L4

Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH: given a persistence module, what can we say about the data that generated it? To study the geometry of the fibers of MPH, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications that are preserved by the map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.

The study of the maximum diameter of $d$-dimensional pseudomanifolds on $n$ vertices was initiated by Criado and Santos as an abstraction of the analogous problem for polytopes, in relation to the Polynomial Hirsch Conjecture.
A series of works by Santos, Criado, Newman and Bohman establish the asymptotics. Here we use a mixture of deterministic and random tools to 
determine the exact value for every large enough $n$ when $d=2$, and for a positive fraction of $n$ when $d\geq 3$.  

Our pseudomanifolds with maximum diameter crucially depend on a surprising connection to the cube of Euler trails in uniform hypergraphs. To this end, for every fixed uniformity $d$ and power $r \geq 2$, we show that satisfying the natural divisibility conditions implies the existence of the $r$th power of an Euler tour/trail in any large enough $d$-uniform hypergraph with large enough codegree.  The talk represents joint work with Stefan Glock, Olaf Parczyk, and Silas Rathke.

TBA

TBA

Calibrated fibrations are expected to play an important role in exceptional holonomy, much as special Lagrangian fibrations do in the SYZ picture for Calabi--Yau manifolds. Singular fibres are expected to be essential, and a natural question is whether they are forced by the geometry. In his PhD thesis, Baraglia showed that coassociative fibrations of compact full-holonomy G_2-manifolds must have singular fibres. The analogous problem for Cayley fibrations remained open for many years and turns out to be substantially more difficult, with its resolution relying on deep results from 4-manifold topology.

The proof takes some unexpected twists and turns, involving a Diophantine equation arising from the Spin(7)-structure, families Seiberg--Witten theory and parametrized homotopy theory. After a short crash course on Spin(7) geometry, I will explain how these pieces fit together. This is joint work with Jianfeng Lin and Viktor Majewski.