Seiberg-Witten monopoles, G2 instantons, and Z/2 harmonic spinors
Mittwoch, 4.1.17, 11:15-12:15, Raum 404, Eckerstr. 1
The gluing formula for the analytic torsion - a new approach
Mittwoch, 4.1.17, 14:15-15:15, Raum 404, Eckerstr. 1
Hypoelliptic Laplacian and its applications
Donnerstag, 5.1.17, 10:15-11:15, Raum 404, Eckerstr. 1
The hypoelliptic Laplacian, constructed by Bismut, is a family of\noperators that interpolates between the ordinary Laplacian and the geodesic\nflow. In this talk, we will describe its construction from geometric,\nanalytic and probabilistic points of view. We explain also some\napplications. One important application is a solution to the Fried\nconjecture which claims an identity between the analytic torsion and the\nzero value of a dynamical zeta function.
Callias-type operators in C^∗ -algebras and positive scalar curvature on noncompact manifolds
Donnerstag, 5.1.17, 13:15-14:15, Raum 404, Eckerstr. 1
A Dirac-type operator on a complete Riemannian manifold is of\nCalliastype if its square is a Schrödinger-type operator with a potential\nuniformly positive outside of a compact set. We present an index theorem for\nCallias-type operators twisted with Hilbert C^∗-module bundles. As an\napplication, we derive an obstruction to the existence of Riemannian metrics\nof positive scalar curvature on noncompact spin manifolds in terms of closed\nsubmanifolds of codimension-one.
Donnerstag, 5.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Modular and automorphic forms & beyond (1)
Montag, 9.1.17, 10:15-11:15, Raum 119, Eckerstr. 1
Is it worth to elaborate a (new) mathematical theory which is a huge generalization of\nthe theory of (holomorphic) modular/automorphic forms, without knowing if at some point\nyou will have fruitful applications similar to those of modular forms? If your answer is yes, this\ntalk might be useful for you. This new theory starts with a moduli space of projective varieties\nenhanced with elements in their algebraic de Rham cohomology and with some compatibility with the Hodge filtration and\nthe cup product. These moduli spaces are conjectured to be affine varieties, and their ring of functions are candidates for\nthe generalization of automorphic forms. Another main ingredient of this theory is a set of certain vector\nfields on such moduli spaces which are named "Gauss-Manin connection in disguise".\nI will explain this picture in three examples.\n1. The case of elliptic curves and the derivation of the algebra of quasi-modular forms (due to Kaneko and Zagier). \n2. The case of Calabi-Yau varieties and the derivation of generating function for Gromov-Witten invariants.\n3. The case of principally polarized abelian surfaces and the derivation of Igusa's generators for the algebra of genus two Siegel\nmodular forms.\n\nIn the follow-up lectures I will try to explain the three cases above in more details.\n\nReferences:\n\nGauss-Manin Connection in Disguise: Calabi-Yau modular forms,\nSurveys in Modern Mathematics, International Press, Boston, 2017.\n\nGauss-Manin connection in disguise: Calabi-Yau threefolds \n(with Murad Alim, Emanuel Scheidegger, Shing-Tung Yau), CMP, 2016.\n\nQuasi-Modular forms attached to elliptic curves: Hecke operators, JNT, 2015.\n\nA course in Hodge Theory: With Emphasis on Multiple Integrals, Book under preparation.
Periods of algebraic cycles
Montag, 9.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
The origin of Hodge theory goes back to many works on elliptic, abelian\nand multiple integrals (periods). In this talk, I am going to explain how Lefschetz\nwas puzzled with the computation of Picard rank (defined using periods)\nand this led him to consider the homology classes of curves inside surfaces.\nThis was ultimately formulated in Lefschetz (1,1) theorem and then the Hodge conjecture. In the second half of the talk\nI will discuss periods of algebraic cycles and will give some applications in identifying\nsome components of the Noether-Lefschetz and Hodge locus. The talk is based on my book\nunder preparation: A course in Hodge Theory: With Emphasis on Multiple Integrals,\n
Modular and automorphic forms & beyond (2)
Dienstag, 10.1.17, 10:15-11:15, Raum 119, Eckerstr. 1
Lecture 1: Ramanujan's relations between Eisenstein series is derived from the Gauss-Manin connection of a family of elliptic\ncurves. A similar discussion will be done for Darboux and Halphen equations. I will also give some applications regarding\nmodular curves.
Monotonicity Formula for Varifolds
Dienstag, 10.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
Modular and automorphic forms & beyond (3)
Mittwoch, 11.1.17, 10:15-11:15, Raum 119, Eckerstr. 1
Lecture 2: I will explain a purely algebraic version of the Bershadsky-Cecotti-Ooguri-Vafa anomaly equation using\na Lie algebra on the moduli of enhanced Calabi-Yau varieties.
Modular and automorphic forms & beyond (4)
Donnerstag, 12.1.17, 10:15-11:15, Raum 119, Eckerstr. 1
Lecture 3: In this lecture, I will explain how automorphic forms, and in particular Siegel modular forms, fit well\nto the geometric theory explained in the previous lectures.
Donnerstag, 12.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Reciprocity functors and class field theory
Freitag, 13.1.17, 10:15-11:15, Raum 404, Eckerstr. 1
Quantum statistical models and inference
Freitag, 13.1.17, 12:00-13:00, Raum 404, Eckerstr. 1
Quantum statistics is concerned with the inference for systems\ndescribed by quantum mechanics. After an introduction to the\nmain mathematical notions of quantum statistics: quantum states,\nmeasurements, channels, we describe nonparametric quantum models.\nWe prove the local asymptotic equivalence (LAE) in the sense of\nLe Cam theory of i.i.d. quantum pure states and a quantum Gaussian\nstate. We show nonparametric rates for the estimation of the quantum\nstates, of some quadratic functionals and for the testing of pure\nstates. The LAE allows to transfer proofs to a different model.\nSurprisingly, a sharp testing rate of order n^{-1/2} is\nobtained in a nonparametric quantum setup.\nThis is joint work with M. Guta and M. Nussbaum.
Equivariant bordism
Montag, 16.1.17, 10:15-11:15, Raum 318, Eckerstr. 1
Maassformen, Besselfunktionen und die Eisensteinreihe für SL(2,Z)
Montag, 16.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
Erfüllt eine glatte Funktion eine geeignete Invarianzeigenschaft unter der Wirkung der Gruppe SL(2,Z) auf der oberen Halbebene, so besitzt sie eine diskrete Fourierentwicklung. Über die Koeffizienten dieser Entwicklung wird die L-Reihe definiert, die in vielen Fällen interessante Eigenschaften wie eine Funktionalgleichung besitzt. Das Standardbeispiel hierfür sind Modulformen.\nIch werde zunächst die Definition einer Maassform geben, sie mit der einer Modulform vergleichen und die Eisensteinreihe für SL(2,Z) als Beispiel für eine Maassform vorstellen. Dann werde ich die Fourierentwicklung einer Maassform herleiten und Eigenschaften und Bedeutung der dort auftretenden (modifizierten) Besselfunktionen diskutieren. Zuletzt werde ich die zugehörige L-Reihe definieren und ihre Funktionalgleichung angeben.
Connection between \(p\)-harmonic functions and the inverse mean curvature flow
Dienstag, 17.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk we will present a generalisation of Roger Mosers method for an alternative proof for the existence of solutions to the IMCF in the case of a complete and closed Riemannian manifold with bounded curvature. In the course of doing so we will also discuss why certain properties of \(p\)-harmonic functions, such as the Hölder-continuity of the gradient, will be preserved and give an outlook on further proceedings in the field.
Redukte und invariante Unterräume
Mittwoch, 18.1.17, 16:30-17:30, Raum 404, Eckerstr. 1
Discrete Alexandroff estimate and pointwise rates of convergence for FEMs
Donnerstag, 19.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
We derive an Alexandroff estimate for continuous piecewise linear functions which states that the max-norm of their negative part is controlled by the Lebesgue measure of the sub-differential of their convex envelope at the contact nodes. We develop a discrete Alexandroff-Bakelman-Pucci estimate which controls the Lebesgue measure of the sub-differential in terms of the discrete Laplacian via gradient jumps. We further apply these estimates in the analysis of three finite element methods (FEMs).\n\nWe first discretize the Monge-Ampere equation (MA) with a FEM based on the geometric interpretation of MA. We next discretize MA with a two-scale FEM which exploits an eigenvalue representation of the determinant of SPD matrices. We finally present a two-scale FEM for linear elliptic PDEs in non-divergence form. We prove rates of convergence in the max-norm for all three FEMs, study their optimality, and check it computationally.\n\nThis is joint work with D. Ntogkas and W. Zhang.\n
Cartier crystals and perverse constructible étale p-torsion sheaves
Freitag, 20.1.17, 10:15-11:15, Raum 404, Eckerstr. 1
In 2004, Emerton and Kisin established an analogue of the Riemann-Hilbert correspondence for varieties over fields with positive characteristic p. It is an anti-equivalence between the derived categories of so-called unit F-modules and etale constructible \(p\)-torsion sheaves, inducing an anti-equivalence between the abelian categories of unit F-modules and Gabber's perverse sheaves.\n\nIn the talk we explain how this Riemann-Hilbert correspondence can be generalized to singular varieties of positive characteristic which admit an embedding into smooth, F-finite varieties, and introduce the notion of Cartier crystals as a suitable alternative for unit F-modules in this context. Furthermore, we discuss possible further generalizations and the current situation with respect to compatibilities of the correspondence with pull-back and push-forward for certain morphisms.
Asymptotic equivalence between density estimation and Gaussian white noise revisited
Freitag, 20.1.17, 12:00-13:00, Raum 404, Eckerstr. 1
Asymptotic equivalence between two statistical models means that they\nhave the same asymptotic properties with respect to all decision\nproblems with bounded loss. A key result by Nussbaum states that\nnonparametric density estimation is asymptotically equivalent to a\nsuitable Gaussian shift model, provided that the densities are smooth\nenough and uniformly bounded away from zero.\n\nWe study the case when the latter assumption does not hold and the\ndensity is possibly small. We further derive the optimal Le Cam distance\nbetween these models, which quantifies how close they are. As an\napplication, we also consider Poisson intensity estimation with low\ncount data. This is joint work with Johannes Schmidt-Hieber.
Topological entropy of Finsler geodesic flows
Montag, 23.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
Regularity for some elliptic equations with orthotropic structure
Dienstag, 24.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
We discuss a variant of the p-Laplacian operator, which arises as the first variation of a suitable Dirichlet integral. The corresponding elliptic equation is much more degenerate/singular than that for the standard p-Laplacian operator and higher regularity of solutions is a difficult issue.\nWe will present some regularity results for the gradient of solutions (differentiability, boundedness and continuity), mainly for the two dimensional case. We will also briefly address the case of nonstandard growth conditions and the higher dimensional case.\nThe results presented are contained in some papers in collaboration with Pierre Bousquet (Toulouse), Guillaume Carlier (Paris Dauphine), Vesa Julin (Jyvaskyla), Chiara Leone (Napoli), Giovanni Pisante\n(Caserta) and Anna Verde (Napoli).
A two-phase free boundary problem for the fractional Laplacian
Dienstag, 24.1.17, 17:15-18:15, Raum 404, Eckerstr. 1
In this talk, I will discuss a non-local free boundary problem of two-phase type, related to the fractional Laplacian. In particular, I will discuss the optimal regularity and the separation of phases. It turns out that certain non-local problems differ from their local siblings, in the sense that the two phases can never meet. This is joint work with Mark Allen and Arshak Petrosyan.\n\n\n\n
Ample Theorien von endlichem Morley-Rang
Mittwoch, 25.1.17, 16:30-17:30, Raum 404, Eckerstr. 1
Donnerstag, 26.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 26.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
G2 manifolds and octonions
Montag, 30.1.17, 16:15-17:15, Raum 404, Eckerstr. 1