The weak efficient market hypothesis can be interpreted as asserting that major market indices should lie near the Markowitz efficient frontier.  This is seen in many years, but not in years before large market downturns.  Most notably, it was not seen in most of 2007 and 2008 leading up the the crash of 2008.  Indeed, all the major indices were found on the Markowitz inefficient frontier right after the crash. More generally, we consider a frontier computed for Markowitz portfolios that hold only long positions, which lies to the right of the classical Markowitz frontier. For many years this so-called long frontier lies close to the Markowitz frontier, but not in years of market volatility. The question will be posed, but not answered, if this separation is a measure of a market exposed to large short positions that could contribute to a systemic downturn.

Let $\Gamma$ be a graph with a weight $\sigma$. Let $d$ and $\mu$ be the distance and the measure associated with $\sigma$ such that $(\Gamma, d, \mu)$ is a doubling space. Let $p$ be the natural reversible Markov kernel associated with $\sigma$ and $\mu$  and $P$ the associated
operator defined by $Pf(x) = \sum_{y} p(x, y)f(y)$. Denote by $L=I-P$ the discrete Laplacian on $\Gamma$. 
In this talk we develop the theory of Hardy spaces associated to the discrete Laplacian $H^p_L$ for $0<p\leq 1$. We then obtain boundedness of certain singular integrals on $\Gamma$ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces $H^p_L$. This is joint work with The Anh Bui.

Kapouleas and Yang have constructed, by gluing methods, sequences of
minimal embeddings in the round 3-sphere converging to the Clifford
torus counted with multiplicity 2. Each of their surfaces, which
they call doublings of the Clifford torus, resembles a pair of
coaxial tori connected by catenoidal tunnels and has symmetries
exchanging the two tori. I will describe an extension of their work
which yields doublings admitting no such symmetries as well as
examples incorporating an arbitrary (finite) number of tori, that
is Clifford torus triplings, quadruplings, and so on.