ANALYTIC NON-ABELIAN HODGE THEORY Introduction in [Sim5]
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School and Workshop on New Trends in Higgs Bundle Theory
School and Workshop on New Trends in Higgs Bundle Theory Research Programme on Moduli Spaces Madrid, 12-16 November 2018 Instituto de Ciencias Matem´aticas(ICMAT) Contents 1 Venue 2 2 Sponsors 2 3 Scientific committee 2 4 Organizing committee 2 5 Speakers 3 6 Programme 4 7 Schedule 4 8 Abstracts 5 9 Meals, coffee breaks, and social dinner 8 10 List of participants 8 11 Wifi 9 This meeting is devoted to several active fields of current research in the theory of Higgs bundles and their interplay with geometry, topology and theoretical physics. It will combine research lectures with a 3 hour mini-course, run in parallel so as to give a balance of research and training. The event is organized within the ICMAT Research Programme on Moduli Spaces (15 September { 15 December 2018). This activity also ties in with the ICMAT Severo Ochoa Donaldson{Hitchin Laboratory chaired jointly by the members of the scientific committee. The organizing committee wishes to thank you for your interest and participation in this con- ference. Web page: https://www.icmat.es/RT/2018/RPMS/hbt.php 1 Venue All Lectures will take place at AULA NARANJA (Orange Lecture Room) of ICMAT. This lecture room is equipped with blackboards and beamer projector. The Instituto de Ciencias Matem´aticas { ICMAT (Institute for Mathematical Sciences) | is a joint research institute of the Consejo Superior de Investigaciones Cient´ıficas{ CSIC (Spanish National Research Council) and three Madrid universities: the Universidad Aut´onoma de Madrid (UAM), the Universidad Carlos III de Madrid, and the Universidad Complutense de Madrid. -
Towards an Approach to Hilbert's Sixth Problem
Towards An Approach to Hilbert’s Sixth Problem: A Brief Review Yuri Ximenes Martins∗, Rodney Josué Biezuner Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Pampulha, CP 702, CEP 31270-901, Belo Horizonte, MG, Brazil Abstract In 1900 David Hilbert published his famous list of 23 problems. The sixth of them - the axiomatization of Physics - remains partially unsolved. In this work we will give a gentle introduction and a brief review to one of the most recent and formal approaches to this problem, based on synthetic higher categorical languages. This approach, developed by Baez, Schreiber, Sati, Fiorenza, Freed, Lurie and many others, provides a formalization to the notion of classical field theory in terms of twisted differential cohomologies in cohesive (1; 1)-topos. Furthermore, following the Atiyah-Witten functorial style of topological quantum field theories, it provides a nonperturbative quantization for classical field theories whose underlying cohomology satisfies orientability and duality conditions. We follow a pedagogical and almost non-technical style trying to minimize the mathematical prerequisites. No categorical background is required. 1. Introduction naive set theory and axiomatic set theory. In both cases (naive or axiomatic) we need a background language (also called In the year of 1900, David Hilbert (a mathematician logic) in order to develop the theory, as in the diagram below. which is mostly known in Physics for his contribution to the In the context of set theory this background language is just Einstein-Hilbert action functional) published a list containing classical logic. 23 problems (usually known as the Hilbert’s problems) which in his opinion would strongly influence the development of 20th naive century mathematics [107]. -
Moduli of Sheaves Nicole Mestrano, Carlos Simpson
Moduli of sheaves Nicole Mestrano, Carlos Simpson To cite this version: Nicole Mestrano, Carlos Simpson. Moduli of sheaves. Development of Moduli Theory—Kyoto 2013, 69, pp.77-172, 2016, Advanced Studies in Pure Mathematics. hal-01298535 HAL Id: hal-01298535 https://hal.archives-ouvertes.fr/hal-01298535 Submitted on 21 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright Moduli of sheaves Nicole Mestrano and Carlos Simpson Dedicated to Shigeru Mukai on the occasion of his 60 th birthday Abstract. We survey classical and recent developments in the theory of mod- uli spaces of sheaves on projective varieties. Contents 1. Introduction 2 2. Moduli of sheaves 4 3. The Bogomolov-Gieseker inequality 9 4. Symplectic structures 11 5. Geography 14 6. Reider's theorem 19 7. Relation with linear systems on curves, I: the Cayley- Bacharach condition 22 8. The boundary of the moduli space 29 9. Relation with linear systems on curves, II: O'Grady's method 32 10. Rationality 39 11. Strange duality 41 12. Jumping curves 43 13. Wall-crossing 45 14. Betti numbers 54 15. Poincar´ebundles and universal families 55 16. -
Mathematisches Forschungsinstitut Oberwolfach Geometry and Physics
Mathematisches Forschungsinstitut Oberwolfach Report No. 23/2019 DOI: 10.4171/OWR/2019/23 Geometry and Physics of Higgs Bundles Organized by Lara B. Anderson, Blacksburg Tam´as Hausel, Klosterneuburg Rafe Mazzeo, Stanford Laura P. Schaposnik, Chicago 12 May – 18 May 2019 Abstract. This workshop focused on interactions between the various per- spectives on the moduli space of Higgs bundles over a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and mathematical physics, and the goal was to promote interactions between these various branches of the subject. The main current directions of research were well represented by the participants, and the talks included many from both senior and junior participants. Mathematics Subject Classification (2010): 14xx, 22xx, 51xx, 53xx, 81xx. Introduction by the Organizers The workshop Geometry and Physics of Higgs Bundles, organized by Lara Ander- son (Virginia Tech), Tamas Hausel (IST, Austria), Rafe Mazzeo (Stanford) and Laura Schaposnik (University of Illinois at Chicago) was attended by 46 partic- ipants, with broad representation from Europe, Asia, North and South America and India. Quite a few of these participants were either young postdocs or grad- uate students, and 16 of the 46 were women. Notably, the two founders of the subject, Nigel Hitchin and Carlos Simpson, both attended. The topic of Higgs bundles is distinctly interdisciplinary, drawing on algebraic geometry, geometric topology, geometric analysis, mathematical physics and beyond. The goal of this meeting was to draw together researchers working on various aspects of this gen- eral area. There were 18 talks in all, with reports from senior mathematician, leaders in the field, to promising young researchers. -
School and Workshop on New Trends in Higgs Bundle Theory Scientific
School and Workshop on New Trends in Higgs Bundle Theory ICMAT (Madrid), 12-16 November 2018 Scientific Programme Mini-course: Nigel Hitchin (Oxford): Higgs bundles and mirror symmetry Workshop lectures: Olivier Biquard (ENS,´ Paris): From HyperK¨ahlermetrics on Higgs moduli spaces to Ricci flat K¨ahlermetrics on complexified symmetric spaces Lucas Branco (MPIM Bonn): Higgs bundles and mirror symmetry | examples via iso- genies Ugo Bruzzo (SISSA, Trieste): A conjecture about Higgs bundles Brian Collier (Maryland): Generalized opers, conformal limits and maximal variations of Hodge structure Emilio Franco (Porto): Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups Sergei Gukov (Caltech): Modular tensor categories from wild Higgs bundles Georgios Kydonakis (Strasbourg): Model Higgs bundles in exceptional components of character varieties Qiongling Li (Nankai): Dominations in moduli space of Higgs bundles Andr´eOliveira (Porto & UTAD): Generalized Cayley correspondence and higher Teich- m¨uller components for SO(p; q)-Higgs bundles Ana Pe´on-Nieto (Gen`eve): Branes on the Hitchin system from torsion line bundles Carlos Simpson (Nice): Higgs bundles in the geometric Langlands correspondence for genus 2 curves Jan Swoboda (M¨unchen): The large scale geometry of Higgs bundle moduli spaces Alexander Thomas (Strasbourg): Geometric higher Teichm¨uller theory | punctual Hilbert schemes in Higgs theory Junho Peter Whang (MIT): Diophantine analysis on moduli of local systems Mini-course syllabus: Nigel Hitchin (Oxford): Higgs bundles and mirror symmetry Lecture 1. Hodge theory, abelian and non-abelian SYZ mirror symmetry and semiflat metric for a Calabi-Yau The integrable system and semiflat metric for Higgs bundles Lecture 2. -
Conformal Limits and the Bialynicki-Birula Stratification of The
CONFORMAL LIMITS AND THE BIALYNICKI-BIRULA STRATIFICATION OF THE SPACE OF λ-CONNECTIONS BRIAN COLLIER AND RICHARD WENTWORTH Abstract. The Bia lynicki-Birula decomposition of the space of λ-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomorphic Lagrangian fibration over a component of the space of complex variations of Hodge structure. In this paper, we generalize known results for the Hitchin section and the space of opers to arbitrary strata. These include the following: a biholomorphic identification of the fibers of the two strata over a stable variation of Hodge structure via the “~-conformal limit” of Gaiotto, a proof that the fibers of the Morse and partial oper stratifications are transverse at the base point, and an explicit parametrization of the fibers as half-dimensional affine spaces, 1. Introduction For a compact Riemann surface X, both Hitchin’s moduli space of Higgs bundles and the de Rham moduli space of holomorphic connections are interesting holomor- phic symplectic manifolds. While the celebrated nonabelian Hodge correspondence provides a homeomorphism between the two, they are in fact quite different as complex analytic spaces. For example, the Higgs moduli space contains positive dimensional subvarieties (e.g. the moduli space of holomorphic bundles), whereas the de Rham moduli space is Stein. Nevertheless, the complex structures are re- lated in the sense that their composition gives a third complex structure which in fact defines a hyperk¨ahler structure.