## The ERC Project QUADAG: an overview

Principal Investigator: Marc Levine

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. This is accomplished by refining the classical integer-valued invariants to live in the Grothendieck-Witt ring of quadratic forms over the given field of definition for the enumerative problem.

Building on these new developments, the goals of the project QUADAG 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 $\mathbb{C}$ and $\mathbb{R}$ and thirdly, to use this new approach to reveal the hidden arithmetic nature of enumerative problems.

## Project Members

The ERC grant provides funding for postdocs and Ph.D. students to work in areas related to the main project goals.

### Postdocs-current members

Dhyan Aranha, 01.09.2020-

Andrés Jaramillo-Puentes, 01.10.2020-

Sabrina Pauli, 01.10.2020-

### Postdocs-past members

Fangzhou Jin, 01.01-31.11.2020

### Ph.D. Students-current members

Ran Azouri, 01.09.2019-

Pietro Gigli, 01.10.2020- (from 01.01.2021 to 31.10.2022 supported by the DFG-Project LE 2259/8-1, "Enumerative geometry with quadratic forms")

## Research

For details on the research activities of past and present members, follow this link.

## Activities

Due to the corona pandemic, many activities supported by this project have been postponed. However, our Motives Seminar continues to present lecture series that are closely related to the project topics. Follow the link at the top of this page to view the current program and the Teaching link for past seminars.