Gauge functions significantly generalize the notion of a norm, and gauge optimization is the class of problems for finding the element of a convex set that is minimal with respect to a gauge. These conceptually simple problems appear in a remarkable array of applications. Their gauge structure allows for a special kind of duality framework that may lead to new algorithmic approaches. I will illustrate these ideas with applications in sparse signal recovery.

Consider the question of how a derivatives security (e.g., a call option) should be priced. Due to their potential utility for hedging and speculation, economists started becoming interested in this question in the early 1900s. In the late 1960s, this emerged as an important concern for finance markets as well, due to a relaxation in regulations that allowed insurance companies and banks to invest in derivatives. After decades of growing interest and research efforts, Fisher Black and Myron Scholes developed a mathematically based pricing strategy in the early 1970s that was later simplified and expanded on by Robert Merton. The resulting formulas revolutionized trading practices worldwide. In 1997, Merton and Scholes received the Nobel prize for this body of work (Black had passed away and therefore was ineligible).

In the talk, we will examine an extremely simplified version of this body of work. In particular, we will analyze the single-period Cox-Ross-Rubenstein (CRR) model using some of the key principles and methodologies that Black, Scholes, and Merton developed to construct their derivatives pricing theory. Despite the fact that this model is very simple, the analysis will illustrate certain essential aspects of their Nobel prize winning work such as dynamic hedging and the risk neutral probability measure. This will give a flavor of the mathematical content of Math 194, being offered in the winter quarter.

We exhibit a numerical method to compute a three-point branched covers
of the complex projective line. We develop algorithms for working
explicitly with Fuchsian triangle groups and their finite index
subgroups, and we use these algorithms to compute power series
expansions of modular forms on these groups. As one application, we
find an explicit rational function of degree 50 which regularly
realizes the group $PSU_3(5)$ as a Galois group over the rationals.
This is joint work with Michael Klug, Michael Musty, and Sam
Schiavone.

In this talk, we summarize the compact coupled interface method and introduce modifications for handling a wider class of Poisson-type problems, as well as comparisons to the coupled interface method and immersed interface method. We pay special attention to the Poisson-Boltzmann equation and its role in electrostatic effects specifically for implicit solvation.

The Langlands-Shahidi method provides us with a constructive
way of studying automorphic L-functions. For the classical groups over
function fields we will present recent results that allow us to obtain
applications towards global Langlands functoriality. This is done via
the Converse Theorem of Piatetski-Shapiro, which we can apply since our
automorphic L-functions have meromorphic continuation to rational
functions and satisfy a functional equation. We lift globally generic
cuspidal automorphic representations of a classical group to an
appropriate general linear group. Then, we express the image of
functoriality as an isobaric sum of cuspidal automorphic representations
of general linear groups, where the symmetric and exterior square
automorphic L-functions play a technical role. As a consequence, we can
use the exact Ramanujan bounds of Laurent Lafforgue for GL(N) to prove
the Ramanujan conjecture for the classical groups. Our results are
currently complete for the split classical groups under the assumption
that characteristic p is different than two.

I will give a brief introduction to Hodge theory and discuss Hodge-theoretic problems involving abelian varieties.