## KWG PhD Prize

**Day 2 (Wednesday 12 April) @ 14:30–15:30**

Every year, the Dutch Royal Mathematical Society (KWG) invites several PhD candidates to present their research at the Dutch Mathematical Congress (NMC). A jury of mathematicians from various fields chooses a winner who receives, besides the prestige attached to winning, a floating trophy and a monetary reward of 1000 euro. The jury, chaired by Prof. Gerard van der Geer, selected 4 candidates.

- Vandana Dwarka (TU Delft)
- Sergej Monavari (Utrecht University)
- Lucas Slot (CWI)
- Sebastiano Tronto (Leiden University)

##### Vandana Dwarka (TU Delft)

**From Proofs to Practice: Scalable Iterative Solvers For The Helmholtz Equation and Its Applications**

At the heart of many applications such as seismic and medical imaging, telecommunications and weather prediction, lies the Helmholtz equation. While the equation itself, which is essentially the shifted Laplace equation, appears simple and elegant, retrieving accurate and scalable numerical solutions leads to a wide array of issues. For some problems, no analytical solution exists and numerical methods are the only available alternative. To ensure accurate numerical solutions, we are required to use fine grids, resulting in large matrices. Due to the shift, also known as the wavenumber, these matrices become non-selfadjoint and indefinite, making the use of state-of-the-art iterative solvers infeasible. Solvers which are appropriate converge slowly and the number of iterations to reach convergence grows with the wavenumber. So how can we design numerical solvers which remain scalable, accurate and robust? In this talk, we will touch upon novel theoretical results and insights to explore some answers to this question. Particularly, we discuss how the use of higher-order parametric curves within multilevel methods could help solve some open problems, and how these methods can bring mathematics to the forefront of modern computational methods for real world wave propagation problems.

##### Sebastiano Tronto (Leiden University)

#### Kummer theory for algebraic groups

Kummer theory is the branch of number theory that studies extensions of a base field, usually a number field such as the field of rational numbers, generated by the n-th roots of some given element. More generally, one can study the extensions generated by n-th division points of some set of points of a commutative algebraic group defined over the base field. Of particular interest in this context are the degrees of such field extensions, that appear for example in the solution to problems related to Artin’s primitive root conjecture. Classical results by Ribet and others provide lower bounds for these degrees, and there has recently been progress in making such results effective and explicit.

In this talk we will see an overview of a technical framework that was developed to generalize the more recent effective results, and that unifies the treatment of different cases that have so far been dealt with separately, such as elliptic curves with or without complex multiplication. The level of generality of this theory allows one to obtain similar results for higher dimensional abelian varieties and other classes of commutative algebraic groups, provided that certain technical conditions are satisfied.

##### Sergej Monavari (Utrecht University)

#### Modern Enumerative Geometry: a tale of Math and Physics

Enumerative Geometry deals with the problem of counting — or better, enumerating — geometric object. Such geometric questions fascinated ancient Greeks, but were classically addressed with naïve techniques. In the 20th Century, the rapid development of Algebraic Geometry offered sufficiently solid foundation for a proper and systematic study of the subject, which was still lacking rigor and heavily relied on the purely geometric intuition. Towards the end of the last Century, the development of Theoretical Physics — String Theory, to be precise — revived the interest in Enumerative Geometry, due to the strong connection of the latter to instanton counting and its efficacy in mathematically formalizing (some instances of) a theory of quantum gravity.

In this short presentation we are going to gently walk from a very classical problem — easily stated in geometric terms but elegantly solved with the powerful machinery of Algebraic Geometry — to one of the results of my thesis, which solves a beautiful conjecture formulated in the Physics literature, but enjoys interpretations in many related mathematical fields, from Algebraic Geometry to Representation Theory and Combinatorics.

##### Lucas Slot (CWI)

#### Sums of squares and approximation hierarchies for polynomial optimization

Sums of squares have a long history in real algebraic geometry, dating back to the work of Hilbert and Artin in the early 1900s. Classical results of Putinar and Schmüdgen from the 1990s show that any polynomial which is positive on a compact semialgebraic set X has a structured representation in terms of sums of squares and the polynomial inequalities defining X. More recently, these so-called Positivstellensätze have found a new application in the field of mathematical programming; specifically in polynomial optimization, where one is tasked with computing the global minimum of a polynomial f on a given semialgebraic set X. Lasserre and Parrilo use sums of squares to define hierarchies of tractable bounds on these hard, non-convex optimization problems.

A key property of their hierarchies is that they converge asymptotically to the true minimum of f on X. A natural question is whether one may estimate the rate of this convergence. In this talk, we present several of such estimates in a variety of settings. Our proofs rely on classical orthogonal polynomials, Fourier analysis and reproducing kernels. From the point of view of optimization, our results may be seen as showing guarantees on the quality of the bounds of Lasserre and Parrilo. From the point of view of algebraic geometry, we show upper bounds on the minimum degree of the sums of squares required to represent positive polynomials in Putinar’s and Schmüdgen’s Positivstellensätze.