The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterisation in terms of subgroups whose centraliser is not p-nilpotent. Recent work with Greenlees has highlighted a connection with group actions and the singularity category of the cochains on the classifying space. My plan is to give an introduction to these ideas, and present some recent theorems and conjectures.

Abstract: Let G be a finite group acting linearly on the polynomial ring  with invariant ring R. We assign, to a  linear representation of G, a corresponding quotient scheme over Spec R, and we show how to reconstruct the action from the quotient scheme. This works in particular in the case of a reflection group, where Spec R itself is an affine space, in contrast to the Auslander correspondence, where one has to assume that the basic action is small, i.e. contains no pseudo reflection.  These quotient schemes exhibit rich geometric features which mirror properties of the representation. In order to understand the image of this construction, we encounter module schemes (a forgotten notion of Grothendieck), module schemes up to modification and fiberflat bundles.

Abstract: Alexander Pope once said "To err is human, to forgive is divine."  Sometimes, though, finding errors is also divine.  In this talk I will discuss the problem of finding many small integral elements of a new family of number fields that can be used as ``errors" in homomorphic encryption schemes.  A key part of the construction is making explicit the group theory behind the construction of infinite class field towers by Golod and Shafarevitch.  This is joint work with Frauke Bleher.

Abstract: We prove Quillen's conjecture for finite simple unitary groups and obtain, as a consequence of the Aschbacher-Smith work, Quillen's conjecture for all odd primes.

Abstract: Let  k  be a field, G  an affine  k-group scheme of finite type, and  V  a small finite-dimensional G-module (small means that when we remove G-stable closed subsets of codimension two or more from  V  and  V//G, the canonical map  V -> V//G  is a G-torsor).  Is it true that the ring of invariants  k[V//G]=k[V]^G  is quasi-Gorenstein if and only if  G->GL(V) factors through  SL(V) (naive Watanabe's theorem)?  There has been affirmative answers by Watanabe, Braun, Fleischmann-Woodcock, and Liedtke-Yasuda.  On the other hand, Knop studied actions of (not necessarily connected) reductive groups in characteristic zero, and gave a negative answer to this question.  He clarified the contribution of the top exterior power of the adjoint representation.  This is generalized to the (dual of) the canonical module  k[G_ad]  of  G  with the adjoint action, pulled back to the unit element of G (it is a one-dimensional representation of  G).  We show that if  G  is a finite group scheme, then this one-dimensional representation of   G  is trivial if and only if the  k-algebra  k[G]^*, where  k[G] is the coordinate ring of  G, is a symmetric algebra.  As a corollary, we recover the results of Braun, Fleischmann-Woodcock, and Liedtke-Yasuda.  We also show an example that naive Watanabe's theorem does not hold for a finite group scheme  G.

Abstract: We will discuss the problem of lifting a smooth projective curve with an action of a group of automorphisms from prime characteristic to characteristic 0. We will give a criterion based on the Harbater-Katz-Gabber compactification of local actions, which allows us to decide whether a local action of a semidirect product of a cyclic p-group by a cyclic prime-to-p-group lifts or not. In particular, we give an example of dihedral local action that can not lift and in this way we give a stronger obstruction to the lifting than the  so-called KGB-obstruction, and disprove the generalized Oort conjecture.

Abstract: Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. $K^b({}_{kG}\mathbf{triv})$ is the bounded homotopy category of $p$-permutation $kG$-modules, and is tensor-triangulated. Due to recent work of Balmer and Gallauer, $K^b({}_{kG}\mathbf{triv})$ admits a Verdier quotient equivalent to $D^b({}_{kG}\mathbf{mod})$, the bounded derived category of $kG$-modules, and its Balmer spectrum has been described.

Given a tensor-triangulated category, it is generally a question of interest to determine its invertible objects. Endotrivial complexes are the invertible objects of $K^b({}_{kG}\mathbf{triv})$, and are analogous to endotrivial modules, the invertible objects of the stable module category $\mathbf{stmod}(kG)$. These complexes also connect to other objects of interest to modular representation theorists, such as endo-$p$-permutation modules, splendid Rickard equivalences, and the trivial source ring. In this talk, we will introduce these complexes and describe some of their connections to other topics of interest. Unlike the endotrivial module case, where the task of classifying all of these modules is open, we have a complete classification of endotrivial complexes, which we will discuss in the talk as well.

Abstract: I will discuss a recent computation of the rings of of invariants for the defining representations of the finite orthogonal groups of plus type in odd characteristic. This is joint work with Eddy Campbell and David Wehlau.

Abstract: Let k be a field, let G be a smooth affine k-group of finite type and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V ; say V is geometrically rigid (resp. absolutely rigid ) if V is rigid after base change of G and V to an algebraic closure of k (resp. any field extension of k). We show that all simple G-modules are geometrically rigid, though they are not in general absolutely rigid. More precisley, we show that if V is a simple G-module, then there is a finite purely inseparable extension k_V /k naturally attached to V such that V_{k_V} is absolutely rigid as a G_{k_V} -module. The proof for connected G turns on an investigation of algebras of the form K \otimes_k E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension k_V /k through an analogous version for artinian algebras.