Special Year Learning Seminar

Using loops of contactomorphisms generated by vector fields that are positively transverse to the contact distribution, Eliashberg and Polterovich introduced the notion of orderability for contact manifolds. Recent work seems to suggest that the...

Consider a 𝐶𝑘 function 𝐹:𝑀𝑚×𝑁𝑛⟶ℝ defined on the product of the manifold 𝑀𝑚

of dimension 𝑚 with the compact manifold 𝑁𝑛 of dimension 𝑛.

The function defined by inf𝑁𝑛𝐹:𝑀𝑚⟶ℝ is locally Lipschitz. Yomdin proved (in 1984) that if 𝑘𝑘(𝑚,𝑛), where the integer...

Embedded contact homology (ECH) is a powerful invariant of contact 3-manifolds due to Hutchings and Taubes. It is key to the proofs of the Weinstein conjecture (existence of Reeb orbits), Arnold's chord conjecture, Irie's closing lemma...

I will discuss the wonderful theory of intersection homology, introduced by Goresky and MacPherson, and its connections to the combinatorics of convex polytopes and cellular spaces. I will mention results and questions about "toric g-vectors"...

Let $f: X \to S$ be a quasi-projective family of varieties defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. We show that the points of $S(\overline{\mathbb{Q}})$ that are Hodge generic for the variation of Hodge structures associated to $f$...

Let A be a simple abelian variety, and X and Y be two subvarieties. We say X and Y are in the Bezout range if \dim X + \dim Y >= \dim A, and outside of the Bezout range otherwise. It is known that two varieties in the Bezout range in A always...

I will discuss how the Lang–Trotter conjecture for pairs of elliptic curves implies new cases of the Zilber–Pink conjecture for curves in 3. Unlike previous results for curves in g, our result does not rely on any assumption on intersections with...

Around 1921 Polya asked about the algebraicity of the indefinite integral of an algebraic function, in terms of the integrality of the coefficients in a power-series expansion. Polya nicely answered the special case of rational functions, using...

Every o-minimal structure determines a collection of "tame" or "definable" subsets of bbRn. We can then ask about the fragment of complex geometry present in the structure: Which holomorphic functions are definable, and which spaces are cut out by...

Ben Bakker explained to us how to construct moduli spaces of polarized Hodge structures, and then produced period maps associated to families of smooth projective varieties. However, in practice one often encounters a family of smooth varieties...