L6

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A d-dimensional TQFT is a topological invariant which assigns (d-1)-dimensional manifolds to vector spaces and d-dimensional cobordisms to linear maps. In the early 90s, Reshetikhin and Turaev constructed examples of these in the case d=3, using the data of certain types of linear categories. In this talk, I will provide an overview of this construction, and then explore how this might be meaningfully extended downwards to assign 1-manifolds to "2-vector spaces". Minimal knowledge of category theory assumed!

An important invariant in the complex representation theory of reductive p-adic groups is the wavefront set, because it contains information about the character of such a representation. In this talk, Mick Gielen will introduce a new invariant called the canonical dimension, which can be said to measure the size of a representation and which has a close relation to the wavefront set.  He will then state some results he has obtained about the canonical dimensions of compactly induced representations and show how they teach us something new about the wavefront set. This illustrates a completely new approach to studying the wavefront set, because the methods used to obtain these results are very different from the ones usually used.

I will present short, new proofs of Dowker duality using various poset fiber lemmas. I will introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. Using the relational product complex, I will then discuss generalizations of Dowker duality to settings of relations among three (or more) sets.

Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance d_I , which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but as proposed by Gasparovic et al., we explore intrinsic metrics d_I and d_B realized by infinitesimal path length in merge tree space, which do indeed coincide. This result suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too.