topology@ub

13 May 2019, 12:00, IMUB

Abstract: We will recall the definition of Lawvere theories and, more generally, enriched theories and theories with arities. Conditions will be given under which an idempotent functor preserves models of such theories.

20 May 2019, 12:00, IMUB

Abstract: The calculus of homotopy functors is a method invented by Goodwillie which consists of decomposing a homotopy functor F: C ⟶ D between two nice model categories into a "Taylor tower" F ⟶ ... ⟶ P_n F ⟶ P_n-1 F ⟶ ... ⟶ P₀ F, where F ⟶ P_n F is the universal n-excisive approximation of F. In this talk, I will specialize to the case when C = Alg_O is the category of algebras over a reduced operad O, and D = Ch(k) is the category of chain complexes over a field k of characteristic zero. I will give an explicit description of the Taylor tower of "simplicial functors" F: Alg_O ⟶ Ch(k).

29 May 2019, 11:00, Aula T1

Abstract: Persistent homology has been developed as an important tool of Topological Data Analysis, with numerous applications in science and engineering. However, for very large data sets this tool can be very expensive to compute, both in terms of computational time and hard-disk memory. We will present a new distributive algorithm which takes part directly on the input data. This has some theoretical difficulties since we need to work within the category of persistence modules. In particular, we will see a solution to the extension problem for the persistent Mayer-Vietoris spectral sequence. At the end we speculate that this approach might give us more information than ordinary persistent homology.

29 May 2019, 12:30, Aula T1

Abstract: Persistent homology is a versatile technique to analyse data from geometric and topological perspectives. It aims at extracting novel information from data. With this motto in mind, we present practical examples of medical knowledge originated via persistent homology, and ways to enhance the power of this technique via classical tools in algebraic topology.