(Un)likely intersections

CMSA EVENTS: CMSA MATHEMATICAL PHYSICS AND ALGEBRAIC GEOMETRY SEMINAR

When: November 14, 2024
10:00 am - 11:00 am
Where: CMSA, 20 Garden St, G10
Address: 20 Garden Street, Cambridge, MA 02138, United States
Speaker: Tom Scanlon (UC Berkeley)

The Zilber-Pink conjectures predicts that for an ambient special variety S (such as an abelian variety or a Shimura variety), if  X \subseteq S is an irreducible algebraic subvariety which is not contained a proper special subvariety of S (e.g. a proper algebraic subgroup in the abelian variety case or a variety of Hodge type in the case of Shimura varieties), then the union of the unlikely intersections X \cap S' as S' ranges over the special subvarieties of S with \dim X + \dim S' < \dim S is not Zariski dense in X.  While various instances of this conjecture have been proven, it remains open in most cases of interest.  In this lecture, I will describe some of my work with Jonathan Pila in which we prove an effective function field version of this conjecture along with a counterpart to the Zilber-Pink conjecture proven with Sebastian Eterović:  after accounting for some geometric obstructions, the likely intersections, i.e. the union of the intersections X \cap S' with S' \subseteq S special and \dim X + \dim S' \geq \dim S,  are dense in the Euclidean topology in X.   Our techniques for both results come from o-minimal complex analysis and differential algebra.

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