Seminar on Complex Analysis and Complex Geometry - Fall 2013

Wednesday, October 23, 2013, 10am-11am, PGH 646

Title: The number of rational points on curves over global fields

Speaker: Felipe Voloch, UT Austin

Abstract: I will provide some detail and explore further some of the questions discussed in my colloquium, covering also the function field case.

Wednesday, October 23, 2013, 3pm, PGH 646 (Colloquium)

Title: How many rational points does an algebraic curve have?

Speaker: Felipe Voloch, UT Austin

Abstract: A theorem of Faltings ensures that an algebraic curve defined by equations with rational coefficients have finitely many rational points provided its genus is at least two. How many are these finitely many? Can this this number grow, for fixed genus, by varying the curve? How about the typical curve, how many points does it have? These are open questions but there has been some recent progress in understanding them, which I will discuss.

Wednesday, November 20, 2013, 10am, PGH 651G

Title: Mapping B^n to B^{3n-3}

Speaker: Jared Andrews, UH

Abstract: Poincar\'e found that a biholomorphic map between two open pieces of the unit sphere in $\mathbb C^2$ is the restriction of an automorphism of $\mathbb B^2$, and Tanaka showed that this result holds for all $n > 2$. Alexander then showed that any proper holomorphic self-mapping of the ball in $\mathbb C^n, n > 1$, is an automorphism. This began a line of research that led eventually to Xuang's 1999 result, classifying all proper holomorphic CR mappings from $\mathbb B^n$ to $\mathbb B^N$ that are twice continuous up to the boundary, where $n \leq N \leq 2n-2$, up to automorphism on the boundary of $\mathbb B^n$ or $\mathbb B^N$. This is the so-called First Gap Theorem.

The case where $N = 2n-1$ proved to be much harder, but in 2001, Xuang and Ji proved that the set of equivalence classes of $Rat(n,2n-1)$ has two elements, one totally geodesic and one consisting of mappings equivalent to the Whitney map.

For $2n \leq N \leq 3n-4$, $n \geq 4$, the result came in 2006 from Huang, Ji, and Xu. This was the Second Gap Theorem. We now cover the boundary case of this theorem, where $N = 3n-3$. This case is more difficult, and we use some results and techniques from the other gap theorems, including the Third.

Wednesday, December 4, 2013, 10am, PGH 646

Title: Embeddability Problem of a CR Hypersurface

Speaker: Brandon Lee, UH

Abstract: Under certain conditions on the dimensions of a CR hypersurface of revolution M and the Gaussian curvature of its associated domain, we show that the hypersurface cannot be embedded into a sphere.