We will discuss three stories revolving around convex programming. The first is of a new algorithmic framework for a century old problem in physics called phase retrieval, which involves recovering
vectors from quadratic measurements and naturally connects to questions in quantum mechanics and theoretical CS. The second is on recovering the 3D structure of a scene from a collection of images, a fundamental task in computer vision which requires algorithms that are robust to a large fraction of arbitrary corruptions in the input data. Lastly, we will present new non-convex guarantees for solving certain semidefinite programs quickly by exploiting parsimony in their solutions.

The dynamics of complex biological systems is often driven by multiscale, rare but important events. In this talk, I will introduce the numerical methods for computing transition states, and then give two examples in distinct biological systems: one is a multiscale stochastic model to investigate a novel noise attenuation mechanism that relies on more noises in different cellular processes to coordinate cellular decisions during embryonic development; the other is a phase field model to study the neuroblast delamination in Drosophila.

We will discuss some new density results about the $2$-primary part of
class groups of quadratic number fields and how they fit into the framework
of the Cohen-Lenstra heuristics. Let $\mathrm{Cl}(D)$ denote the class
group of the quadratic number field of discriminant $D$. The first result
is that the density of the set of prime numbers $p\equiv -1\bmod 4$ for
which $\mathrm{Cl}(-8p)$ has an element of order $16$ is equal to $1/16$.
This is the first density result about the $16$-rank of class groups in a
family of number fields. The second result is that in the set of
fundamental discriminants of the form $-4pq$ (resp. $8pq$), where $p\equiv
q \equiv 1\bmod 4$ are prime numbers and for which $\mathrm{Cl}(-4pq)$
(resp. $\mathrm{Cl}(8pq)$) has $4$-rank equal to $2$, the subset of those
discriminants for which $\mathrm{Cl}(-4pq)$ (resp. $\mathrm{Cl}(8pq)$) has
an element of order $8$ has lower density at least $1/4$ (resp. $1/8$). We
will briefly explain the ideas behind the proofs of these results and
emphasize the role played by general bilinear sum estimates.
\newline\newline

Note: The speaker will give a prep-talk for graduate students in
AP&M 7421 at 1:15pm. All graduate students interested in number theory
are strongly encouraged to attend.

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", and discuss its properties. If time permits, I'll also outline a surprising connection to string theory and mirror symmetry.

I will discuss Gromov's non squeezing theorem in symplectic topology,
the notion of Gromov radius, and the notion of monotonic symplectic invariant
due to Ekeland and Hofer. Then I will report on recent progress in the subject,
concerning the relations of these results to the existence of symplectic embeddings.