AGGITatE 2023

10am: Morning talk 1

Mima Stanojkovski: Smooth cuboids in group theory 

A smooth cuboid can be identified with a 3x3 matrix of linear forms, with coefficients in a field K, whose determinant describes a smooth cubic in the projective plane. To each such matrix, one can associate a group scheme G over K. In particular, when K is the field of rationals and F is the field of p elements, the F-points G(F) of G form a finite p-group, and, as p varies, one obtains an infinite family of groups. In this talk, I will present joint work with Josh Maglione and Christopher Voll on the geometric study of automorphisms and isomorphism types of groups defined from smooth cuboids. I will also explain a connection with Higman's PORC conjecture and show that our geometric approach yields faster isomorphism testing of groups in this context.

 

11am: Break

 

11:30am: Morning talk 2

Susanna Zimmermann: Real birational involutions of rational conic fibrations

It is an old problem to classify the finite subgroups of Cremona groups. It was initiated by Bertini, and the most complete classification so far has been achieved by Dolgachev-Iskovskikh and Blanc over an algebraically closed field. Over the field of real numbers, the classification is almost complete, with a large part completed by Robayo and Yasinsky in 2017. In this talk, we will focus on birational involutions of the real plane; we will discuss the general attack to the classification and then focus on a particular subclassification - one of the birational involutions of the plane preserving a pencil of rational curves. This is joint work with I. Cheltsov, F. Mangolte and E. Yasinsky.

 

12:30pm: Lunch break 

 

2pm: Afternoon talk 1

Michael Bate: Edifices

Jacques Tits established the theory of buildings about fifty years ago as a way to bring combinatorial and geometric techniques to the study of semisimple algebraic groups (especially exceptional groups, where no "obvious" geometry immediately presents itself). In this talk I will report on some recent work which shows how to attach a building-like structure to an arbitrary smooth connected algebraic group over a field; we have christened these new structures "edifices". Edifices share some, but not all, of the nice geometric and combinatorial properties of spherical buildings - they are complete metric spaces, for example, but may not be simplicial complexes - and are well-behaved with respect to natural operations on algebraic groups, like homomorphisms and base change.

In this talk, I'll discuss how to construct an edifice, some basic properties and examples, and some of the motivation for studying them.

 

3pm: Break

 

3:30pm: Afternoon talk 2

Christian Urech: Representation dimension of finite subgroups of Cremona groups

Studying finite subgroups of Cremona groups is a very rich subject with a long history. Already in rank two there is no complete classification of finite subgroups of Cremona groups. Instead of giving a complete classification, one can try to find bounds on the complexity of these subgroups. In this talk, we will consider the minimal dimension of faithful representations of finite groups as a measure for their complexity. This is joint work with A. Duncan and B. Heath.

7pm: Conference dinner (Brook Red Lion Hotel)

09:30am: Morning talk 1

Andrea Fanelli: On the Cremona group over imperfect fields

In this talk I will present an ongoing joint project with Fabio Bernasconi, Julia Schneider and Susanna Zimmermann aimed to extend some structural results on the rank-two Cremona group to the case of imperfect fields. Our motivation comes from the study of algebraic subgroups of the rank-three Cremona group over algebraically closed fields and our approach is birational geometric: our strategy requires a careful study of the Sarkisov program for rational surfaces over imperfect fields.

 

10:30am: Morning talk 2

Anne Lonjou: Regularisable subgroups of the Cremona group

The Cremona group, the group of birational transformations of the projective plane is a group which is nowadays well understood. A birational transformation is said regularisable if it is conjugated to an automorphism of a projective surface. Such elements are classified but the following natural question about regularisation is still open: Consider a finitely generated subgroup G of the Cremona group such that any element is regularisable. Is G conjugated to a subgroup of an automorphism of a projective surface ? In this talk, we will see how this question is related to understand elliptic elements and subgroups for the action of the Cremona group on some CAT(0) cube complexes and in which cases we are able to solve this question. This talk is based on several works with C. Urech, A. Genevois and P. Przytycki.

 

11:30am: Break

 

12pm: Morning talk 3

Frédéric Mangolte: Comessatti’s Theorem on Rational Surfaces and Real Fano threefolds

From the classification of real rational surfaces worked out by Comessatti at the beginning of the 20th century we get the following striking characterization of real rational surfaces: a geometrically rational real surface is rational if and only if its real locus is non-empty and connected. In a work in progress with Andrea Fanelli, we explore real loci of geometrically rational Fano threefolds and study the rationality of these.

 

2pm: Afternoon talk 1

David Stewart: Absolute rigidity of simple modules for algebraic groups (Joint work with Michael Bate)

Let k be a field and G be a smooth affine k-group of finite type. We show that the simple G-modules are absolutely rigid; that is, if V is a simple G-module, then the socle and radical series for V_E as a G_E-module coincide for any field extension E/k. Central to the proof is the authors’ recent classification of simple modules for connected G by highest weight. We use this to describe the endomorphism ring of simple G-modules and thereby give a dimension formula. For most of our results, the key case to consider is when G is a pseudo-split pseudo-reductive group and E/k is finite and purely inseparable.

 

3pm: Break

 

3:30pm: Afternoon talk 2

Anna Bot: Realisation of birational maps

Given an invertible matrix F, we investigate under which conditions it can be realized as a birational map f of ℙ², meaning if there exists a blow-up to a rational surface X on which f lifts to an automorphism whose action on the Picard group of f is precisely F. Based on ideas by McMullen and Uehara, we give sufficient conditions on points on a cuspidal cubic, and indicate how this can be used to prove that the ordinal of all dynamical degrees of all birational maps of ℙ² is ω^ω.

 

5-7pm: Drinks reception

10am: Morning talk 1

Paula Lins: Bivariate zeta functions and twisted conjugacy  

In this talk we will discuss bivariate zeta functions of a large class of finitely generated nilpotent groups encoding numbers of (twisted-)conjugacy classes of each cardinality of congruence quotients of the associated groups.

Twisted conjugacy is a generalisation of usual conjugacy in which one considers a group automorphism φ of Γ and the action g · x = gxφ(g)^-1 on Γ. The orbits of such action are called twisted conjugacy classes (also known as Reidemeister classes). We will discuss how bivariate zeta functions can be used to determine the possible sizes of twisted-conjugacy classes of finite nilpotent groups.

 

10:30am: Morning talk 2

Sokratis Zikas: Equivariant geometry of quadric bundles and maximal subgroups of Cremona groups

Maximal connected algebraic subgroups of the group of birational transformations Bir(X) of a variety X appear as automorphism groups of Mori fiber spaces birational to X. When X = ℙⁿ for n = 2, 3 we have a classification of such groups and the models they act on, while in dimension 4 and more this is an open problem. In this talk we will explore the equivariant geometry of a class of quadric fibrations over ℙ¹. As a result, we will see how their automorphism groups gives us families of maximal subgroups of Bir(ℙⁿ) for any n at least 4. This is work in progress, joint with Enrica Floris.

 

11:30am: Break

 

12:00pm: Afternoon Talk 1

Ivan Cheltsov: Equivariant birational geometry of the three-dimensional projective space

In this talk, we will discuss G-equivariant birational geometry of the three-dimensional projective space for a finite group G acting biregularly on the projective space. In particular, we will describe all possibilities for the group G such that the projective space is not G-equivariantly birational to fibrations into rational curves or surfaces.

 

13:00pm: End

Participant	Affiliation
Michael Bate	University of York
Marta Benozzo	LSGNT
Anna Bot	University of Basel
Tiago Duarte Guerreiro	University of Essex
Erroxe Etxabarri Alberdi	University of Nottingham
Andrea Fanelli	Université de Bordeaux
Simon Hart	University of York
Liana Heuberger	University of Bath
Paula Lins de Araujo	University of Lincoln
Alastair Litterick	University of Essex
Anne Lonjou	UPV & Ikerbasque
Jesus Martinez Garcia	University of Essex
Mima Stanojkovski	Università di Trento
David Stewart	University of Manchester
Harvey Sykes	Durham University / University of Essex
Adam Thomas	University of Warwick
Marc Truter	University of Warwick
Christian Urech	EPFL
Gerald Williams	University of Essex
Elena Denisova	University of Edinburgh
Sokratis Zikas	University of Poitiers
Susanna Zimmermann	Université Paris-Saclay
Arman Sarikyan	University of Edinburgh
Ivan Cheltsov	University of Edinburgh
Mangolte	Aix-Marseille Université
Alexandra Ciotau	University of Manchester
Xuantong Qu	University of Nottingham
Joseph Malbon	University of Edinburgh
Antoine Pinardin	University of Edinburgh
Michael Chitayat	University of Ottowa