Geometrische Multiplizität des zweiten Schrödinger-Eigenwerts auf geschlossenen zusammenhängenden Flächen
Monday, 1.7.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Ein Schrödinger-Operator auf einer Fläche \(S\) ist definiert als Summe aus dem Laplace-Operator mit einem Potential \(V \bin C_0(S)\). Wir interessieren uns hierbei speziell für die Multiplizität des zweiten Eigenwerts über geschlossenen zusammenhängenden Flächen. Y. Colin de Verdière hat die Vermutung aufgestellt, dass sich deren Supremum explizit über die Färbungszahl der Fläche ausdrücken lässt. Wir wollen dies mit einer Abschätzung über die Eulercharakterisik untermauern, in dem wir uns spezielle zweifache Überlagerungen für die Flächen betrachten und dafür eine verwandte Version des Borsuk Ulam Theorems zeigen.\n
On Caccioppoli's inequalities of Stokes and Navier-Stokes equations up to boundary
Tuesday, 2.7.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We are concerned with Caccioppolis inequalities of the non-stationary Stokes system and Navier- Stokes equations. It is known that the Caccioppolis inequalities of the Stokes system and the Navier-Stokes equations are true known in the interior case. We prove that the Caccioppolis inequalities of the Stokes system and the Navier-Stokes equations may, however, fail near boundary, when only local analysis is considered at the at flat boundary. This is a joint work with Dr. Tong-Keun Chang.\n\n\n
Stability of graph tori with almost nonnegative scalar curvature
Tuesday, 2.7.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Nonlocal evolution transport densities
Wednesday, 3.7.19, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
One-sided exact categories and glider representations.
Thursday, 4.7.19, 10:30-11:30, Raum 403, Ernst-Zermelo-Str. 1
Quillen exact categories provide an excellent framework to do homological algebra and algebraic K-theory. A Quillen exact category is an additive category together with a chosen class of kernel-cokernel pairs (called conflations) satisfying 8 axioms. These 8 axioms can be partitioned into two dual sets of axioms referring to either the kernel-part of a conflation (called an inflation) or the cokernel-part of a conflation (called a deflation). However, 2 of the axioms were quickly found to be redundant. These two dual axioms are known as Quillen's obscure axioms.\n\nA one-sided exact category is defined by keeping either the set of axioms referring to inflation or deflations, however, one might wonder whether the obscure axiom needs to be included. In this talk, we will provide several homological interpretations of the obscure axiom. Moreover, any one-sided exact category can naturally be closed under the obscure axiom and is derived equivalent to this obscure closure. As such, we conclude that the obscure axiom may just as well be included into the definition. \n\nWe apply the theory of one-sided exact categories to obtain a categorical framework for glider representations. Glider representations are a type of filtered representations of a filtered ring. We end by concluding that glider representations remember more information on the original ring than ordinary representation theory.
Thursday, 4.7.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Localizing (one-sided) exact categories
Friday, 5.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
One-sided exact categories are a generalization of Quillen exact categories; they satisfy many desirable homological properties and provide a comparable framework for K-theory. Similar to the exact setting, one can consider the derived category of a one-sided exact category by taking the Verdier quotient of the homotopy category by the subcategory of acyclic complexes.\n\nMimicking the setting of a Serre subcategory of an abelian category, we introduce percolating subcategories of exact categories. One can show that the corresponding localization is, in general, not exact, but merely one-sided exact.\n\nIn this talk, we will discuss these localizations and the corresponding Verdier localizations on the bounded derived categories.\n(Based on joint work with Ruben Henrard.)
Fakultätsfest und Abschlussfeier 2019
Friday, 5.7.19, 15:00-16:00, Großer Hörsaal Physik, Hermann-Herder-Str. 3a
Geometrische Reduktionen in algebraisch abgeschlossenen bewerteten Körpern
Monday, 8.7.19, 14:15-15:15, Raum 404, Ernst-Zermelo-Str. 1
Viele Phänomene in der Modelltheorie henselsch bewerteter Körper lassen sich auf Fragen über die Wertegruppe \(\bGamma\) und den Restklassenkörper \(k\) zurückführen, die a priori einfacher zu verstehen sind. Der Prototyp eines solchen Resultats ist das Ax-Kochen-Ershov-Prinzip.\n\nIm Vortrag werde ich eine Reihe von geometrischen Reduktionen in nichttrivial bewerteten algebraisch abgeschlossenen Körpern vorstellen. Deren Theorie ACVF eliminiert Quantoren, und die Imaginären sind durch höherdimensionale Analoga von \(\bGamma\) und \(k\) klassifiziert.\nHrushovski-Loeser haben die Modelltheorie von ACVF verwendet, um topologische Eigenschaften von Analytifizierungen algebraischer Varietäten auf definierbare Räume in \(\bGamma\), d.h. stückweise lineare Räume, zurückzuführen.Im Vortrag werde ich dies skizzieren, sowie eine äquivariante Version hiervon für semiabelsche Varietäten eingehen. Letzteres ist eine gemeinsame Arbeit mit Ehud Hrushovski und Pierre Simon.
Orientation problems for PDEs and instanton moduli spaces
Monday, 8.7.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Moduli spaces of solutions to non-linear elliptic PDE's such as instantons in\ngauge theory are fundamental for the construction of counting invariants. Using\ninformation about the solution space provided by the index theory of an\napproximating family of linear differential operators, we explain our results\non orientations for moduli spaces, including new developments in G2-holonomy.
Sobolev embeddings of higher order, isoperimetric inequalities and Frostman measures
Tuesday, 9.7.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Sobolev embeddings of higher order, isoperimetric inequalities and Frostman measures
Tuesday, 9.7.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Das gymnasiale Lehramtsstudium – Ansätze zur Gestaltung unter widerstreitenden Anforderungen
Tuesday, 9.7.19, 19:30-20:30, Hörsaal II, Albertstr. 23b
Das gymnasiale Lehramtsstudium im Fach Mathematik unterliegt zahlreichen Anforderungen, die oft schwer ins Gleichgewicht zu bringen sind. Neben bekannten Standardproblemen, die durch die Natur des Fachs bedingt sind und bereits von Felix Klein und Otto Toeplitz erkannt wurden, spielen dabei auch aktuelle Problemverschärfungen eine Rolle. Der Vortrag bietet eine Bestandsaufnahme zu dieser Problematik, arbeitet Zielvorstellungen für das gymnasiale Lehramtsstudium heraus und stellt Ansätze zur Gestaltung vor, die der Vortragende erprobt hat.
Around the residue symbol
Thursday, 11.7.19, 16:15-17:15, Hörsaal II, Albertstr. 23b
Everybody knows the “residue" from complex analysis and Cauchy's residue\nformula. One can regard this as a one-dimensional theorem in the sense\nthat the complex plane has complex dimension one. There are several\ndifferent theories of multi-dimensional residues, all essentially\ncompatible, but in complicated ways. I will explain a picture due to A.\nParshin.\nWhereas Cauchy's residue formula implies a statement of the form “the\nsum of residues at all points of a fixed curve is zero", Parshin's\n2-dimensional generalization provides a nice analogous result stating\nthat “the sum of residues along all curves passing through a fixed\npoint" is zero. This talk will focus on the down-to-earth geometric\napproach of the Soviet school to these issues, which is not so\nwell-known in the Western world.\n (I will not talk about the following, because it would be much too\ntechnical, but of course the same result also immediately follows from\nGrothendieck's residue symbol, the approach more popular in the Western\nworld, but only after introducing f!, derived categories, local\ncohomology, etc.; in fact Grothendieck's theory even in the classical\none-dimensional case already relies on local cohomology).
Zero cycles on moduli spaces of curves
Friday, 12.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Tautological zero cycles form a one-dimensional subspace of\nthe set of all algebraic zero-cycles on the moduli space of stable curves. The full group of zero cycles can in general be infinite-dimensional, so not all points of the moduli space will represent a tautological class. In the\ntalk, I will present geometric conditions ensuring that a pointed curve does define a tautological point. On the other hand, given any point Q in the moduli space we can find other points P1, ..., Pm such that Q+P1+ ... +\nPm is tautological. The necessary number m is uniformly bounded in terms of g,n, but the question of its minimal value is open. This is joint work with R. Pandharipande.
Cohomogeneity one Spin(7)-manifolds
Monday, 15.7.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Spin(7) is one of the exceptional holonomy groups. Spin(7)-manifolds are in particular Ricci flat. The condition for a Spin(7)-structure to be torsion-free gives rise to a complicated system of non-linear differential equations. One of the most fundamental ways to solve differential equations is to use symmetries to reduce the number of variables and complexity. For exceptional holonomy manifolds this can only be used in the non-compact setting. I will explain the construction of Spin(7)-manifolds with cohomogeneity one. Here the non-linear PDE system is reduced to a non-linear ODE system. I will give an overview over previous work and mention recent progress.
Semi-local and global Properties of Jacobi-related Geometries
Tuesday, 16.7.19, 09:00-10:00, Raum 127, Ernst-Zermelo-Str. 1
After a short introduction to Jacobi related geometries, such as Poisson,\nsymplectic, contact and generalized complex/contact manifolds, and their\nappearance in mathematical physics, I want to present some results on their\n(semi-)local structure around transversal submanifolds, so-called "Normal\nforms". They can be seen as generalization of the Weinstein splitting theorem\nfor Poisson manifolds and they induce in fact a very explicit local\ndescription of Jacobi-related structures.\n\nThe second part of the talk is intended to focus on a special Jacobi related\ngeometry: generalized contact bundles, the odd-dimensional counterparts of\ngeneralized complex manifolds. I want to show that their global existence is\ncohomologically obstructed by means of a spectral sequence. At the end I want\nto give some classes of examples of generalized contact structures. \n\n
Essential self-adjointness of powers of first order differential operators on noncompact manifolds with low regularity metrics
Tuesday, 16.7.19, 11:00-12:00, Raum 127, Ernst-Zermelo-Str. 1
The problem of determining the essential self-adjointness of a\ndifferential operator on a smooth manifold, and its powers, is an\nimportant and well studied topic. One of the primary motivations for studying\nthe essential self-adjointness of a differential operator \(D\),\ncomes from the fact that it allows one to build a functional calculus (of Borel\nfunctions) for the closure of that operator. Such a\nfunctional calculus is then used to solve partial differential equations on a\nmanifold, defined through the operator.\nIn this talk, I will present joint work with L. Bandara where we consider the\nquestion of essential self-adjointness of first order differential operators,\nand their\npowers, in the context of non-smooth metrics on noncompact manifolds. Using\nmethods from geometry and operator theory we are able to show\nthat essential self-adjointness, at its heart, is an operator theoretic\ncondition which requires minimal assumptions on the geometry\nof the manifold. Applications to Dirac type operators on Dirac bundles will be\ndiscussed.\n
Nicht-Eindeutigkeit von Entropielösungen der Euler-Gleichungen
Tuesday, 16.7.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Removable singularities of Kähler metrics of constant holomorphic sectional curvature
Tuesday, 16.7.19, 14:15-15:15, Raum 318, Ernst-Zermelo-Str. 1
Let n>1 be an integer, and B^n be the unit ball in C^n. K\bsubset B^n is a compact subset or {z1=0=z2}. By using developing map and Hartogs' extension theorem, we show that a Kaehler metric on B^n\bK with constant holomorphic sectional curvature uniquely extends to the ball. This is a\njoint work with Si-en Gong and Hongyi Liu.
Interior estimate for scalar curvature equations
Tuesday, 16.7.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Motivated by the isometric embedding problem and fully\nnonlinear PDE theory, we study the apriori estimate for scalar curvature equations. Joint with Prof. Pengfei Guan, we proved that there is an interior second order estimate for isometrically embedded hypersurfaces with positive scalar curvature. By employing Warren and Yuan's integral method and my new observation in three-dimensional hypersurface with positive scalar curvature, I give an affirmatively answer to the interior second order estimate to this fully nonlinear PDE in dimension three.
Wall crossing morphisms for moduli of stable pairs
Friday, 19.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Consider a moduli space M parametrizing stable pairs of the form (X, \bsum ai Di) with ai n positive rational numbers. Consider n positive rational numbers bi with bi \ble ai, and assume that the objects on the interior of M are pairs with KX +\bsum bi Di big. Then on the interior of M one can send a pair (X, \bsum ai Di) to the canonical model of (X, \bsum bi Di). If N is a moduli space of stable pairs with coefficients bi this gives a set theoretic map from an open substack of M to N. We investigate when such a map can be extended to the whole M. Our main result is if the interior of M parameterizes klt pairs we can extend the map, up to replacing M and N with their normalizations. The extension does not exist if above we replace the word normalization with seminormalizaton instead. This is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi.
Singular hyperbolic metrics on Riemann surfaces
Friday, 19.7.19, 14:15-15:15, Raum 318, Ernst-Zermelo-Str. 1
J. Nitsche showed that an isolated singularity of a hyperbolic metric is either a cone singularity or a cusp one. M. Heins proved on compact Riemann surfaces a classical existence theorem about singular hyperbolic metrics where the Gauss-Bonnet formula is the necessary and sufficient condition. We prove that a developing map of a singular hyperbolic metric on a compact Riemann surface has a Zariski dense monodromy group in PSL(2;R). Moreover, we also provide\nsome evidences to the conjecture that it be also the case on a noncompact Riemann surface which admits no non-trivial negative subharmonic function. This is a joint work with Yu Feng, Yiqian Shi, Jijian Song.
A finite element method for the surface Stokes equation
Monday, 22.7.19, 12:00-13:00, Bibliothek Rechenzentrum, R216 Hermann-Herder-Str. 10
In this talk we will consider the Stokes system posed on a\nsurface along with the main challenges associated with its\ndiscretization. These include the inability to formulate a conforming\nfinite element method and the possibility of degeneracies in the system\ndue to the presence of Killing fields (rigid motions of the surface).\n We then describe a finite element method for the system and discuss\nits interactions with these challenges. \n
Computational aspects of orbifold equivalence
Monday, 22.7.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Landau-Ginzburg models are a family of quantum field theories characterized by a polynomial (satisfying some conditions) usually called ‘potential’. Often appearing in mirror-symmetric phenomena, they can be collected in categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called `orbifold equivalence’. We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.
Riemann-Roch-Grothendieck theorem for families of curves with hyperbolic cusps and its applications to the moduli space of curves
Tuesday, 23.7.19, 09:00-10:00, Raum 127, Ernst-Zermelo-Str. 1
We’ll present a refinement of Riemann-Roch-Grothendieck theorem on\nthe level of differential forms for families of curves with hyperbolic cusps.\nThe study of spectral properties of the Kodaira Laplacian on a Riemann surface,\nand more precisely of its determinant, lies in the heart of our approach.\n\nWhen our result is applied directly to the moduli space of punctured stable\ncurves, it expresses the extension of the Weil-Petersson form (as a current) to\nthe boundary of the moduli space in terms of the first Chern form of a\nHermitian line bundle, which provides a generalisation of a result of\nTakhtajan-Zograf. \n\nIf time permits, we will explain how our result implies some bounds on the\ngrowth of the Weil-Petersson form near the compactifying divisor of the moduli\nspace of punctured stable curves. This would permit us to give a new approach\nto some well-known results of Wolpert on the Weil-Petersson geometry of the\nmoduli space of curves.\n\n\n
Optimierung mit fraktionellen Differentialoperatoren bei der Regularisierung und Zerlegung von Bildern
Tuesday, 23.7.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Maximum principles for a fully nonlinear nonlocal equation on unbounded domains
Tuesday, 23.7.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Thursday, 25.7.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Moduli of special cubic 4-folds
Friday, 26.7.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Monopoles with arbitrary symmetry breaking
Monday, 29.7.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Monopoles are pairs formed of a connection and an endomorphism of the bundle that satisfy the Bogomolny equation. There is ample literature on the study of monopoles on R3 under the constraint that the eigenvalues of the endomorphism on the sphere at infinity are distinct, the so-called maximal symmetry breaking case. In joint work with Ákos Nagy, we are exploring monopoles with arbitrary symmetry breaking on R3, and in particular their Nahm transform.