Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of
intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds
in a Riemannian manifold (1968)], an important study of the very idea of curvature
was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades,
B.-Y. Chen’s fundamental inequalities have been investigated by many authors in the
context of various geometric structures. In this talk, we start by presenting B.-Y. Chen’s
fundamental inequality for Kähler submanifolds in complex space forms, and we recall
why the case of Kähler surfaces in C^3 satisfying scal(p) = 4 inf sec(π^r ) appears
naturally and is important. Then we provide several characterizations of strongly minimal
complex surfaces in the complex three dimensional space. We focus our study on the question
of finding further examples of strongly minimal Kähler surfaces, as the question of a
complete classification of these geometric objects is still open.

Cryptology provides a real-world application of many topics in number theory: integer factorization, primality testing, quadratic reciprocity, and elliptic curves, just to name a few. For these applications to cryptology, it is important to know whether or not a given procedure can be performed quickly. How does one convey to students that, for example, primality testing is relatively fast while integer factorization is relatively slow? We will present labs designed for UC Irvine Math 173AB: Introduction to Cryptology. These labs use Sage to introduce relevant cryptology topics, and at the same time they enable students to work with numbers at the limits of what their computers can handle computationally. For such numbers, the difference between a "fast" algorithm and a "slow" algorithm is striking, and as a result, students learn a key principle justifying the security of many modern cryptosystems.

The absolute Galois group of a number field is a mysterious
object, that one can try to understand by means of its representations. It
is known that holomorphic cuspidal modular forms are, in a suitable sense,
a source of many p-adic Galois representations. More generally, it is
conjectured that also torsion classes in the coherent cohomology of
Shimura varieties have attached Galois representations, with prescribed
local properties. I will give an introduction to these themes, and present
results obtained in collaboration with Matthew Emerton and Liang Xiao
toward the conjecture.

Low-rank structures are common in modern data analysis and signal processing, and they usually
play essential roles in various estimation and detection problems. It is challenging to recover the underlying low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce convex and nonconvex optimization methods for low-rank recovery by two examples.

The first example is community detection in network data analysis. In the literature, it has been formulated as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However, the statistical advantages of convex optimization approaches over other competitive methods, such as spectral clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world network dataset with a heterogeneous distribution of degrees.

Although SDP methods are provably effective and robust, the computational complexity is usually high and there is an issue of storage. For the problem of phase retrieval, which has various applications and can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far less storage and has much higher rate of convergence compared to convex methods

Consider a holomorphic line bundle L over a compact Kahler manifold. Motivated by mirror symmetry, I will define an equation on L that is the line bundle analogue of the special Lagrangian equation, which can be studied even when the base is not a Calabi-Yau manifold. I will show solutions are unique global minimizers of a positive functional. To address existence, I will introduce a line bundle analogue of the Lagrangian mean curvature flow, and prove convergence in certain cases. This is joint work with S.-T. Yau.