Seminar on Complex Analysis and Complex Geometry - Fall 2011

Wednesday, August 31, 2011, 11am, PGH 646

Title: Working Seminar

Speaker: Gordon Heier, UH

Wednesday, September 7, 2011, 11am, PGH 646

Title: Working Seminar

Speaker: Gordon Heier, UH

Wednesday, September 14, 2011, 11am, PGH 646

Title: Linear independence of Heegner points.

Speaker: Alexandru Buium, University of New Mexico

Abstract: This joint work with Bjorn Poonen. The main result is that if f is a map from a modular curve X into an elliptic curve E then the image by f of the CM locus of X meets any finite rank subgroup of E in finitely many points. A proof based on ``arithmetic differential equations" will be explained.

Wednesday, September 14, 2011, 3pm, AH 106 (Colloquium)

Title: Calculus with integer numbers.

Speaker: Alexandru Buium, University of New Mexico

Abstract: One can develop an arithmetic analogue of differential calculus in which functions are replaces by integer numbers and the derivative operator is replaced by a Fermat quotient operator. Then an arithmetic analogue of the Lie-Cartan geometric theory of differential equations can be developed. This arithmetic theory can then be applied to prove statements in diophantine geometry over number fields in the same way in which usual differential equations are being used to prove results in diophantine geometry over function fields.

THURSDAY, September 29, 2011, 3pm, AH 2 (SPECIAL ROOM)

Title: Surfaces of low degree containing a canonical curve.

Speaker: Izzet Coskun, University of Illinois at Chicago

Abstract: If C is a general canonical curve of genus g>23, then a theorem of Ciliberto and Harris says that the minimal degree surface S containing C has degree 2g-3 and S is a cone over C with vertex a point on C. In this talk, I will describe what can be said about canonical curves that are contained in a surface of lower degree. The main theorem is that if g>3k+12, (k,g) is different from (-1, 10) and C is contained in a surface of degree g+k, then k has to be odd and C is a double cover of a curve of genus (k+3)/2.

Wednesday, October 12, 2011, 11am, PGH 646

Title: Probability that random positive integers are k-wise relatively prime.

Speaker: Jerry Hu, UH-Victoria

Abstract: The integers a_1, a_2,...,a_m are k-wise relatively prime if any k of them have no common divisor other than 1. In this talk we will discuss how to find the probability that m positive integers are k-wise relatively prime. In particular, we will show our method for the case k=3. The case k=2 was solved by L. Toth in 2002.

Wednesday, October 26, 2011, 11am, PGH 646

Title: Negatively curved Finsler metric and Kobayashi hyperbolicity of moduli spaces of canonically polarized algebraic manifolds.

Speaker: Sai-Kee Yeung, Purdue University

Abstract: It was known classically from the work of Ahlfors, Royden and Wolpert that the Weil-Petersson metric on a moduli space of hyperbolic Riemann surface has holomorphic sectional curvature bounded from above by a negative constant and hence is Kobayashi hyperbolic. It is natural to ask if similar properties hold in a higher dimensional analogue. The main purpose of the talk is to explain the joint work of Wing-Keung To and myself on the construction of such a negatively curved Finsler metric on any moduli space of Kaehler Einstein manifolds with negative scalar curvature, from which Kobayashi hyperbolicity follows naturally.

Wednesday, October 26, 2011, 3pm, AH 106 (Colloquium)

Title: Fake structures and exotic structures.

Speaker: Sai-Kee Yeung, Purdue University

Abstract: An exotic structure on a differentiable manifold is one which is homeomorphic but not diffeomorphic to the manifold. A fake structure on a complex manifold is one which has the same Betti numbers but not biholomorphic to the manifold. The main purpose of the talk is to explain the joint work of Gopal Prasad and myself on classification of fake projective planes. Some observations related to exotic structures will be mentioned. Fake projective plane was first introduced by David Mumford. It has the smallest possible Euler number among all smooth surfaces of general type. Related work of Klingler, myself, Prasad and Cartwright-Steger will also be discussed.

Wednesday, November 09, 2011, 11am, PGH 646

Title: Techniques in the study of complex hyperbolicities: The use of the Logarithmic Derivative Lemma, I.

Speaker: Min Ru, UH

Abstract: In 1925, R. Nevanlinna proved what is known as the lemma of the logarithmic derivative for a meromorphic function $f$ defined on the whole complex plane. The lemma is the main technical tool in the study of his celebrated Second Main Theorem for meromorphic functions. It turns out to be very important in the study of complex hyperbolicities, together with the notion of jet-differentials (higher order differentials). In this series of talks, I'll introduce the results of Green-Griffiths, McQuillan, Siu-Yeung, and K. Yamanoi, using the approach of the lemma of the logarithmic derivative.

Wednesday, November 16, 2011, 11am, PGH 646

Title: Techniques in the study of complex hyperbolicities: The use of the Logarithmic Derivative Lemma, II.

Speaker: Min Ru, UH

Abstract: In 1925, R. Nevanlinna proved what is known as the lemma of the logarithmic derivative for a meromorphic function $f$ defined on the whole complex plane. The lemma is the main technical tool in the study of his celebrated Second Main Theorem for meromorphic functions. It turns out to be very important in the study of complex hyperbolicities, together with the notion of jet-differentials (higher order differentials). In this series of talks, I'll introduce the results of Green-Griffiths, McQuillan, Siu-Yeung, and K. Yamanoi, using the approach of the lemma of the logarithmic derivative.