First-order elliptic boundary value problems beyond self-adjoint induced boundary operators
Monday, 7.1.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The Bär-Ballmann framework is a comprehensive framework to consider elliptic boundary value problems (and also their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that the induced operator on the boundary is symmetric. Many operators satisfy this requirement including the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general operators with the quintessential example being the Rarita-Schwinger Dirac operator, which is an operator that fails to satisfy this hypothesis.\n\nIn this talk, I will present recent work with Bär where we dispense the symmetry assumption and consider general elliptic operators. The ellipticity of the operator still allows us to understand the spectral theory of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to recover many of the results of the Bär-Ballman framework. \n\nIf time permits, I will also touch on the non-compact boundary case, and potential extensions of\nthis to the L^p setting and Lipschitz boundary. \n
Thursday, 10.1.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
p-adic variations of automorphic sheaves
Friday, 11.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Elliptic modular forms are a special kind of functions on the Poincare' upper half space and have played an increasingly important role in modern Number Theory. Starting with the works of J.P. Serre and N. Katz more than 30 years ago, it was discovered that, given a prime number p, such modular forms have also a p-adic nature and, especially, live in p-adic families. This phenomenon is the counterpart of the theory fo p-adic deformations of Galois representations and has become a basic tool for number theorists. I will present joint work with A. Iovita and V. Pilloni providing a geometric explanation of this, purely p-adic, phenomenon.\n
Superconformal algebras for twisted connected sums and \(G_2\) mirror symmetry
Monday, 14.1.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Early hints of mirror symmetry for Calabi-Yau manifolds arose from studying certain vertex operator algebras, intimately connected to string dynamics on these geometries. In my recent article 1809.06376, which the talk will be about, I perform a similar analysis, replacing Calabi-Yau manifolds by 7-dimensional \(G_2\) holonomy spaces constructed via the so-called "twisted connected sum" method of Corti, Haskins, Nordstrom, and Pacini. Besides connecting nicely with recent results on the conjectured "mirror symmetry" for \(G_2\), this work is a necessary step for applications of twisted connected sums in string theory.\n\n
Wirksamer Mathematikunterricht
Tuesday, 15.1.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Dass der Mathematikunterricht wirksam sein soll, würde als Ziel wohl kaum in Abrede gestellt werden. Bei näherer Betrachtung stellt sich jedoch die Frage, was "wirksam" eigentlich bedeutet und worin sich eine solche Wirksamkeit zeigt. Um Antworten zu finden, wurden in einer Erhebung insgesamt 19 Lehrkräfte an Hochschulen, Studienseminaren und Schulen zur Wirksamkeit von Mathematikunterricht befragt. Die Antworten wurden auf Gemeinsamkeiten hin untersucht, um so übergreifende Überzeugungen zu wirksamem Mathematikunterricht herauszuarbeiten. Im Vortrag werden diese Gemeinsamkeiten in acht Teilaspekten vorgestellt.
Boundedness results for singular Fano varieties, and applications to Cremona groups
Wednesday, 16.1.19, 10:30-11:30, Hörsaal II, Albertstr. 23b
A normal, projective variety is called Fano if a negative\nmultiple of its canonical divisor class is Cartier and if the associated line bundle is ample. Fano varieties appear throughout geometry and have been studied intensely. The Minimal Model Programme predicts in an appropriate sense that Fanos are one of the fundamental classes of\nvarieties, out of which all other varieties are built.\n\nWe report on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov, asserting that Fano varieties with mild singularities form a bounded family once their dimension is fixed. This has immediate consequences for our understanding of Cremona groups.\nFollowing Prokhorov-Shramov, we explain how Birkar’s boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property; this answers a question of Serre in the\npositive.
Thursday, 17.1.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Computing classes of admissible covers
Friday, 18.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Let Adm(g,h,G) be the space of degree admissible G covers C → D of a genus h curve D by genus g curves C. There is a natural map f : Adm(g,h,G) → Mgnbar into the moduli space of stable curves taking the source curve of an admissible cover and forgetting everything else. When the class [f(Adm(g,h,G))] is tautological we can try to express this class in terms of a known basis for the tautological ring of Mgnbar. We will discuss several strategies for making these computations and give a number of examples.\n
TBA
Friday, 18.1.19, 13:00-14:00, Raum 218, Ernst-Zermelo-Str. 1
Manifolds with singular Riemannian foliations by aspherical leafs
Monday, 21.1.19, 14:30-15:30, Raum 318, Ernst-Zermelo-Str. 1
Singular Riemannian foliations are generalizations of smooth isometric group\nactions. In the setting of compact group actions, torus actions by isometries\non a fixed Riemannian manifold have been studied to understand the topology of\nthe manifold or properties the Riemannian metric might have.\n\nWe extend this study to the setting of singular Riemannian folaitons by tori.\nWe show that some techniques developed for the study of torus actions can be\ncarried to the foliated setting. In particular we focus on the case where the\nfoliation has codimenision 2, in order to fix notions.\n\n\nIn this particular case we obtain the following result:\n\nIf (M,F) is an singular Riemannian foliation of codimension 2 by tori, on a\ncompact, simply-connected Riemannian manifold, then the foliation is induced by\na smooth torus action.\n
The Pontryagin product and geodesic loops on Riemannian 2-spheres (following A.Nabutovsky/R.Rotman)
Monday, 21.1.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
wird noch bekanntgegeben
Tuesday, 22.1.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
On Talagrand's Solution to Maharam's Problem
Wednesday, 23.1.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Abstract:\nEin Submaß \(\bnu\) auf einer Booleschen Algebra \(\bmathcal{B}\) heißt\nausschöpfend, wenn jede disjunkte Folge in \(\bmathcal{B}\) unter \(\bnu\) eine\nNullfolge ist. Zwei Submaße heißen äquivalent, falls sie dieselben\nNullfolgen haben. Eines der beiden Maharamprobleme ist die folgende Frage:\n\n Ist jedes auschöpfende Submaß äquivalent zu einem endlich additiven Maß?\n\n Dieses Problem geht auf eine Frage zurück, die John von Neumann 1937 im\n“Scottish Book” stellte, nämlich ob jede vollständige schwach\n\(\bomega\)-distributive Algebra mit höchstens abzählbaren Antiketten ein\npositives \(\bsigma\)-additives Wahrscheinlichkeitsmaß trägt.\n\n2008 gab Michel Talagrand eine negative Antwort auf das Problem von Maharam\nund damit auch auf von Neumanns Frage. In seinem Beweis konstruiert\nTalagrand zunächst ein pathologisches Submaß, also ein Submaß, das kein\nnichttriviales Maß dominiert.\n\nZiel dieses Vortrags ist, dieses Submaß zu betrachten und zu zeigen, dass\neine ähnliche Konstruktion auf der Cantoralgebra nicht pathologisch ist.
On manifolds with a degree of kinship
Thursday, 24.1.19, 14:15-15:15, Raum 318, Ernst-Zermelo-Str. 1
In 1956, John Milnor exhibited the first examples of manifolds\nhomeomorphic, but not diffeomorphic to spheres, since then called exotic\nspheres. Interesting results on exotic manifolds were obtained through explicit\ngeometric constructions. Here we present a new construction that relates exotic\nmanifolds, such as exotic spheres, flag manifolds and connected sums of them,\nto their standard counterparts. Such relation is established through a Morita\nequivalence of action groupoids and is used to produce/reproduce metrics with\npositive Ricci and almost non-negative curvature. Joint work with L. Cavenaghi.
Evolutionary Gamma convergence for gradient systems
Thursday, 24.1.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Many ordinary and partial differntial equations can be written as a gradient flow, which means that there is an energy functional that drives the evolution of the the solutions by flowing down in the energy landscape. The gradient is given in terms of a dissipation structure, which in the simplest case is a Riemannian metric. We discuss classical and nontrivial new examples in reaction-diffusion systems or friction mechanics. We will emphasize that having a gradient structure for a given differential equation means that we add additional physical information.\n\nConsidering a family of gradient systems depending on a small parameter, it is natural to ask for the limiting (also called effective) gradient system if the parameter tends to 0. This can be achieved on the basis of De Giorgi's Energy-Dissipation Principle (EDP). We discuss the new notion of "EDP convergence" and show by examples that this theory is flexible enough to allow for situations where starting from quadratic dissipation potentials we arrive at physically relevant, effective dissipation potentials that are no\nlonger quadratic, namely exponential laws for transmission at membranes or slip-stick motion on rough surfaces.
Rigidity for equivariant K-theory
Friday, 25.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
This talk is a report on joint work with Jeremiah Heller and Paul Arne Østvær. The Gabber-Gillet-Thomason rigidity theorem asserts that the natural map from a henselian local ring to its residue field induces an isomorphism on algebraic K-theory with finite coefficients (coprime to the exponential characteristic). We establish a version of this rigidity theorem in the setting of homotopy invariant equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic K-theory and presheaves with equivariant transfers.
Characterizing Borcherds-Kac-Moody algebras
Monday, 28.1.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Already in 1995, physicists Harvey and Moore suggested a multiplication on the space of so-called Bogomol'nyi-Prasad-Sommerfield (BPS) states in certain string theories and claimed to obtain Borcherds-Kac-Moody Lie algebras. These were introduced as a generalization of finite-dimensional semisimple Lie algebras, and I will present a theorem showing that they are indeed the widest generalization retaining existence and some well-behavedness of a grading, (Killing) bilinear form and (Cartan) involution. One direction of this theorem was proven and used by Borcherds in his work on monstrous moonshine. The other direction is considered known as well, but its proof has seemingly never been written down completely due to analogy to the existing literature on Kac-Moody algebras. Its verification, however, turned out not to be as easy as expected. I will sketch the main points of what I felt should be added compared to the literature and give some background on the relation to string theory.
wird noch bekanntgegeben
Tuesday, 29.1.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case
Tuesday, 29.1.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we shall present a new result about prescribing\nGaussian curvature on a closed Riemann surface with conical\nsingularities in the negative case. This is a joint work Prof. Yunyan Yang.
Förderung der Raumvorstellung
Tuesday, 29.1.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Unter räumlichem Vorstellungsvermögen versteht man die Fähigkeit, in der Vorstellung räumlich zu sehen und räumlich zu denken. Für viele Bereiche der Mathematik ist eine gute Raumvorstellung notwendig oder zumindest hilfreich. Im Vortrag werden praxiserprobte Möglichkeiten für verschiedene Klassenstufen vorgestellt, wie sich die Raumvorstellung bei den Schülerinnen und Schüler fördern lässt. Dazu zählen neben Übungen mit konkreten Körpern oder Papierfaltübungen auch Übungen, die weitgehend im Sinne einer Kopfgeometrie in der Vorstellung durchgeführt werden. Darüber hinaus wird verdeutlicht, wie sich mit den Übungen die Prozesskompetenzen Kommunizieren, Argumentieren und Problemlösen fördern lassen.
Nonstandard methods in Ramsey Theory.
Wednesday, 30.1.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Abstract: I will present a new technique in nonstandard analysis\nthat has been recently applied in Ramsey Theory of numbers.\n\nTo illustrate the methods, I will present nonstandard proofs\nof fundamental results, such as Ramsey's Theorem and Hindman's Theorem.\nI will also present a couple of simple examples of new results\nabout partition regularity of Diophantine equations.\n\nIn the second part of the talk, I will briefly discuss the\n(discrete) topological dynamics as given by the\nhypernatural numbers of nonstandard analysis\nendowed with the shift operator, and present\na new proof of van der Waerden's Theorem: In any finite\ncoloring of the natural numbers there exist monochromatic\narithmetic progressions of arbitrary length.\n
Numerical Approximation of the Stochastic Cahn-Hilliard Equation Near the Sharp Interface Limit
Thursday, 31.1.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I discuss a stable time discretization of the stochastic\nCahn-Hilliard equation with an additive noise term \(\bvarepsilon^{\bgamma}\n\bdot{W}\), where \(\bgamma >0\), and \(\bvarepsilon>0\) is the interfacial\nwidth parameter. For sufficiently small noise (i.e., for \(\bgamma\) sufficiently\nlarge) and sufficiently small time-steps \(k \bleq k_0(\bgamma)\), I detail\narguments which lead to strong error estimates where the\nparameter \(\bvarepsilon\) only enters polynomially -- avoiding Gronwall's lemma.\n
Diffusion in strongly layered domains
Thursday, 31.1.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Inspired by chemical signalling in cell organelles\nwe analyse the effect of strongly layered domains\non diffusion. Homogenisation\nreveals memory effects and splitting into PDE-ODE systems, i.e. the specific geometry of\nthe domain has strong qualitative effects on\nthe solution of the heat equation. We will discuss these results\nin context with phenomena observed in cell biology.