# The 19th Graduate Colloquium of the Swiss Doctoral Program in Mathematics

Good to know

• All talks take place in Hörsaal 119 at Kollegienhaus (address: Petersplatz 1, 4051 Basel).
• We will have the social dinner at Restaurant Union (address: Klybeckstrasse 95, 4057 Basel) on Tuesday evening. It’s a burger place with nice vegetarian (and vegan) options.
• If the weather is nice, bring your swimsuit! Those who want might take a dip in the Rhine.
• The talks should take around 45 min.
• Organiser: Julia Schneider (Julia.noemi.schneider@unibas.ch)

The Plan

Monday, June 4

 14:00 Welcome Coffee 14:30 Linda 15:30 Luc 16:30 Tour de Bâle

Tuesday, June 5

 9:30 Good Morning Coffee 10:00 Mattias 11:00 Christian 12:00 Lunch 14:00 Lucas 15:00 Roman 19:00 Dinner

Wednesday, June 6

 9:00 Good Morning Coffee 9:30 Gabriel 10:30 Giacomo 11:45 Birkhäuser Prize

## Abstracts

Linda Frey (University of Basel)

Introduction to heights, the Bogomolov property and elliptic curves

We will see some beauty of number theory without any technicalities. This talk will introduce even applied math PhD students to some hardcore algebraic number theory without even noticing it. We will introduce the notion of the height of an algebraic number, mention some properties and learn about the Bogomolov property for fields. Using elliptic curves we will construct such a field.

Luc Pétiard (University of Neuchâtel)
TBA

Mattias Hemmig (University of Basel)
TBA

Christian Schulze (University of Basel)
Cellular mixing with bounded palenstrophy

We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on the two dimensional unit square. The scalar solves the continuity equation with a divergence-free velocity field $u$ with uniform-in-time bounds on the palenstrophy. We measure the degree of mixedness of the tracer $\rho$ via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scale. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.

Lucas Dahinden (University of Neuchâtel):

Counting moduli spaces of circular linkages

A (planar) linkage is a collection of bars of fixed length in the plane that are connected through joints around which the bars can freely rotate. Since a linkage has a certain liberty from the joints and rigidity from the barlengths, it is an interesting task to study the space of its possible positions. As an exercise, try to find the space of positions of a quadrilateral (= circular linkage with four bars) with side lengths (4,3,3,1). Surprisingly, we can find topological invariants of this space by simple combinatorics. When we go a level up and look at the set of possible circular linkages, and count the different moduli spaces that can arise this way, the "simple combinatorics" develop into a hard problem.

Roman Prosanov (University of Fribourg)

A variational approach to Alexandrov-type results

How can we describe the boundary of a 3-dimensional Euclidean polytope from the intrinsic metric viewpoint? One can easily find that it is a metric on the 2-sphere, which is flat everywhere except vertices where it has conical singularities of total angle less than 2pi (we call it positive curvature). A natural question is if this description is complete. It was answered positively by Alexandrov, who proved that every flat metric on the 2-sphere with conical singularities of positive curvature can be uniquely (up to isometry) realized as the induced metric on the boundary of a 3-dimensional convex polytope.

In 90's Igor Rivin obtained a similar result for hyperbolic cusp-metrics on the 2-sphere and ideal hyperbolic 3-polytopes. Ten years after Jean-Marc Schlenker gave a proof for the case of hyperbolic cusp-metrics on surfaces of genus > 1 and ideal Fuchsian 3-polytopes. All their original proofs were indirect.

In our talk we will discuss a more constructive new approach to the results of this type. It is based on resolving singularities in polytopal manifolds by a variational technique. We will also consider some another perspectives of this method.

Gabriel Dill (University of Basel)
Unlikely intersections: a fairy tale

Once upon a time, there were two polynomials G(T) and H(T). They liked roots of unity very much and were always happy when in the forest of complex numbers they found some number at which both their values were roots of unity. They wondered if they could always find another such number or if at some point there would be no more new ones left. They went to see three wise men called Ihara, Serre and Tate who told them the answer. And they lived happily ever after.

Giacomo Elefante (University of Fribourg)
QMC method for integration on manifolds with mapped low-discrepancy points and greedy minimal k_s-energy points

To integrate with the Quasi-Monte Carlo method (QCM) on two-dimensional manifolds we consider two sets of points.

The first is the set of mapped low-discrepancy sequence by

a measure preserving map.

Low-discrepancy points are best choice to integrate functions through QCM in the unit cube [0,1]^d but to use them to integrate functions on a manifold we need to preserve their uniformity with respect to the Lebesgue measure.

The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold.

We chose greedy minimal energy points since thanks to the Poppy-seed Bagel Theorem (cf. Saff) we know that the class of points with minimal Riesz $s$-energy, under

suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure.

On the other hand, we do not know if the greedy extraction produce a set of points that are

a good choice to integrate functions with the QCM on manifolds.

Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions.

Bibliography:
D. P. Hardin and E. B. Saff, Minimal Riesz energy point configurations for rectifiable $d$-dimensional manifolds., Adv. Math., vol. 193, no. 1, pp. 174-204, 2005.