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Arbeitstagung Bonn 2013

In Memory of Friedrich Hirzebruch

herausgegeben von: Werner Ballmann, Christian Blohmann, Gerd Faltings, Peter Teichner, Don Zagier

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume contains selected papers authored by speakers and participants of the 2013 Arbeitstagung, held at the Max Planck Institute for Mathematics in Bonn, Germany, from May 22-28. The 2013 meeting (and this resulting proceedings) was dedicated to the memory of Friedrich Hirzebruch, who passed away on May 27, 2012. Hirzebruch organized the first Arbeitstagung in 1957 with a unique concept that would become its most distinctive feature: the program was not determined beforehand by the organizers, but during the meeting by all participants in an open discussion. This ensured that the talks would be on the latest developments in mathematics and that many important results were presented at the conference for the first time. Written by leading mathematicians, the papers in this volume cover various topics from algebraic geometry, topology, analysis, operator theory, and representation theory and display the breadth and depth of pure mathematics that has always been characteristic of the Arbeitstagung.

Inhaltsverzeichnis

Frontmatter

The Hirzebruch Signature Theorem for Conical Metrics

Abstract

Exactly 60 years ago the young Fritz Hirzebruch came up with two spectacular theorems [H53, H54] which set the scene for the future development of algebraic geometry and topology.

Michael Atiyah

Depth and the Local Langlands Correspondence

Abstract

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Anne-Marie Aubert, Paul Baum, Roger Plymen, Maarten Solleveld

Guide to Elliptic Boundary Value Problems for Dirac-Type Operators

Abstract

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contain local elliptic boundary conditions in the sense of Lopatinski and Shapiro as well as the Atiyah–Patodi–Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

Christian Bär, Werner Ballmann

Symplectic and Hyperkähler Implosion

Abstract

We review the quiver descriptions of symplectic and hyperkähler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a similar description for hyperkähler implosion.

Andrew Dancer, Brent Doran, Frances Kirwan, Andrew Swann

Kazhdan–Lusztig Conjectures and Shadows of Hodge Theory

Abstract

We give an informal introduction to the authors’ work on some conjectures of Kazhdan and Lusztig, building on work of Soergel and de Cataldo–Migliorini. This article is an expanded version of a lecture given by the second author at the Arbeitstagung in memory of Frederich Hirzebruch.

Ben Elias, Geordie Williamson

Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights

Abstract

Let \(\Gamma \subset \mathrm{ PSL}_{2}(\mathbb{R})\) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane \(\mathbb{H}\). Consider the d-dimensional space of cusp forms \(\mathcal{S}_{2k}^{\Gamma }\) of weight 2k for \(\Gamma\), and let {f ₁, …, f _d} be an orthonormal basis of \(\mathcal{S}_{2k}^{\Gamma }\) with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity \(S_{2k}^{\Gamma }(z):=\sum _{ j=1}^{d}\vert f_{j}(z)\vert ^{2}\,\mathrm{Im}(z)^{2k}\) is bounded as \(O_{\Gamma }(k)\) in the cocompact setting, and as \(O_{\Gamma }(k^{3/2})\) in the cofinite case, where the implied constants depend solely on \(\Gamma\). We also show that the implied constants are uniform if \(\Gamma\) is replaced by a subgroup of finite index.

Joshua S. Friedman, Jay Jorgenson, Jürg Kramer

Fivebranes and 4-Manifolds

Abstract

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d \(\mathcal{N} = (0,2)\) theories, we obtain a number of results, which include new 3d \(\mathcal{N} = 2\) theories T[M ₃] associated with rational homology spheres and new results for Vafa–Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0, 2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines/walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d \(\mathcal{N} = (0,2)\) theories and 3d \(\mathcal{N} = 2\) theories, respectively.

Abhijit Gadde, Sergei Gukov, Pavel Putrov

Higgs Bundles and Characteristic Classes

Abstract

Sixty years ago Hirzebruch observed how the vanishing of the Stiefel–Whitney class w ₂ led to integrality of the \(\hat{A}\)-genus of an algebraic variety [Hirz1]. This was one motivation for the Atiyah–Singer index theorem but also for my own thesis about Dirac operators and Kähler manifolds. Indeed the interaction between topology and algebraic geometry which he developed has been a constant theme in virtually all my work.

Nigel Hitchin

Hirzebruch–Milnor Classes and Steenbrink Spectra of Certain Projective Hypersurfaces

Abstract

We show that the Hirzebruch–Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g., in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch–Milnor classes of complete intersections.

Laurentiu Maxim, Morihiko Saito, Jörg Schürmann

On Lusztig’s q-Analogues of All Weight Multiplicities of a Representation

Abstract

The ground field \(\mathbb{k}\) is algebraically closed and of characteristic zero. Let G be a connected semisimple algebraic group, and T a maximal torus inside a Borel subgroup B.

Dmitri I. Panyushev

The Triangulation of Manifolds: Topology, Gauge Theory, and History

Abstract

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.

Frank Quinn

Elliptic Calabi–Yau Threefolds over a del Pezzo Surface

Abstract

We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are Calabi–Yau threefolds. We will discuss especially the generating functions of Gromov–Witten and Gopakumar–Vafa invariants.

Simon Rose, Noriko Yui

Remarks on Cohomological Hall Algebras and Their Representations

Abstract

The aim of this paper is to discuss a class of representations of Cohomological Hall algebras related to the notion of framed stable object of a category. The paper is an extended version of the talk the author gave at the workshop on Donaldson–Thomas invariants at the University Paris-7 in June 2013 and at the conference “Algebra, Geometry, Physics” dedicated to Maxim Kontsevich (June 2014, IHES). Because of the origin of the paper it contains more speculations than proofs.

Yan Soibelman

A Stratification on the Moduli of K3 Surfaces in Positive Characteristic

Abstract

We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.

Gerard van der Geer

The Right Adjoint of the Parabolic Induction

Abstract

We extend the results of Emerton on the ordinary part functor to the category of the smooth representations over a general commutative ring R, of a general reductive p-adic group G (rational points of a reductive connected group over a local non-archimedean field F of residual characteristic p). In Emerton’s work, the characteristic of F is 0, R is a complete artinian local \(\mathbb{Z}_{p}\) -algebra having a finite residual field, and the representations are admissible. We show:

The smooth parabolic induction functor admits a right adjoint. The center-locally finite part of the smooth right adjoint is equal to the admissible right adjoint of the admissible parabolic induction functor when R is noetherian. The smooth and admissible parabolic induction functors are fully faithful when p is nilpotent in R.

Marie-France Vignéras

Titel: Arbeitstagung Bonn 2013
herausgegeben von: Werner Ballmann
Christian Blohmann
Gerd Faltings
Peter Teichner
Don Zagier
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-43648-7
Print ISBN: 978-3-319-43646-3
DOI: https://doi.org/10.1007/978-3-319-43648-7