My research topics are part of a field called nonlinear algebra. This is the application-driven study of nonlinear equations, with a view towards computation.
In my PhD research, I have developed methods for solving systems of polynomial equations numerically. The focus was on the important class of so-called 0-dimensional systems (i.e. with finitely many solutions). In some approaches such a system is reformulated as an eigenvalue problem. Other algorithms solve the problem via numerical homotopy continuation.
I am particularly interested in the important role of toric geometry in the solution of sparse systems of polynomial equations, and its applications in other fields of science. Recently, I've been exploring methods that go beyond toric varieties, exploiting for instance Khovanskii bases.
Solving nonlinear equations has many applications. For example, we use our methods to decompose tensors, which are multidimensional generalizations of matrices. My group also addresses mathematical questions coming from particle physics. This includes solving scattering equations for the evaluation of scattering amplitudes and computing the singularity loci of Feynman integrals.
Upcoming Conferences & Workshops
- Applied and Computational Algebraic Geometry, Isaac Newton Institute for Mathematical Sciences, January 22-26
- Positive Geometry in Particle Physics and Cosmology, MPI MiS, February 12-16
- New Frontiers in Landau Analysis, Higgs Centre, Edinburgh, April 24-26
- Combinatorial Algebraic Geometry from Physics, MPI MiS, May 13-17
- European Congress of Mathematics, Sevilla, July 15-19
- MEGA 2024, Leipzig, July 29 - August 2
- Combinatorics of Fundamental Physics, IAS Princeton, November 18-22