We present an overview of elementary methods to study extensions
of modular representations of various types of "groups". We shall
begin by discussing actions of an elementary
abelian pp-group, E=(Z/p)rE=(Z/p)r, on finite dimensional
vector spaces...
What combinatorial properties are likely to be satisfied by a
random subspace over a finite field? For example, is it likely that
not too many points lie in any Hamming ball of fixed radius? What
about any combinatorial rectangle of fixed side...
Let C be a class of groups. (For example, C is a class of all
finite groups, or C is a class of all finite symmetric groups.) I
give a definition of approximations of a group G by groups from C.
For example, the groups approximable by symmetric...
An extension of Gromov compactness theorem ensures that any
family of manifolds with convex boundaries, uniform bound on the
dimension and uniform lower bound on the Ricci curvature is
precompact in the Gromov-Hausdorff topology. In this talk,
we...
What can we say on a convex body from seeing its projections? In
the 80s, Lutwak introduced a collection of measurements that
capture this question. He called them the affine quermassintegrals.
They are affine invariant analogues of the classical...
(work in progress with R. Rouquier) I will present a
computational (yet conjectural) method to determine some
decomposition matrices for finite groups of Lie type. These
matrices encode how ordinary representations decompose when they
are reduced to...
The problem of classification of perverse sheaves on the
quotient h/W for a semisimple Lie algebra g has an explicit answer
which turns out to be related to the algebraic properties of
induction and restriction operations for parabolic
subalgebras...
The Hecke algebra admits an involution which preserves the
standard basis and exchanges the canonical basis with its dual.
This involution is categorified by "monoidal Koszul duality" for
Hecke categories, studied in positive characteristic in my...
