### Mathematics Colloquium

** Oishee Banerjee
**

IAS

**Title:** Cohomology and arithmetic of Mapping spaces

**Date:** Monday, February 03, 2023

**Place and Time:** fsu.zoom.us/s/97976878227, 3:05-3:55 pm

**Abstract.**

Abstract: How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X--->Y from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g the geometric Batyerv-Manin type conjectures) and algebraic geometry (e.g. moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc). I will show how one can fruitfully attack such problems by incorporating techniques from homotopy theory to the holomorphic/algebraic world (e.g. by constructing a new spectral sequence). Most of this talk should be understandable to first year graduate students.