in \(R^n\). The main idea are the uniform gradient estimates and second derivative estimates. Then we use these estimates to study the Green function in a \(k-1\) convex domain in \(R^n\). This is joint works with Zhang dekai and Gao zhenghuan.
Metric inequalities with positive scalar curvature
Monday, 22.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We will discuss various situations where a certain perturbation of the Dirac operator on spin manifolds can be used to obtain distance estimates from lower scalar curvature bounds. \n\nA first situation consists in an area non-decreasing map from a Riemannian spin manifold with boundary \(X\) into the round sphere under the condition that the map is locally constant near the boundary and has nonzero degree. Here a positive lower bound of the scalar curvature is quantitatively related to the distance from the support of the differential of f and the boundary of \(X\). \n\nA second situation consists in estimating the distance between the boundary components of Riemannian “bands” \(M×[−1,1]\) where \(M\) is a closed manifold that does not carry positive scalar curvature. Both situations originated from questions asked by Gromov. \n\nIn the final part, I will compare the Dirac method with the minimal hypersurface method and show that if \(N\) is a closed manifold such that the cylinder \(N \btimes \bmathbb{R}\) carries a complete metric of positive scalar curvature, then \(N\) also carries a metric of positive scalar curvature. This answers a question asked by Rosenberg and Stolz. Based on joint work with Daniel Raede and Rudolf Zeidler.
"Can you escape?" Escape-Aktivitäten im Mathematikunterricht
Tuesday, 23.5.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Escape Rooms haben sich seit einiger Zeit als ein sehr erfolgreiches Unterhaltungsformat etabliert. In der Regel werden Besucher in kleinen Gruppen in einen speziell präparierten Raum "eingesperrt" und müssen eine Vielzahl von Rätseln lösen, um zu entkommen. Aufgrund ihrer interaktiven und herausfordernden Natur sind Escape Rooms besonders motivierend für die Teilnehmer*innen. Ein erfolgreicher Fluchtversuch erfordert jedoch die Anwendung von Problemlösungsstrategien, Frustrationstoleranz und die Zusammenarbeit im Team. In diesem Vortrag werden verschiedene Escape Rooms vorgestellt, die sich auf mathematische Rätsel konzentrieren. Es werden theoretische Grundlagen zur Konzeption und Gestaltung eines Escape Rooms behandelt und die Möglichkeiten und Grenzen eines sinnvollen Einsatzes im Bildungsbereich diskutiert.