**The Principal Speaker of the 2019 Workshop will be Professor Ciprian Manolescu of the University of California, Los Angeles.** He will be speaking on *Triangulations, homology cobordisms, and Floer theories.*

Titles and abstracts of the individual talks are as follows:

**Talk 1: Triangulations of manifolds**

Which manifolds are triangulable? Can any two triangulations of the same manifold be made equivalent after subdivision? I will survey the history of these questions, and explain the role of the three dimensional cobordism group in the study of triangulations.

**Talk 2: Seiberg-Witten theory and homology cobordism **

I will explain how invariants from gauge theory give information about homology cobordism. In particular, Pin(2)-equivariant Seiberg-Witten Floer homology can be used to prove that there are no elements of order 2 and Rokhlin invariant one in homology cobordism. This implies that non-triangulable manifolds exist in all dimensions at least five.

**Talk 3: Involutive Heegaard Floer homology**

I will describe an invariant of 3-manifolds called involutive Heegaard Floer homology, which gives further information about homology cobordism. The invariant was constructed in joint work with Kristen Hendricks, and has recently led to a proof (by Dai-Hom-Stoffregen-Truong) that the homology cobordism group has an infinite-rank summand.