Descripción del proyecto
Soluciones algebraicas a problemas enumerativos en geometría compleja y real
La geometría enumerativa, disciplina matemática que se ocupa de contar el número de soluciones a problemas geométricos, analiza los problemas geométricos al calcular invariantes numéricas. Esta rama de la geometría algebraica ha logrado aportar soluciones a problemas de recuento en geometría de los números complejos. El proyecto QUADAG, financiado con fondos europeos, emplea la geometría algebraica y la teoría de la homotopía motívica para desarrollar nuevos métodos puramente algebraicos a fin de tratar problemas enumerativos con números reales, números racionales o campos finitos. El proyecto se basará en exitosos trabajos anteriores efectuados por los investigadores que han llevado al desarrollo de un enfoque puramente algebraico para tratar problemas de la geometría enumerativa, lo cual aportará más información sobre soluciones reales y complejas de forma unificada.
Objetivo
Enumerative geometry, the mathematics of counting numbers of solutions to geometric problems, and its modern descendents, Gromov-Witten theory, Donaldson-Thomas theory, quantum cohomology and many other related fields, analyze geometric problems by computing numerical invariants, such as intersection numbers or degrees of characteristic classes. This essentially algebraic approach has been successful mainly in the study of problems over the complex numbers and other algebraically closed fields. There has been progress in attacking enumerative problems over the real numbers; the methods are mainly non-algebraic. Arithmetic content underlying the numerical invariants is hidden when analyzed by these non-algebraic methods. Recent work by the PI and others has opened the door to a new, purely algebraic approach to enumerative geometry that recovers results in both the complex and real cases in one package and reveals this arithmetic content over arbitrary fields. Building on these new developments, the goals of this proposal are, firstly, to use motivic homotopy theory, algebraic geometry and symplectic geometry to develop new purely algebraic methods for handling enumerative problems over an arbitrary field, secondly, to apply these methods to central enumerative problems, recovering and unifying known results over both C and R and thirdly, to use this new approach to reveal the hidden arithmetic nature of enumerative problems. In 2009 R. Pandharipande and I applied algebraic cobordism to prove the degree zero MNOP conjecture in Donaldson-Thomas theory. More recently, I have developed several aspects of the theory of quadratic invariants using motivic homotopy theory.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-ADG - Advanced GrantInstitución de acogida
45141 Essen
Alemania