Abstracts

Abstract: file

Abstract: The number 3876 is one more than the dimension of the second smallest representation of E8. It has long been known that there exists a commutative nonassociative algebra on that representation, but its concrete construction is quite recent. In fact, Maurice Chayet and Skip Garibaldi have constructed a whole new class of such commutative nonassociative algebras, one for each simple Lie algebra. 

The explicit description of these algebras is quite involved and the formulas seem unnatural and lack an easy interpretation. 

In recent work with Louis Olyslager, we have discovered that these algebras have a beautiful interpretation inside the degree 2 component of an affine vertex algebra.

Abstract: In recent years there have been a number of developments towards a classification of primitive axial algebras of Jordan type. These can be regarded as the smallest non-trivial case of axial algebras, but the known examples remained limited to either Jordan algebras or Matsuo algebras until recently.

In this talk, we will outline how solid subalgebras, a concept devised by Sergey Shpectorov, can be used to unravel the structure of such algebras: it turns out that any primitive axial algebra of Jordan type is in some sense an extension of a Matsuo algebra by a Jordan algebra.

Time permitting, we will present a construction for non-trivial examples of such extensions, and we will outline some open questions towards a full classification.

This is joint work with Sergey Shpectorov

Abstract: Axial algebras of Monster type were introduced in 2015 by Hall, Rehren, and Shpectorov in order to axiomatise some key features of certain classes of algebras related to large families of finite simple groups.

In this talk I will report on the current status of the classification of the 2-generated primitive axial algebras of Monster type. This is a joint project with Mario Mainardis, Justin McInroy and Sergey Spectorov.

Abstract: The Killing-Cartan-Weyl classification of finite dimensional semisimple Lie algebras is elegant and brief. Existence of the exceptional algebras is perhaps the most difficult part. Starting from 3-transposition groups and a result of Cuypers-Hall-Segev-Shpectorov, we revisit the classification and smooth this out. An earlier solution is due to Ernst Witt. Both use extraspecial groups and their related orthogonal geometry.

Abstract: Saturated fusion systems are categories modelling the p-local structure of finite groups (i.e. the structure of the normalizers of non-trivial p-subgroups of finite groups). They play a role in Aschbacher’s program to revisit the proof of the classification of finite simple groups, in the modular representation theory of finite groups and in certain parts of homotopy theory. Linking systems associated to fusion systems were introduced by Broto, Levi and Oliver to study “p-completed classifying spaces of fusion systems”, but recent results suggest that they are also important from a purely algebraic point of view. Chermak showed that every linking system corresponds to a group-like structure called a locality. After an introduction to the subject, I will report in this talk on an attempt at simplifying Aschbacher’s program using the theory of localities.

Abstract: file

Abstract: I will discuss the group and geometry in the title.

Abstract: As a PhD student is was my honour to be invited by Sergey to collaborate on a revision on Phan's theorems, to be used in the second

generation proof of the classification of the finite simple groups. Not only did the team that Sergey formed establish all possible Phan-type

theorems that can exist: An (Phan, Bennett-Shpectorov), Bn (Bennett, Hoffman, Horn, Nickel, Shpectorov, K.), Cn (Hoffman, Shpectorov, K.), Dn

(Hoffman, Nickel, Shpectorov, K.), En (Phan, Devillers, Hoffman, Mühlherr, Shpectorov, Witzel, K.), F4 (Hoffman, Mühlherr, Sphectorov, Witzel, K.). 

It also linked to the study of finite presentations of centralizers of involutions in Kac-Moody-groups over finite fields (Devillers, Horn, Mühlherr, Witzel, K.), which in the affine case contributed to the study of finiteness properties of arithmetic lattices in semisimple Lie groups of positive characteristic (Bux, Witzel, K.), answering a question posed by Borel and Serre in 1976.

Abstract: I’ll present a joint work with Clara Franchi in which we use Majorana representations to study the subalgebras of the Griess algebra that have shape (2B,3A,5A) and whose associated Miyamoto groups are isomorphic to A_n. We prove that these subalgebras exist only if n∈{5,6,8}. The case n=5 was already treated by Ivanov, Seress, McInroy, and Shpectorov. In case n=6 we prove that these algebras are all isomorphic and provide their precise description. In case n=8 we prove that these algebras do not arise from standard Majorana representations.

Abstract: Axial algebras are a class of non-associative algebras with a strong natural link to groups which have recently received much attention. They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law. Of primary interest are the axial algebras with the Monster type $(\alpha, \beta)$ fusion law, of which the Griess algebra (with the Monster sporadic simple group as its automorphism group) is an important motivating example.

By previous work of Yabe, and Franchi and Mainardis, any symmetric 2-generated axial algebra of Monster type $(\alpha, \beta)$ is either in one of several explicitly known families, or is a quotient of the infinite-dimensional Highwater algebra $\mathcal{H}$, or its characteristic 5 cover $\hat{\mathcal{H}}$. We complete this classification by explicitly describing the infinitely many ideals and thus quotients of the Highwater algebra (and its cover), which turn out to have a simple combinatorial description. As a consequence, we find that there exist 2-generated algebras of Monster type $(\alpha, \beta)$ with any number of axes (rather than just $1,2,3,4,5,6,\infty$ as we knew before) and of arbitrarily large finite dimension.

In this talk, we will not assume any knowledge of axial algebras.

This is joint work with: Clara Franchi (Catholic University of the Sacred Heart, Brescia campus) and Mario Mainardis (University of Udine)

Abstract: file 

Abstract: Hilbert’s 17th problem (affirmatively resolved by Artin-Schreier) can be formulated as the existence, for any globally nonnegative 2n-ary homogeneous polynomial (a.k.a. fom) f, of a sum of squares (s.o.s., for short) form q so that qf is a sum of squares. For n=3, Hilbert has shown a quadratic, in the degree d of f, bound on the degree of q. In general, the best known degree bounds are huge. 

The next interesting case is n=4, and f of degree 4. In this case there exists a product q of two non-negative quadrics so that qf is an s.o.s. of quartics. There is a reduction to the case where f has a real projective zero; then f becomes a quadratic polynomial f=ax^2+2bx+c in one of the variables, x. Its discriminant D=ac−b^2 must be nonnegative, and allows one to write a decomposition for af=(ax+b)^2+D. Then either D is an s.o.s., or D−tb^2 is a non-exposed element of the boundary of the cone of nonnegative sextic 3-variate forms, for a nonnegative constant t. We discuss setbacks and progress towards resolving the question of minimal degree of q in this case (we know it is either 2 or 4).

Abstract: Matsuo algebras, introduced by Matsuo in 2007 are an important class of algebras of Jordan type. Every flip (an automorphism of order 2) σ of a Matsuo algebra M defines a flip subalgebra of M generated by all single and double axes fixed by σ. These flip subalgebras can be viewed as twisted versions of Matsuo algebras and they belong to the class of algebras of Monster type (2η, η). 

One of the key properties of an axial algebra is whether it is (semi)simple and if not then what is its radical. It turns out that both Matsuo algebras and their flip subalgebras are generically (i.e., for all but finitely many values of the parameter η) (semi)simple, that is, they have trivial radical. The exceptional values of η, for which the radical is non-zero, are called critical. Hall and Shpectorov suggested a method of finding the critical values for an arbitrary Matsuo algebra and finding the dimension of the radical. 

In the talk we will present some methods developed for determining key properties (simplicity and the dimension of radical) of flip subalgebras in Matsuo algebras. It turns out that flip subalgebras have the same critical values as their ambient Matsuo algebras and the dimension of the radical can be found by solving a simple system of linear equations.

Abstract: Recently I. Gorshkov, S. Shpectorov, A. Staroletov, and J. Desmet proved results describing when two-generated algebras of an axial algebra are solid. The premise of this talk is that these results  can be understood in terms of idempotental identities of axial algebras. This enables us to reconstruct universal primitive axial algebras, either with or without a Frobenius form, in a rather straightforward manner. 

Abstract: file 

Abstract: I will report on the work of Capdeboscq, Gorenstein, Lyons, Stroth and myself (CGLSS) on our revision of the classification proof for finite simple groups (CFSG). I will highlight the important role of Phan Theory, as developed by Sergey Shpectorov, Ralf Köhl and their collaborators. I will also state a conjecture related to the Z*p problem and highlight a result of George Glauberman and Geoff Robinson on this problem, which we use in our ongoing work extending Aschbacher’s important papers on groups of characteristic 2-type with e(G)=3 to groups of even type with e(G)=3.

Abstract: Hsiang algebras are a class of nonassociative algebras defined in terms of a relation quartic in elements of the algebra. This class arises naturally in relation to the construction of real algebraic minimal cones. It is known that any Hsiang algebra is also an axil algebra (normally with an infinite automorphism group). Furthermore, there are several intimate relations between Hsiang, Jordan and Clifford algebras, In particular, one can define a certain cubic form on Hsiang algebra A, analogues to the generic norm of a Jordan algebra (the zero locus of the generic norm is a minimal cone in the ambient vector space). In my talk I will discuss some remarkable properties of this generic norm. In particular, for almost all known Hsiang algebras A, the generic norm can be written (in a certain orthonormal basis) as the so-called Steiner triple form. Furthermore, the replication number of the corresponding Steiner triple system is uniquely determined by the Peirce dimensions of A. We shall discuss this construction and some of its applications.

The talk is based on an ongoing project with Daniel J. Fox (Madrid).

Abstract: Let G be a finite group and D be a normal set of involutions in G. We call (G,D) a k-transposition group if D generates G and the product of any two elements in D has at most order k. The 3-transposition groups were first investigated by Fischer, where he found three new sporadic groups, before a complete classification was made by Cuypers and Hall. Also, Fischer predicted the Baby Monster and Monster sporadic groups when looking at 4- and 6-transposition groups respectively. In this talk, we will discuss the previous work around transposition groups before giving some basic results. For the rest of the talk, we will be looking at the progress so far on classifying the almost quasisimple 6-transposition groups. This is joint work with Chris Parker.

Abstract:  Suppose that a finite group G acts on a set and that the maximum number of fixed points of non-trivial elements of G is constrained.         In this talk, we will see why this might be an interesting situation to look at from a group theoretic perspective and what type of results we can prove. We will also discuss some motivation and possible applications.