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Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely…

For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We…

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our…

A new type of semigroups which appears while dealing with $N=1$ superconformal symmetry in superstring theories is considered. The ideal series having unusual abstract properties is constructed. Various idealisers are introduced and…

For p odd, the Lie group SO_0(p+1,p+1) has a family of unitary degenerate principal series representations realized on the space of real (p+1) by (p+1) skew symmetric matrices, similar to the Stein's complementary series for SL(2n,C) or…

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in…

We define various formal moduli spaces of p-divisible groups which are regular, and morphisms between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture of the third author…

Let $G$ be a finite group and $p$ be a prime divisor of $|G|$. An irreducible $p$-Brauer character $\varphi$ of $G$ is called super-monomial if every primitive $p$-Brauer character inducing $\varphi$ is linear. The group $G$ is said to be a…

This survey is about old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer's Main Theorems, fusion, Rickard equivalences. In the…

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is…

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any…

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygon. Using this method, I give a new proof that there exist supersingular curves of genus $4$ for every…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

A method to construct irreducible unitary representations of a hyperspecial compact subgroup of a reductive group over p-adic field with odd p is presented. Our method is based upon Cliffods theory and Weil representations over finite…

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of…

Many of the conjectures of current interest in the representation theory of finite groups in characteristic $p$ are local-to-global statements, in that they predict consequences for the representations of a finite group $G$ given data about…

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make…

We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of…

We show that there exists a family of irreducible representations R_i (of finite groups G_i) such that, for any constant t, the average of R_i over t uniformly random elements g_1, ..., g_t of G_i has operator norm 1 with probability…