Invited speakers include Boris Adamczewski (CNRS, U Lyon), Valentijn Karemaker (UU), Claire Mathieu (CNRS) and Benjamin Wesolowski (CWI).

### Contributing a talk

PhD students and postdocs are warmly welcomed to submit a contributed talk (25-30 mins.) to the symposium. In the registration form there is an option to submit title and abstract. Alternatively, after registration, a title+abstract can be sent to Robin de Jong at a later date.

### Conference fee + accommodation

There is full DIAMANT support for DIAMANT members. DIAMANT members are full and associate professors listed here, as well as their research group members.

### Programme

The programme starts Thursday at 11:00 AM. For more information on the topics being presented, see the ‘Speakers’ section of this page.

#### INVISIBLE

#### Thursday

11:00-11:25 | Arrival and coffee |

11:25-11:30 | Welcome |

11:30-12:25 | Claire Mathieu (CNRS) – Rank aggregation |

12:30-13:30 | Lunch |

13:30-13:55 | Sander Gribling (CWI) – Zero-error communication via quantum channels |

14:00-14:25 | Christos Pelekis (Institute of Mathematics, Prague) – A continuous analogue of Erdös’ k-Sperner theorem |

14:30-14:55 | Carlo Verschoor (UU) – A Bailey type factorization of Horn’s H4 hypergeometric function |

15:00-15:30 | Tea |

15:30-15:55 | Stefano Marseglia (UU) – Isomorphism classes of abelian varieties over finite fields |

16:00-16:25 | Jared Asuncion (UL) – Computing Hilbert class fields of quartic fields using complex multiplication |

16:45-17:40 | Benjamin Wesolowski (CWI) – Discrete logarithms in quasi-polynomial time in finite fields of small characteristic |

17:45-19.00 | Drinks |

19:00 | Dinner |

+- 20:30 | Discussion about mastermath courses 2020-2021 proposed by Diamant |

#### Friday

9:00-9:55 | Boris Adamczewski (U Lyon) – Expansions in independent bases, automata, and Mahler’s method |

10:00-10:25 | Jana Sotakova (UvA) – Adventures in Supersingularland |

10:30-11:00 | Coffee |

11:00-11:25 | Sebastiano Trento (UL) – Kummer Theory for Elliptic Curves |

11:30-11:55 | Djurre Tijsma (U Düsseldorf) – Automata and finite order elements in the Nottingham group |

12:00-13:00 | Lunch |

13:00-13:25 | Wouter Cames van Batenburg (ULB) – Large independent sets in triangle-free graphs: beyond planarity |

13:30-13:55 | Francesca Bianchi (RUG) – Explicit quadratic Chabauty over number fields |

14:00-14:25 | Jatin Batra (CWI) – Geometry of Scheduling on Multiple Machines |

14:30-15:00 | Tea |

15:00-15:55 | Valentijn Karemaker (UU) – Galois theory, dynamics, and combinatorics of Belyi maps |

16:00 | End of the symposium |

### Speakers

#### INVISIBLE

#### Claire Mathieu (CNRS)

##### Rank aggregation

How do you synthesize ranking information into a single most representative global ranking? This can be modeled as an optimization problem. There are a few possibilities for the objective. Related to the feedback arc set problem in directed graphs, its theoretical formulation is difficult to solve exactly. We will show sinple heuristics and analyze them from the viewpoint of quality of approximation.

#### Sander Gribling (CWI)

##### Zero-error communication via quantum channels

A basic question in information theory is how many distinct messages can be communicated through a (noisy) channel, with zero-error probability. If the channel is classical, then this number equals the stability/independence number of a graph associated to the channel. Duan, Severini, and Winter first studied the same question for quantum channels. They introduced the notion of quantum graphs, which turn out to be certain subspaces of matrix algebras. Parameters like the stability number and the Shannon capacity can be extended to this setting. I will motivate/explain the above notions and I will present a new bound on the Shannon capacity of quantum channels: a quantum version of the Haemers bound. Based on joint work with Yinan Li.

#### Christos Pelekis (Institute of Mathematics, Prague)

##### A continuous analogue of Erdös’ k-Sperner theorem

A chain in the unit *n*-cube is a set *C* having the property that for any two points in *C*, one is coordinate-wise dominating the other. I will show that the 1-dimensional Hausdorff measure of a chain in the unit *n*-cube is at most *n*, and that the bound is sharp. Moreover, I will discuss the problem of maximising the *n*-dimensional Lebesgue measure of a measurable subset, *A*, of the unit *n*-cube, subject to the constraint that the 1-dimensional Hausdorff measure of its intersection with every chain is at most *r*, where *r* is a fixed real number from the interval *(0,n]*. Our main result provides a sharp upper bound on the *n*-dimensional Lebesgue measure of *A* and may be seen as a continuous analogue of Erdös’ theorem on *k*-Sperner families of finite sets. This is joint work with Themis Mitsis and Václav Vlasák.

#### Carlo Verschoor (UU)

##### A Bailey type factorization of Horn’s H4 hypergeometric function

A well known identity by Bailey states that Appell’s F4 function can be written as the product of two Gauss hypergeometric functions under a suitable specialization of its parameters. Other identities of this type are known for Appell’s F2 and F3, which are closely related to Bailey’s identity. The aim of this talk is to show that the same can be done for Horn’s H4 function.

#### Stefano Marseglia (UU)

##### Isomorphism classes of abelian varieties over finite fields

We explain how to compute the isomorphism classes of abelian varieties in ordinary square-free isogeny classes in terms of (non necessarily invertible) fractional ideals of orders in étale algebras over Q. We present a concrete algorithm to perform this tasks and, for the ordinary case, to compute the polarizations and also the automorphisms of the polarized abelian variety.

#### Jared Asuncion (UL)

##### Computing Hilbert class fields of quartic fields using complex multiplication

The Hilbert class field H_K(1) is the maximal unramified abelian extension of K. For imaginary quadratic number fields K, it can be generated using special values of certain analytic, modular functions. For quartic CM-fields K, the corresponding construction yields only a subfield of H_K(1). Ray class fields are generalizations of Hilbert class fields. For a positive integer m > 0, the ray class field H_K(m) is obtained by relaxing the ramification conditions for ideals of O_K dividing m. It turns out that there is a particular subfield L(m) of H_K(m) which can be generated using special values of higher-level modular functions and Stark’s conjectures. For some values of m, this L(m) contains the Hilbert class field H_K(1). Thus, we can compute the Hilbert class field as a subfield of L(m). In this talk, we find an upper bound for such an integer m. If time allows, we concretize the theory by computing the Hilbert class field for a particular example.

#### Benjamin Wesolowski (CWI)

##### Discrete logarithms in quasi-polynomial time in finite fields of small characteristic

We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. In 1987, Pomerance proved that this problem can be solved in expected subexponential time L(1/2). The following 30 years saw a number of heuristic improvements, but no provable results. The quasi-polynomial complexity has been conjectured to be reachable since 2013, when a first heuristic algorithm was proposed by Barbulescu, Gaudry, Joux, and Thomé. We prove this conjecture, and more generally that this problem can be solved in the field of cardinality p^n in expected time (pn)^{2 log_2(n)+O(1)}.

#### Boris Adamczewski (U Lyon)

##### Expansions in independent bases, automata, and Mahler’s method

It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinary difficult to confirm this naive heuristic principle in some way or another. As always when mathematicians have to face such an enormous gap between heuristic and knowledge, it becomes essential to find out good problems. By that, we mean problems which, on the one hand, formalize and express the general heuristic, and, on the other hand, whose solution does not seem desperately out of reach. One aim of this talk is to suggest problems that, hopefully, belong to this category, as well as to present some partial solutions. At first, our problems involve automata theory, but we will see that they eventually fall into the scope of a method in transcendence number theory initiated by Mahler at the end of the 1920s. This method is concerned with algebraic independence of values at algebraic points of analytic functions satisfying some special kind of linear difference equations. This is a joint work with Colin Faverjon.

#### Jana Sotakova (UvA)

##### Adventures in Supersingularland

The problem of path finding in supersingular isogeny graphs has recently been proposed as a hard problem underlying the post-quantum key exchange scheme SIKE. So far, both quantum and classical attacks have exponential time complexity. However, for a small subset of the vertices, the Spine, path-finding is much easier. We will look at many examples of isogeny graphs to show how the Spine sits inside the supersingular isogeny graph. Moreover, we will determine the possible shape based on understanding a related isogeny graph over Fp. This is joint work with Sarah Arpin, Catalina Camacho-Navarro, Kristin Lauter, Joelle Lim, Kristina Nelson and Travis Scholl.

#### Sebastiano Trento (UL)

##### Kummer Theory for Elliptic Curves

Consider an elliptic curve E over a number field K, and a point a in E(K) of infinite order. We are interested in the field extensions of K that are generated by the coordinates of the N-division points of a, for any integer N. In particular, we would like to compute their degree over the N-torsion field of E. It is known that the ratio between N^2 and this degree is bounded independently of N. In a joint work with D. Lombardo we give an explicit value for this bound, in case End_K(E) is trivial, in terms of the l-adic Galois representations associated with E/K and of simple divisibility properties of the point a.

#### Djurre Tijsma (U Düsseldorf)

##### Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series t+a_2t^2+… in the variable t with coefficients a_i from the field of two elements, where the group operation is given by composition of power series. Only a handful of such power series of finite order are explicitly known through a formula for their coefficients. We argue that it is advantageous to describe such series in closed computational form through automata, giving explicit automaton-theoretic descriptions of new order 4 elements, an embedding of the Klein Four group into the Nottingham group at 2. We also relate the existence of `closed formulas’ to sparseness properties of the series and of the related automata. This is joint work with Gunther Cornelissen en Jakub Byszewski.

#### Wouter Cames van Batenburg (ULB)

##### Large independent sets in triangle-free graphs: beyond planarity

The independence number \alpha(G) of a graph G is the maximum size of a set of vertices that are pairwise nonadjacent. Even though determining \alpha(G) is NP-hard in general, one can sometimes derive an optimal bound that holds for all graphs in a given well-structured graph class. For instance: due to the four-colour theorem, every n-vertex planar graph has an independent set of size at least n/4. Zooming in further, Albertson, Bollob\’as and Tucker (1976) conjectured that every n-vertex planar triangle-free graph with maximum degree at most 3 has an independent set of size at least 3/8n. Long before Heckman and Thomas (2008) were able to prove this, it was conjecured by Fraughnaugh and Locke (1995) that the planarity requirement could actualy be relaxed into just forbidding six specific nonplanar induced subgraphs. In recent work, we have proved this much stronger conjecture. Joint work with Jan Goedgebeur and Gwenael Joret.

#### Francesca Bianchi (RUG)

##### Explicit quadratic Chabauty over number fields

The quadratic Chabauty method is a p-adic approach to finding solutions of certain types of Diophantine equations over the rationals. I will describe a generalisation to number fields of some explicit versions of this approach. This is joint work with Balakrishnan, Besser and Müller.

#### Jatin Batra (CWI)

##### Geometry of Scheduling on Multiple Machines

We consider the following general scheduling problem: there are m identical machines and n jobs all released at time 0. Each job j has a processing time p_j, and an arbitrary non-decreasing function f_j that specifies the cost incurred for j, for each possible completion time. The goal is to find a preemptive migratory schedule of minimum cost. This models several natural objectives such as weighted norm of completion time, weighted tardiness and much more. We give the first O(1) approximation algorithm for this problem, improving upon the O(log log nP) bound due to Moseley (2019). To do this, we first view the job-cover inequalities of Moseley geometrically, to reduce the problem to that of covering demands on a line by rectangular and triangular capacity profiles. Due to the non-uniform capacities of triangles, directly using quasi-uniform sampling loses a O(log log P) factor, so a second idea is to adapt it to our setting to only lose an O(1) factor. Our ideas for covering points with non-uniform capacity profiles (which have not been studied before) may be of independent interest.

#### Valentijn Karemaker (UU)

##### Galois theory, dynamics, and combinatorics of Belyi maps

A (genus zero) Belyi map is a finite map from the projective line to itself, branched exactly at *0, 1,* and infinity. Such maps admit combinatorial descriptions by their generating systems. If we further assume that *0, 1,* and infinity are fixed points as well as the unique ramification points above *0, 1,* and infinity respectively, then the resulting maps can be iterated and will therefore exhibit dynamical behaviour. We call these dynamical Belyi maps, and study their properties using a mixture of Galois theory, group theory, and combinatorics. In this talk, we will discuss several results on the monodromy, reductions, and dynamics of dynamical Belyi maps, and the interplay between these. In particular, we fully determine the arboreal Galois representations of dynamical Belyi maps, and prove a result about the density of prime divisors of the corresponding dynamical sequences. Part of this talk is joint work with I. Bouw and O. Ejder, and another part also with J. Anderson, N. Girgin, and M. Manes.

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