Gauge theory and algebraic geometry

• Grant number: 18/21391-1
• Support Opportunities: Research Projects - Thematic Grants
• Start date: May 01, 2019
• End date: April 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Marcos Benevenuto Jardim
• Grantee: Marcos Benevenuto Jardim
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Pesquisadores principais: Ethan Guy Cotterill ; Henrique Nogueira de Sá Earp
• Associated researchers: Alan do Nascimento Muniz ; Pietro Speziali ; Rafael de Freitas Leão ; Simone Marchesi ; Tiago Jardim da Fonseca
• Associated research grant(s): 22/09853-5 - The Aluffi Algebras and Free Divisors, AV.EXT; 22/10067-4 - Asymptotic and polynomial stabilities in Bridgeland stability conditions, AV.EXT
• Associated scholarship(s): 24/05296-0 - On the quiver descriptions of Hilbert and Quot schemes and beyond, BP.PD; 23/06502-0 - Stability of sheaves on projective varieties, BP.PD; 22/16369-2 - Algebraic curves and their applications to Coding Theory, BP.PD; + associated scholarships 23/03565-0 - Derived Categories and Fourier-Mukai Transforms, BP.IC; 22/15788-1 - Categories and applications on Algebraic Geometry and Algebraic Topology, BP.IC; 22/03283-2 - Free plane algebraic curves, BP.MS; 21/11603-4 - The generalized Ricci curvature on contact Calabi-Yau 7-manifolds, BP.MS; 21/10550-4 - Logarithmic sheaves for complete intersection schemes, BP.DR; 21/07249-0 - Symmetries in exceptional holonomy problems, BP.PD; 21/08026-5 - Special geometries and calibrated submanifolds, BP.PD; 21/07190-6 - Algebraic topology via problems, BP.IC; 21/05051-9 - Octonions and cross product, BP.IC; 20/15525-5 - Supersymmetric Yang-Mills theory on contact Calabi-Yau 7-manifolds, BP.MS; 21/02706-4 - Classification of polyhedra, BP.IC; 20/03499-0 - Stability conditions on higher dimensional varieties and moduli spaces, BP.PD; 20/15054-2 - Spectrum of the Laplacian on Hermitian Vector Bundles over Homogeneous Spaces, BP.DD; 20/16173-5 - Schemes and algebraic varieties, BP.IC; 20/06938-4 - Geometry of moduli spaces of sheaves via wall-crossing, BP.PD; 19/23499-7 - Noether-Lefschetz theory in toric varieties, BP.PD; 19/20843-9 - Moduli problems in algebraic geometry, BP.MS; 19/22453-3 - Spectra of the Laplacian on homogeneous spaces, BP.MS; 19/21140-1 - Moduli spaces of pfaffian representations of cubic three-folds and instanton bundles, BP.PD - associated scholarships

Abstract

Gauge theory is the study of differential equations involving connections on vector bundles over manifolds using methods from differential geometry, algebraic geometry and geometric analysis. This is area of research emerged in the late 1970s, especially after works by Michael Atiyah, Nigel Hitchin and simon Donaldson, among others, as a mathematical model for classical field theory in mathematical physics, and had inumerous applications to more tradicional areas of mathematics, like differential topology, algebraic geometry, complex geometry, Riemannian geometry, geometric analysis and representation theory.At the core of the theory is a 1-1 correspondence between solutions of certain equations (e.g. the Yang--Mills anti-self-duality equation) and holomoprhic vector bundles with certain properties (e.g. stability). The research group in Gauge Theory and Algebraic Geometry of the Department of Mathematics at IMECC-UNICAMP focuses on the interaction between these two areas, also working on specific problems in each of them. (AU)

Scientific publications

(23)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MIYAMOTO, HENRIQUE K.; COSTA, SUELI I. R.; EARP, HENRIQUE N. SA. Constructive Spherical Codes by Hopf Foliation. IEEE TRANSACTIONS ON INFORMATION THEORY, v. 67, n. 12, p. 7925-7939, DEC 2021. (18/21391-1, 16/05126-0, 17/20007-0, 13/25977-7)

FONTES, AISLAN LEAL; JARDIM, MARCOS. Monads and moduli components for stable rank 2 bundles with odd determinant on the projective space. Geometriae Dedicata, v. 217, n. 2, p. 25-pg., 2023-04-01. (18/21391-1)

JARDIM, M.; SILVA, D. D.. Instanton sheaves and representaTIONS OF QUIVERS. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v. 63, n. 4, p. 984-1004, NOV 2020. (18/21391-1)

ALMEIDA, C.; JARDIM, M.; TIKHOMIROV, A. S.; TIKHOMIROV, S. A.. New moduli components of rank 2 bundles on projective space. SBORNIK MATHEMATICS, v. 212, n. 11, p. 1503-1552, NOV 1 2021. (16/14376-0, 14/08306-4, 18/21391-1, 16/03759-6)

FRANCO, EMILIO; JARDIM, MARCOS. Mirror symmetry for Nahm branes. EPIJOURNAL DE GEOMETRIE ALGEBRIQUE, v. 6, p. 29-pg., 2022-02-28. (12/16356-6, 16/03759-6, 18/21391-1)

FADEL, DANIEL; LOUBEAU, ERIC; MORENO, ANDRES J.; SA EARP, HENRIQUE N.. Flows of geometric structures. JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, v. N/A, p. 86-pg., 2024-09-03. (21/04065-6, 18/21391-1, 21/08026-5)

LIU, WANMIN; LO, JASON; MARTINEZ, CRISTIAN. Fourier-Mukai Transforms and Stable Sheaves on Weierstrass Elliptic Surfaces. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 55, n. 4, p. 38-pg., 2024-12-01. (18/21391-1, 20/06938-4)

CORREA, MAURICIO; JARDIM, MARCOS; MUNIZ, ALAN. Moduli of distributions via singular schemes. MATHEMATISCHE ZEITSCHRIFT, v. 301, n. 3, p. 23-pg., 2022-03-05. (18/21391-1)

LOUBEAU, ERIC; MORENO, ANDRES J.; EARP, HENRIQUE N. SA; SAAVEDRA, JULIETH. Harmonic Sp(2)-Invariant G(2)-Structures on the 7-Sphere. JOURNAL OF GEOMETRIC ANALYSIS, v. 32, n. 9, p. 49-pg., 2022-09-01. (17/20007-0, 18/21391-1)

CALVO-ANDRADE, OMEGAR; CORREA, MAURICIO; JARDIM, MARCOS. Codimension one distributions and stable rank 2 reflexive sheaves on threefolds. Anais da Academia Brasileira de Ciências, v. 93, n. 3, 2021. (18/21391-1)

FAENZI, DANIELE; JARDIM, MARCOS; VALLES, JEAN. Saito criterion and its avatars. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, v. 73, n. 6, p. 11-pg., 2024-06-10. (18/21391-1)

DWIVEDI, SHUBHAM; LOUBEAU, ERIC; EARP, HENRIQUE SA. Harmonic flow of Spin(7)-structures. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, v. 25, n. 1, p. 65-pg., 2024-01-01. (18/21391-1)

AGGARWAL, DAATTAVYA; HE, YANG-HUI; HEYES, ELLI; HIRST, EDWARD; EARP, HENRIQUE N. SA; SILVA, TOMAS S. R.. Machine learning Sasakian and G₂ topology on contact Calabi-Yau 7-manifolds. Physics Letters B, v. 850, p. 10-pg., 2024-02-20. (21/04065-6, 22/09891-4, 18/21391-1, 20/09838-0)

CAVALCANTE, ALANA; JARDIM, MARCOS; SANTIAGO, DANILO. Foliations by curves on threefolds. Mathematische Nachrichten, v. 296, n. 2, p. 22-pg., 2023-02-01. (18/21391-1)

PORTILLA, LUIS E.; SA EARP, HENRIQUE N.. Instantons on Sasakian 7-manifolds. QUARTERLY JOURNAL OF MATHEMATICS, v. N/A, p. 57-pg., 2023-03-28. (17/20007-0, 18/21391-1)

VON FLACH, RODRIGO A.; JARDIM, MARCOS; LANZA, VALERIANO. Obstruction theory for moduli spaces of framed flags of sheaves on the projective plane. JOURNAL OF GEOMETRY AND PHYSICS, v. 166, AUG 2021. (15/07766-4, 18/21391-1)

JARDIM, MARCOS; MACIOCIA, ANTONY; MARTINEZ, CRISTIAN. Vertical Asymptotics for Bridgeland Stability Conditions on 3-Folds. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, v. N/A, p. 53-pg., 2022-08-31. (18/21391-1, 20/06938-4)

COMASCHI, GAIA; JARDIM, MARCOS; MARTINEZ, CRISTIAN; MU, DAPENG. Instanton sheaves: the next frontier. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. N/A, p. 37-pg., 2023-11-27. (20/03499-0, 19/21140-1, 18/21391-1, 20/06938-4, 22/09063-4)

LOTAY, JASON D.; EARP, HENRIQUE N. SA; SAAVEDRA, JULIETH. Flows of G(2)-structures on contact Calabi-Yau 7-manifolds. ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, v. 62, n. 2, p. 23-pg., 2022-06-21. (18/21391-1)

GNERI, P. O.; JARDIM, M.; SILVA, D. D.. Derived categories of functors and quiver sheaves. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 21, n. 07, p. 26-pg., 2022-07-01. (18/21391-1)

COMASCHI, GAIA; JARDIM, MARCOS. Instanton sheaves on Fano threefolds. MANUSCRIPTA MATHEMATICA, v. 175, n. 1-2, p. 51-pg., 2024-05-14. (18/21391-1, 19/21140-1)

FOWDAR, UDHAV; EARP, HENRIQUE N. SA. Harmonic Flow of Quaternion-Kähler Structures. JOURNAL OF GEOMETRIC ANALYSIS, v. 34, n. 6, p. 48-pg., 2024-06-01. (21/07249-0, 21/04065-6, 18/21391-1)

JARDIM, MARCOS; NEJAD, ABBAS NASROLLAH; SIMIS, ARON. THE BOURBAKI DEGREE OF A PLANE PROJECTIVE CURVE. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. N/A, p. 23-pg., 2024-09-10. (22/09853-5, 18/21391-1)