Tutte Colloquium - Kevin PurbhooExport this event to calendar

Friday, September 28, 2018 — 3:30 PM EDT

Title: The Shapiro-Shapiro Conjecture

Speaker: Kevin Purbhoo
Affiliation: University of Waterloo
Room: MC 5501

Abstract:

Given four lines in 3-space, can you find a fifth line that intersects the other four? How many?

This is the smallest non-trivial example of a "Schubert problem". The answer, in this case, is not hard to compute: there are two such lines. Generalizations of this fact date back to 19th century work of Schubert.
However, this answer comes with the usual caveats of enumerative geometry: not always, not necessarily over the reals, and what exactly do you mean by "find"?  Addressing these finer points is much harder, and until
fairly recently not much was known.

In 1993, Boris and Michael Shapiro formulated a remarkable conjecture about the existence of real solutions to Schubert problems. It was proved in 2009 by Mukhin, Tarasov and Varchenko, using high powered machinery from quantum integrable systems. Since then, many applications and generalizations have been found or conjectured.

In this talk, I will tell the story of the Shapiro-Shapiro conjecture, and then tell you about a new theorem (joint work with Jake Levinson), a not-so-obvious generalization. Our result is independent of Mukhin-Tarasov-Varchenko, implies the conjecture, and naturally lends itself to a much more intuitive proof.

Location 
MC - Mathematics & Computer Building
5501
200 University Avenue West

Waterloo, ON N2L 3G1
Canada

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