# Student talks: AustMS practice

#### Ivo Vekemans: The slice filtration and spectral sequence

A Postnikov tower from classical algebraic topology is built via localisation functors from a filtered sequence of localising subcategories $\tau_{\geq n}$ each generated by spheres $S^k$ for $k \geq n$. Similarly, in the equivariant setting slice towers are built via localising functors from the slice filtration of localising subcategories generated by suitable representation spheres. In this talk I will introduce the slice tower and its spectral sequence (by contrast, the spectral sequence of a Postnikov tower is boring, ie. stable on the first page).

#### Edmund Heng: Braid group action on triangulated categories

It is known that braid groups act on the curves of punctured disk and through this, Thurston gave a classification theory of braid groups by looking at the dynamics of this action. By a paper written by Khovanov and Seidel, these curves on the punctured disk can be viewed as objects in the homotopy category of bounded chain complexes of projective modules over the zig-zag algebra. Furthermore, the braid group acts on it just as how it acts on curves.

In this talk I’ll describe the recipe to relate the geometrical objects (curves) to the algebraic objects (complexes) and how this gives us an action on a triangulated category.

#### Dominic Weiller: Algebras in Braided Tensor Categories and Embedded Surfaces

Given a monoidal category, the algebra objects, their bimodules and

bimodule intertwiners form a bicategory. If we start with a braided

tensor category (BTC), the braiding produces a monoidal product of

algebras making the category of algebras, bimodules and intertwiners

into a tricategory with a single 0-morphism (also called a monoidal

bicategory). Using the theory of surface diagrams for Gray categories

with duals, one expects to produce an `extended TQFT of embedded

surfaces' given a special Frobenius algebra in a BTC.

We will consider a non-extended version and see how to obtain invariants

of embedded surfaces up to isotopy by drawing `fine enough' trivalent

graphs on a surface and choosing a special Frobenius algebra.

Interesting examples must come from non-symmetrically braided categories

as algebras in symmetric tensor categories give boring invariants. For

example, picking the $E6$ algebra in $A_{11}$ (a non-symmetric BTC) we

can distinguish the unlinked union of two tori from two concentrically

nested tori.