The Schedule
10th of December
10th of December
10:15 - 10:30
Reception
Reception
André Carvalho and Beatriz Santos
10:30 - 11:20
Endomorphisms of Automatic Groups
Endomorphisms of Automatic Groups
André Carvalho
Abstract: Automatic groups were introduced by Epstein et al. and have been since then studied by many others. Despite being a rich and important class in the theory of groups and the fact that the study of endomorphisms plays an important role in the theory of finitely generated groups, not much is known about endomorphisms of automatic groups.In this talk, we will discuss how the well-known bounded reduction property (BRP) can be phrased in this setting and see that, in some sense, a synchronous version of it holds for all endomorphisms of finite kernel with quasiconvex image, generalizing a result known for hyperbolic groups. Finally, if time permits, we will apply these techniques to obtain some results on the equalizer of two endomorphism, giving an alternative proof to a result by Gersten-Short.Break 11:20 - 11:30
11:30 - 12:20
Symplectic Keys - Direct Way
Symplectic Keys - Direct Way
João Santos
Abstract: The right and left key maps for Kashiwara-Nakashima tableaux in type C are used to describe type C Demazure and opposite Demazure crystals, respectively. For instance, Fu-Lascoux non-symmetric Cauchy kernels expand into products of opposite Demazure characters and Demazure atoms. These key maps are related via the Lusztig involution and it has been shown that each type C key map can be computed using the Lecouvey-Sheats symplectic jeu de taquin (Lecouvey-Jacon 2020, S. 2021). In fact, each vertex of a crystal of straight shaped Kashiwara-Nakashima tableaux in type C is attached to a cocrystal of type A, similarly to Lascoux's double crystal graph construction, where, in our case, symplectic jeu de taquin slides on consecutive columns are used as crystal operators. These cocrystals contain all the needed information to calculate the left and right key maps. On the other hand, motivated by Willis' direct way for computing type A right and left keys (2011), we also give a way of computing symplectic right and left keys without the use of jeu de taquin.Lunch 12:20 - 14:00
14:00 - 14:50
Moduli spaces through the stack of triangles
Moduli spaces through the stack of triangles
Pedro Silva
Abstract: When classifying mathematical objects, one often encounters a continuum of different classes. These classes can sometimes be described by a space of parameters, whose topology reflects the classification itself. These are the Moduli spaces. In Geometry, for example, they are spaces of geometric objects, such as points, lines or curves. Their topology retains information about how these objects are geometrically similar to one another.In this talk, we will give an informal introduction to the study of Moduli problems, using the example of the stack of triangles as a guide. We will also try to understand how non-trivial automorphisms prevent the existence of fine moduli spaces, as observed by Grothendieck. Finally, time permitting, we will sketch a way out of this problem, by introducing the notion of stack in a very concrete setting, following Deligne-Mumford and Artin (yet opposing their level of abstraction).
Break 14:50 - 15:00
15:00 - 15:50
The star-shaped and convex transform orders
The star-shaped and convex transform orders
Beatriz Santos
Abstract: Given two systems, whose lifetimes are represented by random variables, being able to decide which system ages faster is a problem of special interest in reliability theory. Stochastic orders provide a convenient way to order random variables and they are usually defined through relationships between the respective distribution, survival, failure rate functions, etc. Two very popular order notions that capture the meaning of a system aging faster than another are the star-shaped and convex transform orders, which are defined by suitable transformations of the survival functions. The direct verification of these orders may raise technical difficulties, either because the survival functions do not have explicit formulas or because the systems under comparison belong to families of distributions indexed by several parameters. In this talk, we will present some criteria that allow us to establish star-shaped and convex transform comparibility between different families of distributions.Break 15:50 - 16:00
16:00 - 16:50
Automaton and Non-Automaton Semigroups
Jan Philipp Wächter
Abstract: Finite automata provide a different means to present groups and semigroups. In group theory, many famous dichotomy conjectures could be disproved using groups generated this way as counter-examples (such as the Milnor problem on growth, the Burnside problem on infinite torsion groups and Day’s problem of finding amenable but not elementary amenable groups). This underlines that groups generated by an automaton – or “automaton groups” for short – often have quite exotic properties.Still automaton groups also share some – sometimes quite mundane – common properties not satisfied by all groups (such as being finitely generated), which trivially proves that not every group is an automaton group. However, the knowledge on groups which cannot be generated using automata is very limited; in particular, on non-automaton groups that still share the typical common properties of automaton groups. In fact, the only known results in this respect seem to be for semigroups rather than groups.
The talk will provide an introduction to the topic and present known results of non-automaton semigroups. Eventually, we will also have a look at the directions this research could head to.
Coffee Break