Pure Maths Colloquium: Algebraic cycles and theta functions
3 November 2014 16:00 in CM221
Given two subvarieties Y and Z of a complex projective variety X, when is there a family interpolating between them? This question can be rephrased in terms of certain invariants of X known as groups of algebraic cycles. These groups are extremely interesting and are the subject of some famously difficult conjectures that describe their structure quite precisely. Although they generate great interest, the conjectures for general X seem wide open.
In the last years, however, progress has been made for certain locally symmetric varieties X by several authors. These developments link the structure of algebraic cycles on $X$ to the theory of theta functions. In turn, these functions are very well understood from the point of view of representation theory and so methods from the theory of automorphic forms become available to study algebraic cycles.
We will define all the relevant objects, review recent results in this area and discuss possible future directions.
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