Vorstellung der Schwerpunktgebiete des Mathematischen Instituts
(für Bachelor-Studierende ab dem 3. Semester)
Donnerstag, 13.10.22, 12:00-13:00, Hörsaal Weismann-Haus, Albertstr. 21a
Vorstellung der Schwerpunktgebiete des Mathematischen Instituts: Worum geht es in den Gebieten und welche Vorlesungen sollte man gehört haben, um eine Bachelor-Arbeit in dem Gebiet schreiben zu können?
Yamabe Metriken mit Nullwertiger Skalarkrümmung
Montag, 24.10.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Wir werden eine bestimmte\nEigenschaft auf nicht-kompakten Riemannschen Mannigfaltigkeiten definieren, welche die\nExistenz vollständiger Yamabe Metriken mit nullwertiger Skalarkrümmung impliziert.\nDarüber hinaus werden wir zeigen, dass diese Eigenschaft auf asymptotisch flachen\nMannigfaltigkeiten mit einer gewissen Abfallsrate für die Skalarkrümmung immer erfüllt\nist. Zum Abschluss dieses Vortrags werden wir einen Ausblick zur Gültigkeit dieser\nEigenschaft auf asymptotisch lokal flachen Mannigfaltigkeiten geben.\n
Shape recognition in 3D point clouds
Dienstag, 25.10.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Used in many applications such as 3D-vision, logistics, and mapping, the point cloud has become one of the most important data types in recent years.\nHowever, due to their nature of being unstructured and unordered data, we face difficulties while processing them e.g. for shape recognition purposes.\nOur goal will be to develop techniques to classify point clouds into a predefined number of shapes. We will tackle this problem with a machine-learning approach and provide three types of neural networks operating on point clouds. Furthermore, we will prove a version of the Universal Approximation Theorem for neural networks operating on point clouds to mathematically prove the foundation of one of our neural network types.\n\nLastly, we will extract information on the data by describing some geometric invariances of the shapes to classify for. We will present our results, and the difficulties we faced as well as provide some tips on how to overcome them and give suggestions for future work and improvement. \n\nThe talk is suitable for anyone who has finished the basic mathematic lectures. It is of use to know the fundamentals of machine learning but we will briefly revise all the definitions and concepts necessary.
Dividing lines in positive model theory
Dienstag, 25.10.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Positive logic is first-order logic where formulas are built without negation and using only existential quantifiers. By choosing the right languages to work with, this turns out to be a proper generalization of first-order logic. It is then natural to ask how much of usual model theory we can transfer to this setting; for example, one might ask about the dividing lines in classification theory, such as stability and simplicity: these notions, mostly introduced by Shelah, have been fruitfully used to classify theories with respect to various properties, for example the number of their models in a given cardinality or the existence of certain independence relations.\nIn this talk I will briefly introduce positive model theory and some of the ideas about dividing lines, before discussing some work in progress (joint with Anna Dmitrieva and Mark Kamsma) about their interplay.
Mathematik-Tag für Schülerinnen und Schüler
Freitag, 28.10.22, 08:30-09:30, Raum 226, Hermann-Herder-Str. 10
Higher multiplier ideals
Freitag, 28.10.22, 10:00-11:00, Hörsaal II, Albertstr. 23b
For any Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many important applications in algebraic geometry. It turns out that this is only a small piece of a larger picture. In this talk, I will discuss the construction of a family of ideal sheaves indexed by an integer indicating the Hodge level, called higher multiplier ideals, such that the lowest level recovers the usual multiplier ideals. We describe their local and global properties: the local properties rely on Saito's theory of rational mixed Hodge modules and a small technical result from Sabbah's theory of twistor D-modules; while the global properties need Sabbah-Schnell's theory of complex mixed Hodge modules and Beilinson-Bernstein’s theory of twisted D-modules from geometric representation theory. I will also compare this with the theory of Hodge ideals recently developed by Mustata and Popa. If time permits, I will discuss some application to the singularity of theta divisors on principally polarized abelian varieties. This is joint work with Christian Schnell.\n
Whittaker Fourier type solutions to differential equations arising from string theory
Montag, 31.10.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk, we find the full Fourier expansion of some special functions describing the graviton scattering in the string theory. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Using\nmanipulations with divergent series, we obtain a class of\nformulas evaluating an infinite sum of divisor functions, including a\nsurprising equality\n\[\n\bsum d(|n_1|) d(|n_2|) ( (n_2-n_1) \blog( | n_1/n_2 | ) + 2 ) =\n(2-\blog(4 \bpi^2 |n|) ) d(|n|),\n\]\nwhere \(\bsum\) denotes the sum over all possible non-zero integers\n\(n_1\) and \(n_2\) such that \(n_1+n_2=n\).\n\nThis is a joint work with Kim Klinger-Logan.