Seminar on Complex Analysis and Complex Geometry - Spring 2016

Wednesday, January 27, 2016, 3pm-4pm, PGH 646, COLLOQUIUM

Title: The (Un)reasonable Effectiveness of Mathematics

Speaker: Alan Huckleberry, Ruhr-University Bochum and Jacobs University Bremen

Abstract: In the early 1960's Eugene Wigner published a work entitled "The unreasonable effectiveness of mathematics in the natural sciences." Our talk will begin with a brief summary of Wigner's remarks. This will be followed by arguments, based on examples, which are aimed at showing just the opposite is true: there is an underlying natural force that guides central themes in mathematics toward goals of understanding basic phenomena in nature as well a wide range of everyday applications. We as mathematicians know this, but perhaps we need to be reminded by reminiscing on some of the beautiful long lines in our history. In the talk we will attempt to do this using several examples, the central one being "From planetary motion to IT-security." Perhaps this will encourage public discussion of serious mathematical themes which result in the "un" being crossed out of the "unreasonable."

Thursday, January 28, 2016, 10am-11am, PGH 646

Title: Limiting distributions of scaled eigensections in a GIT-setting, I

Speaker: Alan Huckleberry, Ruhr-University Bochum and Jacobs University Bremen

Abstract: We present recent joint work with D. Berger on limits of naturally occurring sequences of representations associated to an action of a torus $T$ on a projective variety $X$ equipped with an invariant K\"ahlerian structure. The representations are those on spaces of sections $\Gamma (X,L^n)=V_n$ for a fixed ample line bundle $L$, and one is interested in asymptotic properties of these representations as $n\to \infty$. Since the $T$-representations are far from being irreducible (they are in fact diagonalizable) one considers sequences of irreducible pieces, in this case characters, which are assumed to have certain natural asymptotic properties, and one studies the asymptotic properties of the probability densities $$ \varphi_n:=\frac{\vert s_n\vert^2}{\Vert s_n\Vert_{L^2}^2} $$ associated to their eigenvectors defined by $t(s_n)=\chi_n s_n$ where the $\chi_n$ form the sequence of characters under discussion.

In the first case of importance, where $X$ is a smooth toric variety (work of Tate, Shiffman and Zelditch (STZ)), it was possible use direct calculations to show that for certain sequences of characters (in some sense the generic case) the $\varphi_n$ converge to the Dirac measure on a precisely described $T$-orbit. In a subsequent paper, the speaker and Sebert invested qualitative results on the algebraic geometry of toric varieties to handle the general toric (in particular singular) case. The joint work with Berger, which is the main theme here, gives complete results in the case of $T$-actions where interesting new phenomena arise. In this case the limiting object is a canonically associated invariant current supported a real subvariety which naturally arises in geometric invariant theory.

In the talks a great deal of time will be devoted to background information on the relevant complex geometry, group actions and invariant theory.

Friday, January 29, 2016, 10am-11am, PGH 646

Title: Limiting distributions of scaled eigensections in a GIT-setting, II

Speaker: Alan Huckleberry, Ruhr-University Bochum and Jacobs University Bremen

Abstract: See above.

Wednesday, February 24, 2016, 10am-11am, AH 204 (NOTE THE SPECIAL ROOM)

Title: Quantitative Geometric Results on Projective Surfaces

Speaker: Hungzen Liao, UH

Abstract: This talk is about the improvement in M. Ru's new Second Main Theorem when X is a projective surface. As a consequence, we recovered the sharp qualitative result of degeneracy which was derived by A. Levin earlier. The corresponding arithmetic results in Diophantine approximation has also been obtained. If time is available, I also intend to compare M. Ru's proof with P. Corvaja and U. Zannier's qualitative result.

Wednesday, March 2, 2016, 10am-11am, PGH 646

Title: Unicity Results for Gauss Maps of Minimal Surfaces Immersed in $R^m$

Speaker: Jungim Park, UH

Abstract: We prove a unicity theorem for two conformally diffeomorphic complete minimal surfaces immersed in $\RR^m$ whose generalized Gauss maps $f$ and $g$ agree on the pre-image $\cup_{j=1}^q f^{-1}(H_j)$ for given hyperplanes $H_j, 1\leq j \leq q$, in $\PP^{m-1}(\CC)$, located in general position, under the assumption that $\cap _{j=1}^{k+1} f^{-1}(H_{i_j})$ is empty. In the case when $k=m-1$, the result obtained gives an improvement of the earlier result of Fujimoto.

Wednesday, March 9, 2016, 10am-11am, PGH 646

Title: An improved defect relation for holomorphic curves in projective varieties

Speaker: Charles Mills, UH

Abstract: We improve Ru's previous defect relation (as well as the Second Main Theorem) for holomorphic curves $f: {\Bbb C}\rightarrow X$ intersecting $D:=D_1+\cdots +D_q$, where $D$ is of equi-degree, and $D_1, \dots, D_q$ are big and nef, and have no components in common.

Wednesday, March 23, 2016, 10am-11am, PGH 646

Title: Uniqueness Theorem for Algebraic Curves

Speaker: Gul Ugur, UH

Abstract: In this talk, I will discuss a uniqueness theorem for two algebraic curves sharing hyperplanes into the Projective spaces.

Wednesday, March 30, 2016, 10am-11am, AH 205 (NOTE THE SPECIAL ROOM)

Title: An improved defect relation in Nevanlinna theory and diophantine approximation

Speaker: Saud Hussein, UH

Abstract: In this talk, we improve Ru's defect relation (as well as the Second Main Theorem) for holomorphic curves in projective varieties intersecting $D = D_1+...+D_q$, where $D$ is equidegreelizable. The corresponding results in Diophantine Approximation are also included.

Wednesday, April 6, 2016, 10am-11am, PGH 646

Title: DT invariants for CY3 varieties and birational equivalences

Speaker: John Calabrese, Rice University

Abstract: Donaldson-Thomas numbers are enumerative invariants defined for Calabi-Yau varieties of dimension three, which are closely related to Gromov-Witten curve counts. I will discuss how these numbers behave under birational transformations.

Wednesday, April 20, 2016, 10am-11am, PGH 646

Title: Iterated automorphism orbits and boundary type

Speaker: Joshua Strong, UC Riverside

Abstract: The classification of domains of several complex variables can be related to the geometry of the boundary. If the group of automorphisms of such a domain is noncompact, then we may be able to acquire information about certain boundary points. We will discuss a special case of the Greene-Krantz conjecture, specifically when iterations of automorphisms act nicely.