Seminars in geometry/topology

January 29, 14:15 - 15:15, room 734, Sentralbygg 2

Magnus Bakke Botnan (TU Munich): Multidimensional Persistence and Clustering

Abstract: It is widely known that multidimensional persistent homology is "hard". But what does this really mean? In the first part of the talk I will discuss this from the point of view of representation theory of quivers. Next I will show how multidimensional clustering gives rise to multidimensional persistence modules of a particularly appealing form. Given their simplicity, it is natural to wonder if we are able to carry out persistence computations like in 1-D in the setting of clustering. I will discuss recent work with Ulrich Bauer, Steffen Oppermann and Johan Steen, which shows that this restriction alleviates the aforementioned "hardness" in only very special cases. The last part of the talk will concern how much of this complexity we see in persistence modules coming from data.

May 2, 10:15 - 11:15, room 734, Sentralbygg 2

Rachael Boyd (University of Aberdeen/NTNU): Homological stability for Artin monoids

Abstract: Many sequences of groups satisfy a phenomenon known as homological stability. In my talk, I will report on recent work proving a homological stability result for sequences of Artin monoids, which are monoids related to Artin and Coxeter groups. From this, one can conclude homological stability for the corresponding sequences of Artin groups, assuming a well-known conjecture in geometric group theory called the $K(\pi,1)$-conjecture. This extends the known cases of homological stability for the braid groups and other classical examples. No familiarity with Coxeter and Artin groups, homological stability or the $K(\pi,1)$-conjecture will be assumed.

May 2, 14:15 - 15:15, room 656, Sentralbygg 2

Robin Frankhuizen (University of Southampton): $A_\infty$ resolutions and Massey products on Tor-algebras

Abstract: Tor-algebras of monomial rings have a long and fruitful history in commutative algebra. Massey products on the Tor-algebra were first studied by Golod who proved that the vanishing of all Massey products on the Tor-algebra is equivalent to the maximality of the Poincaré series of the monomial ring. Recently, Tor-algebras of monomial rings have gained increasing attention in topology where they naturally show up as the cohomology algebra of moment-angle complexes.

In this talk, we present a new approach to computing Massey products on Tor-algebras by constructing $A_\infty$ algebra structures on the minimal free resolution of a monomial ring using algebraic Morse theory. Using this approach, we give a necessary and sufficient condition for the vanishing of all Massey products on the Tor-algebra.