Multiplication of differential operators in terms of connections using Lie-Rinehart algebras
Monday, 8.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The multiplication of two differential operators in an open set of \(\bmathbb{R}^n\)\nis explicitly known in terms of their standard symbols: this is a motivating point\nfor the theory of deformation quantization. On a differentiable manifold equipped\nwith a connection ∇ in the tangent bundle the same formula --where partial\nderivatives are replaced with symmetrized covariant derivatives-- will be wrong in\ngeneral, and one has to correct it by terms containing torsion and curvature and\ntheir covariant derivatives. In our work with my PhD student Hamilton Menezes de Araujo\nwe shall give an 'explicit formula' of the corrected formula in the more algebraic framework\nof Lie-Rinehart algebras L (G.S.Rinehart, 1963, which are now being used in the study\nof singular foliations) over a commutative unital K-algebra A (where the ground ring K should\ncontain the rational numbers as a subring). L generalizes the\nLie algebra of vector fields (more generally Lie algebroids) and A the algebra of\nsmooth functions in differential geometry. The enveloping algebra of L over A introduced\nby Rinehart plays the rôle of the differential operator algebra. The arising combinatorial problems\ncan conveniently be treated in terms of the fibrewise shuffle comultiplication in the free algebra\ngenerated by L over A and the associated convolution products. The torsion and curvature\nterms arise in a morphism of Lie-Rinehart algebras Z from the free Lie algebra generated\nby L over A equipped with a Lie-Rinehart bracket isomorphic to the one on M.Kapranov's\n'free path Lie algebroid' (2007) to L which are related to the (infinitesimal)\nholonomy of the connection. Z is obtained by a simple explicit linear recursion.\nThe framework allows to discuss `family theorems' by replacing the ground ring K\nbut the smooth function algebra of the base of a fibered manifold.
Nu invariants of extra twisted connected sums
Monday, 15.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The \(\bnu\) invariant is an invariant of \(G_2\)-structures on closed 7-manifolds. It can be computed in examples and has been used to show that for some closed spin 7-manifolds, the moduli space of \(G_2\)-metrics is not connected.\n\nIn this talk, we will present the computation for extra twisted connected sums and show how to obtain a tractable formula in the end.
Elliptic Genera and \(G_2\)-manifolds
Monday, 29.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
In 1988 Witten showed that the universal elliptic genus of a manifold \(M\) can be interpreted as the index of a twisted Dirac operator on the the loop space of \(M\). Furthermore he discovered, that the index of this Dirac operator has similar modular properties, if one restricts to string manifolds. The resulting modular form is now called the Witten genus.\n\nIn my talk I will give an introduction to modular forms and I will formaly derive the Witten genus from the index theorem.\n\nIf we compare the Witten genus with the elliptic genus in dimension \(8\), there occur characteristic classes, which are connected with the Nu-invariant of \(G_2\)-manifolds.
Recently on arXiv: 'Systole and small eigenvalues of hyperbolic surfaces' / 'Classical KMS Functionals and Phase Transitions in Poisson Geometry'
Monday, 6.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
(i) Let S be a closed orientable hyperbolic surface with Euler characteristic \(\bchi\), and let \(\blambda_k(S)\) be the \(k\)-th positive eigenvalue for the Laplacian on \(S\). According to famous result of Otal and Rosas, \(\blambda_−\bchi >0,25\). In this article, we prove that if the systole of S is greater than \(3,46\), then \(\blambda_{−\bchi−1}>0,25\). This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.\n\n(ii)The authors study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. They discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, they focus on the case of \(\bflat\)-Poisson manifolds, where they provide a complete characterization of the convex cone of KMS measures.
The Index theorem on end-periodic manifolds
Monday, 13.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
Atiyah and Singer published 1963 a formula for the index of an elliptic operator over a closed oriented Riemannian manifold just containing topological terms, known as the Atiyah-Singer index theorem. Forty-one years later they were awarded with the Abel Prize, among other things, for this deep result connecting topology, geometry and analysis.\nIn this talk the Atiyah-Singer index theorem will be formulated and a proof via the heat equation and it's asymptotic expansion will be sketched. Further, a modification of this proof leads to an index theorem for end-periodic Dirac operators discovered by Mrowka, Ruberman and Saveliev in 2014. This end-periodic index theorem and how it is related to the classical Atiyah-Patodi-Singer index theorem for manifolds with boundary are also treated in the talk.
Weak Dual Pairs in Dirac-Jacobi Geometry
Monday, 20.12.21, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. \n\nIn my talk I will give a short introduction to Dirac-Jacobi geometry, introduce the notion of weak dual pairs, explain some cases where they exist and apply this to prove a normal form theorem, which locally in special cases gives the \nlocal structure theorems by Dazord, Lichnerowicz and Marle for Jacobi structures on the one hand, and the Weinstein splitting theorem on the other hand, which are generalizations of the Darboux theorem for contact (resp. symplectic) manifolds.
Fourier expansions of vector-valued automorphic functions
Monday, 10.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
In this talk, I provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We will discuss a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
Die Weierstraßdarstellung von Minimalflächen
Monday, 17.1.22, 16:15-17:15, BBB-Raum (s. Diffgeo-Liste)
In diesem Vortrag wird die Weierstraß-Darstellung für konform parametrisierte Minimalflächen hergeleitet, in welcher eine holomorphe und eine meromorphe Funktion auftreten, die eine solche Fläche unter geringen Zusatzbedingungen beschreiben. Anfangs wird dafür an einige Begriffe der Elementaren Differentialgeometrie und der Funktionentheorie erinnert. Besonderes Augenmerk liegt im weiteren Verlauf auf der Korrespondenz zwischen der Menge der konform parametrisierten Minimalflächen und der Menge der holomorphen, isotropen Funktionen auf demselben Definitionsbereich, da diese Beziehung den Ausgangspunkt der Weierstraß'schen Konstruktion darstellt.
Yamabe Flow on Singular Spaces
Monday, 24.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
I will talk about the Yamabe flow on compact spaces with conical singularities (and more generally: smoothly stratified spaces with iterated cone-edge metrics). I will present the classical Yamabe problem, and talk about why the Yamabe flow exists for all time in our setting. I will end by discussing convergence (and failure thereof). \n\nThis is joint work with Gilles Carron and Boris Vertman, arXiv:2106.01799 .\n\n\n
Giant Gravitons in twisted holography
Monday, 31.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
I will talk about a correspondence between solutions of certain matrix equations and holomorphic curves in SL(2,C). The correspondence is motivated by twisted holography, which is a physical duality between a chiral algebra and topological B-model on SL(2,C). Determinant operators in the chiral algebra are dual to the Giant Graviton branes in the B-model. For each saddle of the correlation functions of determinants, we will define a spectral curve in SL(2,C), which we will identify with the worldsheet of the dual Giant Graviton brane.
Graviton scattering and differential equations in automorphic forms
Monday, 7.2.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
Green, Russo, and Vanhove have shown that the scattering amplitude for gravitons (hypothetical particles of gravity represented by massless string states) is closely related to automorphic forms through differential equations. Green, Miller, Russo, Vanhove, Pioline, and K-L have used a variety of methods to solve eigenvalue problems for the invariant Laplacian on different moduli spaces to compute the coefficients of the scattering amplitude of four gravitons. We will examine two methods for solving the most complicated of these differential equations on \(SL_2(\bmathbb{Z})\bbackslash\bmathfrak{H}\). We will also discuss recent work with S. Miller to improve upon his original method for solving this equation.