yes, the title makes sense: there are (a lot of) connections between logic and operator algebras! and in this talk we are going to cover some. the first goal will be to introduce a generalization of classical first order logic, a ``continuous logic'' which can be used to describe continuous structures like metric spaces, operator algebras, etc. we will see how this can help us to study some questions concerning such structures. in particular, we will cover two natural questions that arise in the context of operator algebras which turn out to be independent of zfc - our everyday axiom system all of set theory is based on!

as shown by n. brown in 2011, for a separable $ii_1-factor n$, the invariant $hom(n,r^u)$ given by unitary equivalence classes of embeddings of $n$ in to $r^u$--an ultrapower of the separable hyperfinite $ii_1-factor$--takes on a convex structure. this provides a link between convex geometric notions and operator algebraic concepts; e.g. extreme points are precisely the embeddings with factorial relative commutant. the geometric nature of this invariant provides a familiar context in which natural curiosities become interesting new questions about the underlying operator algebras. for example, such a question is the following. ``can four extreme points have a planar convex hull?''

the goal of this talk is to present a recent result generalizing the characterization of extreme points in this convex structure. after introducing and discussing this convex structure, we will see that the dimension of the minimal face containing an equivalence class $[\pi]$ is one less than the dimension of the center of the relative commutant of $\pi$. this result also establishes the ``convex independence'' of extreme points, providing a negative answer to the above question. along the way we make use of a version of schur's lemma for this context. no prior knowledge of this convex structure will be assumed.

coupling is a way of constructing markov processes with prescribed laws on the same probability space. it is known that the rate of coupling (how fast you can make two processes meet) of elliptic/riemannian diffusions is connected to the geometry of the underlying space. in this talk we consider coupling of hypoelliptic diffusions (diffusions driven by vector fields satisfying hoermander's condition). s. banerjee and w. kendall constructed successful markovian couplings for a large class of hypoelliptic diffusions. we use a non-markovian coupling of brownian motions on the heisenberg group, and then use this coupling to prove analytic gradient estimates for harmonic functions for the sub-laplacian.

this talk is based on the joint work with sayan banerjee and phanuel mariano.

we will discuss various ways to produce statistics for families of curves over finite fields. this gives us a window into the greater world of arithmetic statistics and some of the tools used in the field, from geometric invariant theory to automorphic forms.

elliptic curves defined over the rational numbers are of great
interest in modern number theory. the rank of an elliptic curve is a
crucial invariant, with many open questions about its behavior.

in particular, there is great interest in the ``average'' rank of an
elliptic curve. the minimalist conjecture is that the average rank
should be 1/2. in 2007, bektemirov, mazur, stein, and watkins [bmsw],
using well-known databases of elliptic curves, set out to numerically
compute the average rank of elliptic curves, ordered by conductor.
they found that ``there is a somewhat more surprising interrelation
between data and conjecture: they are not exactly in open conflict one
with the other, but they are no great comfort to each other either.''

in joint work with ho, kaplan, spicer, stein, and weigandt, we have
assembled a new database of elliptic curves ordered by height. i will
describe the database and examine some of the questions posed by
[bmsw]. i will also discuss ongoing work by a team of undergraduates
at oxford on similar questions about families of elliptic curves.

this is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. the case of compact manifolds is well understood
thanks to seminal work of jurgen moser in the 1960s.