# Schedule for: 17w5162 - Symmetries of Surfaces, Maps and Dessins

Beginning on Sunday, September 24 and ending Friday September 29, 2017

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, September 24
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (Corbett Hall Lounge (CH 2110))
Monday, September 25
07:00 - 08:45 Breakfast
Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
08:45 - 09:00 Introduction and Welcome by BIRS Station Manager (TCPL 201)
09:00 - 10:30 Marston Conder: Theme 1: Regular and Edge-Transitive Maps
Introduction (by Marston Conder and Tom Tucker) to regular maps and edge-transitive maps, including some important open questions in this theme area
(TCPL 201)
10:30 - 11:00 Coffee Break (TCPL Foyer)
11:00 - 12:30 Gabino González-Diez: Theme 2: Belyi theory & the absolute Galois group
Introduction (by Ernesto Girondo and Gabino González-Diez) to Belyi theory and the absolute Galois group, including some important open questions in this theme area
(TCPL 201)
12:30 - 13:30 Lunch (Vistas Dining Room)
14:00 - 14:45 Allen Broughton: Theme 3: Defining equations for a Riemann surface
Introduction (by Allen Broughton and Aaron Wootton) to defining equations for a Riemann surface, including some important open questions in this theme area
(TCPL 201)
14:45 - 15:00 Group Photo
Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!
(TCPL Foyer)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:30 Introduction of Additional Themes, 4-6
Additional themes, including some important open questions in each theme area: (a) Theme 4: Properties of higher rank structures (introduced by Dimitri Leemans); (b) Theme 6: $n$-gonal morphisms of Riemann surfaces (introduced by Mariela Carvacho); (c) Theme 5: Cayley maps, skew morphisms and Hamiltonicity of regular maps (introduced by Robert Jajcay).
(TCPL 201)
15:30 - 15:50 Dimitri Leemans: Theme 4: Introduction to incidence geometry with an emphasis on polytopes, hypertopes, maps and maniplexes (TCPL 201)
15:50 - 16:10 Mariela Carvacho: Theme 6: Overview on $n$-gonal automorphism of Riemann surfaces (TCPL 201)
16:10 - 16:30 Robert Jajcay: Theme 5: Cayley Maps and Skew Morphisms (TCPL 201)
16:30 - 16:50 Jen Paulhus: A database of group actions
In the late 1990s, Thomas Breuer created an algorithm, and wrote corresponding computer code, which determines all groups acting on Riemann surfaces of a fixed genus. His code also determined generating vectors for the actions, but he did not record that data. The results of his program up to genus 15, along with additional information such as if the Riemann surface is hyperelliptic or cyclic trigonal, the decomposition of the corresponding Jacobian Variety, and the dimension of the corresponding Shimura variety, can now be found online at: http://www.lmfdb.org/HigherGenus/C/Aut/. We’ll briefly introduce these pages, discuss features of them, and mention future planned additions.
(TCPL 201)
16:50 - 17:10 Alina Vdovina: Some topics on dessins and ribbon graphs
I will present a number of relevant results which may be not known in the ribbon graph or dessins communities
(TCPL 201)
17:10 - 17:30 Dimitri Leemans: Almost simple groups and polytopes
In this talk we will briefly summarise what is known about almost simple groups acting regularly/chirally on abstract polytopes, and state some open questions
(TCPL 201)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Tuesday, September 26
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:05 - 09:30 Jozef Siran: Theme 7: Super-symmetries of regular maps
An introduction to Theme 7 (on uper-symmetries of regular maps)
(TCPL 201)
09:35 - 10:30 Francisco Javier Cirre Torres: Theme 8: Other aspects of group actions on surfaces
An introduction to Theme 8 (other aspects of group actions on surfaces, such as pseudo-real surfaces and purely non-free actions) by Grzegorz Gromadzki and Javier Cirre, including some important open questions in this theme area
(TCPL 201)
10:30 - 11:00 Coffee Break (TCPL Foyer)
11:00 - 12:30 Working groups (Phase 1)
Working groups on the first four main themes: 1) Regular and Edge-Transitive Maps 2) Belyi theory & the absolute Galois group 3) Defining equations for a Riemann surfaces 4) Properties of higher rank structures.
(TCPL - various rooms)
12:30 - 13:30 Lunch (Vistas Dining Room)
14:00 - 15:00 Working groups (Phase 1, cont.) (TCPL - various rooms)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:10 Open problem session (TCPL 201)
16:10 - 16:30 Alexander Zvonkin: Diophantine invariants of dessins d'enfants
During numerous talks and conferences, a question was often raised: How to find a complete set of combinatorial and/or group-theoretic invariants of the Galois action on dessins. We will show that this is impossible. Indeed, sometimes combinatorial orbits of dessins split into several Galois orbits because of some subtle Diophantine relations between degrees of of vertices and faces.
(TCPL 201)
16:30 - 16:50 Becca Winarski: Homomorphisms between mapping class groups of surfaces
A general problem is to understand all (injective) homomorphisms between (finite index subgroups of) mapping class groups of surfaces. Birman and Hilden proved that if $S\rightarrow X$ is a regular branched covering space of surfaces, there is an embedding of the subgroup of the mapping class group of $X$ consisting of mapping classes that have representatives that lift to $S$ in the mapping class group of $S$ modulo the group of deck transformations. This relationship does not always hold for irregular branched covers. We give a necessary condition and a sufficient condition for when such an embedding exists. We also give new explicit examples that satisfy the necessary condition and examples that do not satisfy the sufficient condition.
(TCPL 201)
16:50 - 17:10 Roman Nedela: Complete regular dessins and skew-morphisms of cyclic groups
In this talk we introduce a surprising correspondence between $(m,n)$-complete regular dessins and admissible pairs of skew-morphisms of the cyclic groups of orders $m$ and $n$. A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. We determine the pairs $(m,n)$ for which there exists exactly one dual pair of $(m,n)$-complete regular dessins, thus generalising an earlier result by Jones, Nedela and \v Skoviera (2008). This is joint work with Y.Q.Feng, Kan Hu and M. {\v S}koviera.
(TCPL 201)
17:10 - 17:30 Charles Camacho: Counting quasiplatonic cyclic n-gonal surfaces
Define QC$(n)$ to be the number of isomorphisms classes of quasiplatonic cyclic $n$-gonal surfaces. We use formulas of R.\ Benim and A.\ Wootton to describe a method for computing QC$(n)$. We also relate QC$(n)$ to the number of regular dessins d'enfants having a cyclic group of automorphisms of order $n$.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Wednesday, September 27
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:30 Working groups (Phase 1, final) (TCPL - various rooms)
10:30 - 11:00 Coffee Break (TCPL Foyer)
11:00 - 12:30 Reports back from Phase 1 working groups (TCPL 201)
12:30 - 13:30 Lunch (Vistas Dining Room)
14:00 - 15:00 Working groups (Phase 2)
Working groups on the next four main themes: 5) Cayley maps, skew morphisms and Hamiltonicity of regular maps 6) n-gonal morphisms of Riemann surfaces 7) "Super-symmetries" of regular maps (duality, etc.) 8) Other aspects of group actions on surfaces, such as pseudo-real surfaces and purely non-free actions.
(TCPL - various rooms)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:30 Working groups (Phase 2, cont.) (TCPL - various rooms)
16:30 - 16:50 Shaofei Du: Nilpotent primer hypermaps with hypervertices of prime valency
A (face-)primer hypermap is a regular oriented hypermap whose orientation-preserving automorphism group $G$ acts faithfully on the hyperfaces. In this paper, we investigate the primer hypermaps for which $G$ is nilpotent and the hypervertex-valency is a prime $p$. (We call these PNp-hypermaps.) We prove that for any PNp-hypermap, the group $G$ must be a finite $p$-group, and the number of hyperfaces is bounded above by a function of the nilpotency class of $G$. Moreover, we show that for any positive integer $c$, there is a unique PNp hypermap $H_c$ of class $c$ attaining the given bound, and every other PNp hypermap of class at most $c$ is a quotient of $H_c$.
(TCPL 201)
16:50 - 17:10 Milagros Izquierdo: Dessins d’enfants and a curve of Wiman
Except very few cases (7 in total), if a regular dessin of genus $g$ has rotational group of order $4g$, then the dessin lies on Wiman's curve with $8g$ automorphisms.
(TCPL 201)
17:10 - 17:30 David Torres: Teichmüller curves and Hilbert modular surfaces
Teichmüller curves are totally geodesic complex curves inside the moduli space of Riemann surfaces. Under a Prym-Torelli map, one can see them as curves in some Hilbert modular variety, which parametrises abelian varieties with real multiplication. We give an overview of how to cut out these curves as the vanishing locus of Hilbert modular forms and the invariants that can be computed from this viewpoint.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Thursday, September 28
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:30 Working groups (Phase 2, cont.) (TCPL - various rooms)
10:30 - 11:00 Coffee Break (TCPL Foyer)
11:00 - 12:15 Open problem session, or Working groups (final) (TCPL 201 - or other rooms)
12:15 - 13:15 Lunch (Vistas Dining Room)
13:45 - 18:00 Excursion to Lake Louise (or free afternoon)
Cost is CAD 30 per person (or CAD35 if feeling generous) for those who indicated they would take part. Free afternoon for those who indicated they would NOT take part.
(Meet out the front of the Professional Development Centre at 13:45)
17:30 - 19:30 Dinner (Vistas Dining Room)
Friday, September 29
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:15 Reports back from Phase 2 working groups (TCPL 201)
10:15 - 10:45 Coffee Break (TCPL Foyer)
10:45 - 11:30 Reports back from working groups, and final discussion on workshop and future plans (TCPL 201)
11:30 - 12:00 Checkout by Noon
5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.
(Front Desk - Professional Development Centre)
12:00 - 13:30 Lunch from 11:30 to 13:30 (Vistas Dining Room)