Deformations of path algebras of quivers with relations from a geometric perspective
Montag, 4.11.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Path algebras of quivers with relations naturally appear in algebraic geometry as endomorphism algebras of so-called tilting bundles. In this setting, deformations of such quotients of path algebras can be used to give a concrete description of deformations of the category of (quasi)coherent sheaves as Abelian category, which are known to combine both "classical" deformations of the variety and "noncommutative" algebraic deformation quantizations.\n\nIn this talk I will present recent joint work with Zhengfang Wang for describing deformations of path algebras of quivers with relations algebraically / combinatorially. I plan on focussing on examples of geometric origin and will try to explain such deformations from a geometric perspective.
Experimentelle Analyse der Energetischen Skalierung einer Einzelnen Falte und eines Stufenversetzten Kreisrings Mittels Minimierung des Elastischen Energiefunktionals
Dienstag, 5.11.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Auf rationale Weise zu irrationalen Zahlen
Dienstag, 5.11.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Die Entdeckung irrationaler Größenverhältnisse in der griechischen Antike hat das damalige mathematische Weltbild grundlegend erschüttert. Dabei gelangt man zu dieser Entdeckung ganz „rational“, wenn man dieses Wort in seiner weit gefassten Bedeutung des verstandesmäßigen Vorgehens und exakten logischen Schließens versteht. Der mathematische Fachbegriff „irrationale Zahl“ hat demgegenüber die enger gefasste Bedeutung von „keine Verhältniszahl“. Eine geeignete Behandlung im Unterricht kann Schülerinnen und Schülern den Blick für Grundlagenfragen öffnen und eine Facette der Mathematik erschließen, die leicht zu kurz kommt. Dabei kann auch aufgezeigt werden, wie weit sich der Begriff „Irrationalität“ im gesellschaftlichen Diskurs mittlerweile von seiner mathematischen Ursprungsbedeutung gelöst hat.\n\n
Dynamic learning based on random recurrent neural networks and reservoir computing systems
Donnerstag, 7.11.19, 18:00-19:00, Hörsaal II, Albertstr. 23b
In this talk we present our recent results on a mathematical explanation for the empirical success of dynamic learning based on reservoir computing.\nMotivated by their performance in applications ranging from realized volatility forecasting to chaotic dynamical systems, we study approximation and learning based on random recurrent neural networks and more general reservoir computing systems. For different types of echo state networks we obtain high-probability bounds on the approximation error in terms of the network parameters. For a more general class of reservoir computing systems and weakly dependent (possibly non-i.i.d.) input data, we then also derive generalization error bounds based on a Rademacher-type complexity.\n\nThe talk is based on joint work with Lyudmila Grigoryeva and Juan-Pablo Ortega.
Uniformization of dynamical systems and diophantine problems
Freitag, 8.11.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
This is joint work with Gareth Boxall (Stellebosch University) and Gareth Jones (University of Manchester). We investigate certain number theoretic properties of polynomial dynamical systems, using the notion of a uniformization at infinity. In this talk I will explain how the ideas involved can be used in order to tackle various related problems\n on diophantine geometry.\n
Learning compositional structures
Freitag, 8.11.19, 12:00-13:00, Raum 403, Ernst-Zermelo-Str. 1
Many data problems, in particular in biogenetics, often come with a highly complex underlying structure. This often makes is difficult to extract interpretable information. In this talk we want to demonstrate that often these complex structures are well approximated by a composition of a few simple parts, which provides very descriptive insights into the underlying data generating process. We demonstrate this with two examples.\n \nIn the first example, the single components are finite alphabet vectors (e.g., binary components), which encode some discrete information. For instance, in genetics a binary vector of length n can encode whether or not a mutation (e.g., a SNP) is present at location i = 1,…,n in the genome. On the population level studying genetic variations is often highly complex, as various groups of mutations are present simultaneously. However, in many settings a population might be well approximated by a composition of a few dominant groups. Examples are Evolve&Resequence experiments where the outer supply of genetic variation is limited and thus, over time, only a few haplotypes survive. Similar, in a cancer tumor, often only a few competing groups of cancer cells (clones) come out on top. \n \nIn the second example, the single components relate to separate branches of a tree structure. Tree structures, showing hierarchical relationships between samples, are ubiquitous in genomic and biomedical sciences. A common question in many studies is whether there is an association between a response variable and the latent group structure represented by the tree. Such a relation can be highly complex, in general. However, often it is well approximated by a simple composition of relations associated with a few branches of the tree. \n \nFor both of these examples we first study theoretical aspects of the underlying compositional structure, such as identifiability of single components and optimal statistical procedures under probabilistic data model. Based on this, we find insights into practical aspects of the problem, namely how to actually recover such components from data.
A (possible) regularization of the Selberg zeta function
Montag, 11.11.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Selberg zeta functions are zeta functions associated with the geodesic flow on a hyperbolic surface and, sometimes, a representation of the fundamental group of the surface. The spectral theory of the Selberg zeta for unitary representations is well-known, but, however, for certain no-unitary representations the Selberg zeta function may even not exist.\n\nIn the talk, I would like to suggest a way of the regularization of the Selberg zeta function to such types of non-unitary representations using the transfer operator approach.
Stable sets and additive combinatorics
Mittwoch, 13.11.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Given a subset A of a finite abelian group G, we denote by A+A the subset of elements of G which are sum of two elements of A. A fundamental question in additive combinatorics is to determine the structure of subsets A satisfying that A+A has size at most K times the size of A, where K is a fixed parameter. It is easy to verify that these subsets are translates of subgroups when K=1. Furthermore, for arbitrary K and for abelian groups of bounded exponent, a celebrated theorem of Ruzsa asserts that A is covered by a finite union of translates of subgroups, whose sizes are commensurable to the size of A. Improvements of this result have been subsequently obtained by many authors such as Green, Tao and Sanders, as well as Hrushovski who obtained an analogous result for non-abelian groups using model theoretic tools.\n\nIn this talk I shall present a model theoretic version of Ruzsa's theorem for subsets A satisfying suitable model theoretic conditions, such as stability. This is joint work Amador Martin-Pizarro and Julia Wolf.\n
Poisson geometry and beyond
Montag, 18.11.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Poisson brackets appear in different fields of mathematics and physics, such as the theory of Lie algebras and classical mechanics. \nThis talk is meant to give a short introduction to the subject of Poisson geometry in general and, afterwards, to discuss the connection of Poisson structures and Lie groupoids/algebroids, symplectic geometry and deformation quantization. \n
On the blow up set for the Seiberg-Witten equations with two spinors
Montag, 18.11.19, 17:00-18:00, Raum 404, Ernst-Zermelo-Str. 1
Simulation elastischer Kurven auf Oberflächen
Dienstag, 19.11.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Vorgestellt wird ein Algorithmus zur Minimierung der Biege- \nund Torsionsenergie einer elastischen Kurve, \ndie auf eine Fläche im Raum eingeschränkt ist.\nDie Gültigkeit eines Energiegesetzes und Fehlerkontrollen \nfür die Nebenbedingungen werden an verschiedenen Beispielen überprüft. \nAnschließend wird noch eine Anwendung präsentiert.\n\n
Apps for understanding - Digitale Verstehensunterstützung im Mathematikunterricht
Dienstag, 19.11.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Apps – also thematisch fokussierte und flexibel nutzbare Programme – versprechen eine Verbesserung des Mathematikunterrichts durch Digitalisierung. Leider sind zurzeit viele solcher Digitalisierungsangebote immer noch zu wenig lerntheoretisch und fachdidaktisch fundiert. Ihr didaktisches Potential können sie erst entfalten, wenn sie nicht nur additives Drill & Practice anbieten, sondern in den Lernprozess eingebunden sind. Anhand von Beispielen und Kriteriensystemen werden Apps in Form interaktiver Simulationen vorgestellt und aufgezeigt, wie sie Verstehensorientierung und kognitive Aktivierung beim Entdecken und Problemlösen im Fach Mathematik unterstützen können – vom Zahlverständnis in Klasse 1 über das funktionale Denken oder Wahrscheinlichkeiten in der Mittel- und Oberstufe, bis hin zur Algebra im Mathematikstudium.
Donnerstag, 21.11.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Automorphisms of foliations
Freitag, 22.11.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We will discuss in various contexts the transverse finiteness of the group of automorphisms/birational transformations preserving a holomorphic foliation. This study provides interesting consequences for the distribution of entire curves on manifolds equipped with foliations and suggest some generalizations of Lang’s exceptional loci to non-special manifolds, in the analytic or arithmetic setting. \nThis is a work in progress with F. Lo Bianco, J.V. Pereira and F. Touzet.
Quaternionic Line Bundles
Montag, 25.11.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In contrast to real and complex line bundles, quaternionic line bundles are not classified by a suitable characteristic class. However, there are results on the classification over four and five dimensional spin manifolds.\n\nThis talk is an introduction to quaternionic line bundles and their classification. We will see why the approach to classification in the real and complex cases does not transfer to the quaternionic setting. Based on this, I would like to present different approaches to the classification over spin manifolds of low dimension.
Klein's Quartic, Fermat's Cubic and Rigid Complex Manifolds of Kodaira Dimension One
Freitag, 29.11.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The only rigid curve is \(\bmathbb P^1\). Rigid surfaces exist in Kodaira dimension \(-\binfty\) and \(2\).\nIngrid Bauer and Fabrizio Catanese proved that for each \(n \bgeq 3\) and for each \(\bkappa = -\binfty, 0, 2,\bldots, n\) there is a rigid \(n\)-dimensional projective manifold with Kodaira dimension \(\bkappa\). In this talk we show that the result also holds in Kodaira dimension one.\n\n