Singularity Theorems over Averages on globally hyperbolic Spacetimes
Montag, 2.12.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Unendlichdimensionale Riemannsche Geometrie als mathematisches Fundament von Figurenanalyse und Fluiddynamik
Dienstag, 3.12.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Figurenanalyse und Fluiddynamik haben dasselbe mathematische Fundament, nämlich Riemannsche Geometrie auf unendlich-dimensionalen nicht-linearen Räumen von Abbildungen. In der Figurenanalyse liefert das geodätische Randwertproblem einen Abstandsbegriff und optimale Punkt-zu-Punkt Korrespondenzen zwischen Figuren. In der Fluiddynamik beschreibt das geodätische Anfangswertproblem die zeitliche Dynamik der Flüssigkeit. Der Vortrag bietet eine Einführung in diese Zusammenhänge.
Tame Geometry in Henselian Valued Fields
Mittwoch, 4.12.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In real algebaric geometry, the objects of study are semi-algebraic sets, i.e., subsets of R^n defined using polynomial inequalities. In the 80s, Pillay and Steinhorn introduced o-minimality, a simple axiomatic description of classes of sets for which "geometry works as for semi-algebraic sets". More precisely, the sets in such a class are those which are first-order definable in a suitable language. This axiomatic approach had a huge impact on geometry in R, and many results known for semi-algebraic sets have then be proved in this much more general framework.\n\nSince the invention of o-minimality, various attempts have been made to come up with an analogous notion in (suitable) valued fields like the p-adics or fields of formal Laurent series. Understanding first-order definable sets in such fields has been crucial to obtain rationality of many kinds of Poincaré series, and in the late 90s, it also became the fundament of motivic integration. In this talk, I will present a new analogue of o-minimality for valued fields (a collaboration with Cluckers and Rideau) which is powerful enough so that all these applications (rationality, motivic integration) can be carried out within that framework.\n\nThe talk will only require some very basic knowledge about (some examples of) valued fields and some vague familiarity with o-minimality and/or model theory.\n
Maximal determinants of Schrödinger operators on finite intervals
Freitag, 6.12.19, 10:15-11:15, Raum 318, Ernst-Zermelo-Str. 1
In this talk I will present the problem of finding extremal \npotentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions. We consider potentials in a fixed \(L^q\) space with \(q\bgeq 1\). Functional determinants of Sturm-Liouville operators with smooth potentials or with potentials with prescribed singularities have been widely studied, I will present a short review of these results and will explain how to extend the definition of the functional determinant to potentials in \(L^q\). The maximization problem turns out to be equivalent to a problem in optimal control. I will explain how we obtain existence and uniqueness of the maximizers. The results presented in the talk are join work with J-B. Caillau (UCDA, CNRS, Inria, LJAD) and P. Freitas (Lisboa).
Special vs Weakly-Special Manifolds
Freitag, 6.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A fundamental problem in Diophantine Geometry is to characterize geometrically potential density of rational points on an algebraic variety X defined over a number field k, i.e. when the set X(L) is Zariski dense for a finite extension L of k. Abramovich and Colliot-Thélène conjectured that potential density is equivalent to the condition that X is weakly-special, i.e. it does not admit any étale cover that dominates a positive dimensional variety of general type. More recently Campana proposed a competing conjecture using the stronger notion of specialness that he introduced. We will review both conjectures and present results that support Campana’s Conjecture (and program) in the analytic and function field setting. This is joint work with Erwan Rousseau and Julie Wang.\n\n\n
How implicit regularization of Neural Networks affects the learned function
Freitag, 6.12.19, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
Today, various forms of neural networks are trained to perform approximation tasks in many fields. However, the solutions obtained are not wholly understood. Empirical results suggest that the training favors regularized solutions.\nThese observations motivate us to analyze properties of the solutions found by the gradient descent algorithm frequently employed to perform the training task. As a starting point, we consider one dimensional (shallow) neural networks in which weights are chosen randomly and only the terminal layer is trained. We show, that the resulting solution converges to the smooth spline interpolation of the training data as the number of hidden nodes tends to infinity. This might give valuable insight on the properties of the solutions obtained using gradient descent methods in general settings.
tba
Montag, 9.12.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
tba
Montag, 9.12.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
tba
Diskretisierung eines Selbstvermeidungspotentials bei der Simulation isometrischer Deformationen elastischer Platten
Dienstag, 10.12.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Bei großen Verformungen elastischer Platten kommt es häufig zum Selbstkontakt. Um bei der Simulation solche Selbstdurchdringungen zu vermeiden, verwenden wir eine Punkt-Tangenten-Energie. Auf dem Finite-Elemente Raum der diskreten Kirchhoffdreiecke definieren wir eine Quadratur der Energie. Mittels Gradientenfluss erhalten wir einen Algorithmus, der die Krümmungs- und Selbstvermeidungsenergie minimiert. Wir testen das so resultierende iterative Verfahren anhand einiger Simulationen auf Stabilität und Anwendbarkeit.
Graphs of bounded shrub-depth and first-order logic
Mittwoch, 11.12.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
We show that the expressive power of monadic second-order logic (MSO)\nand of first-order logic (FO) coincide on classes of graphs of bounded\nshrub-depth. Moreover we explain in what sense these classes are maximal\nclasses with MSO = FO.
Deformations of Hilbert schemes of points via derived categories
Freitag, 13.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Hilbert schemes of points on surfaces, and Hilbert squares of higher-dimensional varieties, are important and basic constructions of moduli spaces of sheaves. As such they provide a class of interesting yet tractable varieties. In a joint work with Lie Fu and Theo Raedschelders, we explain how one can (re)prove results about their deformation theory by studying their derived categories, via fully faithful functors and Hochschild cohomology, which describes both classical and noncommutative deformations.
Asymptotic of twisted Alexander polynomials and hyperbolic volume
Montag, 16.12.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Given a hyperbolic 3-manifold M of finite volume, we study a family of twisted Alexander polynomials of M. We show an asymptotic formula for the behavior of those polynomials on the unit circle, and recover the hyperbolic volume as the limit. It extends previous works of Müller (for M closed) and Menal-Ferrer--Porti. This is a joint work with Jerome Dubois, Michael Heusener (Clermont-Ferrand) and Joan Porti (Barcelona). \n\n
On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators
Dienstag, 17.12.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The so-called Solvability Complexity Index (SCI) can be defined as the number\nof independent limits required to solve a given computational problem.\nWe study the SCI for the spectral problem of unbounded selfadjoint operators\non separable Hilbert spaces and perturbations thereof.\nIn particular, we show that if the extended essential spectrum of a selfadjoint operator is convex,\nthen the SCI for computing its spectrum is equal to 1. This result is then extended to relatively\ncompact perturbations of such operators and applied to Schrödinger operators with\n(complex valued) potentials decaying at infinity to obtain SCI=1 in this case, as well.\n\n
Mathematik ist schön
Dienstag, 17.12.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Nicht jeder denkt, wenn von Mathematik die Rede ist, unbedingt an etwas Schönes, an etwas, an dem man sich erfreuen kann. Dabei hat die Mathematik viele spannende und durchaus auch ästhetisch schöne Aspekte zu bieten. Und wenn man sich mit den Erkenntnissen und Ideen längst verstorbener Mathematiker beschäftigt, dann kommt man oft aus dem Staunen nicht heraus.\nIm Vortrag sollen an einige dieser „schönen“ Einsichten erinnert werden, mit denen der Mathematikunterricht bereichert werden kann.
Cohen reals und P-messbare Mengen
Mittwoch, 18.12.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Eine reelle Zahl heißt Cohen real, falls die Menge ihrer\nendlichen Anfangsstücke einen generischen Filter für das Cohen-Forcing\ndefiniert. Es folgt, dass Cohen reals keine Elemente des Grundmodells sein\nkönnen.\nFür eine Halbordnung P kann man die topologischen Eigenschaften "nirgends\ndicht" und "mager" sowie den Begriff der Messbarkeit verallgemeinern.\nIst die Halbordung P das Cohen Forcing, so entsprechen P-nirgends dicht\nund P-mager gerade ihren topologischen Definitionen und P-messbar der\nBaire-Eigenschaft.\nFür zwei Halbordnungen P und Q ergibt sich die interessante Fragestellung\nnach einem Zusammenhang der beiden Definitionen von Messbarkeit. Wenn Q das\nCohen Forcing ist, scheint es außerdem der Fall zu sein, dass es schon\ngenügt zu wissen, ob P Cohen reals addiert, um beantworten zu können, ob es\neinen Zusammhang zwischen P- und Q-messbar gibt.\n\nIn dem Vortrag stelle ich eine neue Forcinghalbordnung T vor. Ich werde\nexemplarisch an ihr zeigen, wie sich aus dem Nachweis von Cohen reals ein\nZusammenhang von T-messbar und der Baire Eigenschaft herstellen lässt. \nDer Vortrag beruht auf dem Paper "More on trees and Cohen reals", das in\nZusammenarbeit mit Giorgio Laguzzi entstanden ist.\n\n
Donnerstag, 19.12.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Smoothing Normal Crossing Spaces
Freitag, 20.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Given a normal crossing variety \(X\), a necessary condition for it to\noccur as the central fiber \(f^{-1}(0)\) of a semistable degeneration \(f:\n\bmathcal{X} \bto \bDelta\) is \(\bmathcal{T}^1_X \bcong \bmathcal{O}_D\) for the\ndouble locus \(D \bsubset X\). Sufficient conditions have been given\nfamously by Friedman for surfaces and by Kawamata-Namikawa in any\ndimension. We give sufficient conditions for smoothing more general\nnormal crossing varieties with \(\bmathcal{T}^1_X\) only globally generated\nby relaxing the condition that the total space \(\bmathcal{X}\) should be\nsmooth. Our main technical tool is the degeneration of a spectral\nsequence in logarithmic geometry that also settles a conjecture of\nDanilov on the cohomology of toroidal pairs.