Online lectures and online classes are changing the landscape of math education. At New York University, we are creating a hybrid Calculus 1 course which will combine interactive online content with an in-class component involving lectures and problem sessions. We relate the structure of this course to that of other non-traditional calculus classes, and we discuss some potential advantages of this class over the standard lecture format.

There is a joke that when one counts something in linear algebra, the only possible results are 0, 1, and infinity.  What about in number theory?  Well, if we count the primes, we get infinity.  To get a finite number, we could try to count primes less than some fixed N.  It was proven in 1955 that something interesting happens sometime before N = 10^(10^(10^1000)).  Huh?!  What if we count gaps between primes?  Just in 2013, it was proven that something interesting happens for gaps less than 70,000,000.  What if we try to write down a really big prime number?  Right now, nobody can write down a prime bigger than 10^(10^8).  In this talk, we'll introduce some of these big numbers in number theory, we'll carefully say what they mean, and we'll describe progress on making the first two big numbers smaller and on making the third big number bigger.

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.