## 20th Graduate Colloquium

**PRACTICAL INFORMATIONS**

- All talk will take place at the villa Battelle, Route de Drize 9, 1227 Carouge (GoogleMaps)
- We will have a social dinner on Tuesday night
- Talks should last 45 minutes, and be aimed at PhD Students in any area of mathematical research. Typically, this means focusing on explaining the general context of a problem rather than giving its proof.
- Contact for the organizers : solenn.estier(at)unige(dot)ch

**SCHEDULE**

**MONDAY, NOVEMBER 18, 2019**

14:00 - Welcome coffee

14:30 - Sebastian Baader

15:30 - Talk 1

**TUESDAY, NOVEMBER 19, 2019**

9:30 - Good Morning Coffee

10:00 - Talk 2

11:00 - Talk 3

12:00 - Lunch

14:00 - Talk 4

15:00 - Talk 5

19:00 - Dinner

**WEDNESDAY, NOVEMBER 20, 2019**

9:00 - Good Morning Coffee

9:30 - Talk 6

10:30 - Talk 7

11:45 - Birkhäuser Prize

**ABSTRACTS**

**ARTHUR BIK** (University of Bern)

*Strength and polynomial functors*

One of the most important invariants one can associate to a matrix is its rank, which expresses how many pairs of vectors you need to write

down a formula for the matrix. Consider infinite-by-infinite matrices. For such a matrix, we define its rank to be the supremum of all its finite-by-finite submatrices. This can be finite, which is equivalent to the matrix A being able to be expressed using finitely many infinite vectors. Or else it is infinite, which turns out to be equivalent to stating that the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all infinite-by-infinite matrices.

For polynomial series, their strength fulfil a similar role. We define the strength of a polynomial series to be the infimum number of pairs of lower degree series needed to write down a formula for it. It is then true that the strength of a polynomial series is infinite if and only if the set of series obtained from it by finitely many substitutions is Zarisky-dense in the space it lives in. Both infinite-by-infinite matrices and polynomial series are examples of the following dichotomy: either you can express them using a finite amount of lower-dimensional

data or their orbit under some group is dense. This talk is about joint work with Jan Draisma, Rob Eggermont and Andrew Snowden that generalizes this statement to all finite-degree polynomial functors.

**GIACOMO ELEFANTE** (University of Fribourg)

*Some recent results on barycentric rational interpolation*

Barycentric rational interpolation is frequently used to approximate functions. In the setting of rational trigonometric interpolation, it was proved by Baltensperger that it is possible to achieve exponential convergence if the nodes used to interpolate are images of the equidistant nodes under a conformal map. Due to this result, we will first present a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts, and we will use it to achieve a better approximation of a function with a steep gradient.

In the second part of the talk, we will focus on the approximation of a discontinuous function in the rational polynomial interpolation framework. In this case we have that the Gibbs phenomenon appears. In order to cancel this, we will present an extension of the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced by De Marchi, Marchetti, Perracchione and Poggiali. Numerical tests show that it yields an accurate approximation of discontinuous functions.

LIVIO FERETTI (University of Bern)

**TBA**

LAURA GRAVE (University of Neuchâtel)

**TBA**

ALEJANDRO VARGAS (University of Bern)

**Counting with tropical geometry**

Tropical geometry is a young branch of algebraic geometry that rose to prominence by showing how to apply tools from polyhedral geometry to calculate invariants of algebraic varieties. In the first half of this talk I give a basic introduction of tropical geometry. In the second half I give three examples of counts in enumerative algebraic geometry that can be studied using tropical geometry: the number of plane curves of degree-d and genus-g passing through 3d−1+g points; the number of lines on a smooth cubic surface; and the number of degree-d morphisms from a genus-g curve to the projective line when d = g/2 + 1