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Quadratic refinements in algebraic geometry

Objective

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.

Field of science

  • /natural sciences/mathematics/pure mathematics/arithmetic
  • /natural sciences/mathematics/pure mathematics/topology/symplectic topology
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry

Call for proposal

ERC-2018-ADG
See other projects for this call

Funding Scheme

ERC-ADG - Advanced Grant

Host institution

UNIVERSITAET DUISBURG-ESSEN
Address
Universitatsstrasse 2
45141 Essen
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 2 124 663

Beneficiaries (1)

UNIVERSITAET DUISBURG-ESSEN
Germany
EU contribution
€ 2 124 663
Address
Universitatsstrasse 2
45141 Essen
Activity type
Higher or Secondary Education Establishments