Project description DEENESFRITPL Algebraic solutions to enumerative problems in real and complex geometry Enumerative geometry, the mathematics of counting numbers of solutions to geometry problems, analyses geometric problems by computing numerical invariants. This branch of algebraic geometry has successfully provided solutions to counting problems in geometry over the complex numbers. The EU-funded QUADAG project is using algebraic geometry and motivic homotopy theory to develop new, purely algebraic methods for handling enumerative problems over the real numbers, rational numbers or finite fields. The project will build on successful previous work by the researcher that has led to the development of a purely algebraic approach to tackling enumerative geometry problems, shedding light on both the complex and real solutions in a unified way. Show the project objective Hide the project objective 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. Fields of science natural sciencesmathematicspure mathematicstopologysymplectic topologynatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2018-ADG - ERC Advanced Grant Call for proposal ERC-2018-ADG See other projects for this call Funding Scheme ERC-ADG - Advanced Grant Host institution UNIVERSITAET DUISBURG-ESSEN Net EU contribution € 2 124 663,00 Address UNIVERSITATSSTRASSE 2 45141 Essen Germany See on map Region Nordrhein-Westfalen Düsseldorf Essen, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 2 124 663,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITAET DUISBURG-ESSEN Germany Net EU contribution € 2 124 663,00 Address UNIVERSITATSSTRASSE 2 45141 Essen See on map Region Nordrhein-Westfalen Düsseldorf Essen, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 2 124 663,00
