Speaker 1 : Mahir Bilen Can (Tulane University)

Title: Graded locally semialgebraic spaces and graded Nash manifolds. 

Abstract: In this talk, we will present our theory of graded locally semialgebraic spaces and graded Nash manifolds. By adapting Batchelor's theorem to our framework, we show that all graded locally semialgebraic spaces and graded affine Nash manifolds are derived from appropriate vector bundles. Our analysis of graded Nash manifolds involves an investigation of a topology, where closed sets are defined by Nash subsets. Within this context, we establish the softness of the structure sheaf for an affine Nash manifold. Moreover, we establish that under this topology, the higher cohomology of quasi-coherent sheaves on an affine Nash manifold completely vanishes.
These results create opportunities for new directions in cohomological studies on Nash manifolds. 

Speaker 2 : Andrew Lobb (Durham University)

Title: Four-sided pegs fitting round holes fit all smooth holes.

Abstract: Given a smooth closed curve in the plane and a cyclic quadrilateral (a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle) we show that there exist four points on the curve forming the vertices of a quadrilateral similar to the one given.  No prior knowledge of anything assumed.  Joint work with Josh Greene, with thanks to OIST and a global pandemic for affording me the space and time to work on this.