N3.12

The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.

[REMOTE TALK]

In this talk, we will propose a new construction of the virtual fundamental classes of quasi-smooth derived schemes using the vanishing cycle complexes. This is based on the dimensional reduction theorem of cohomological Donaldson—Thomas invariants which can be regarded as a variant of the Thom isomorphism. We will also discuss a conjectural approach to construct DT4 virtual classes using the vanishing cycle complexes.

Zoom link: https://us02web.zoom.us/j/86267335498?pwd=R2hrZ1N3VGJYbWdLd0htZzA4Mm5pd…

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.