After a reminder of the braid groups and of how to obtain all knots and links from braids, we define the Thompson group F and show that it is as effective as all the braid groups in constructing knots and links. The construction is motivated by the relationship between subfactors and conformal field theory. Indeed in a ``naive'' attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups for any subfactor. The Thompson group elements are the ``local scale transformations'' of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new ``oriented'' subgroups, which allows us to produce all oriented knots and links
In integrable field theory it is important to describe vacuum expectation values of all local fields. As underlined by Al. Zamolodchikov, they encode all non-perturbative characteristics of the theory. For this purpose an appropriate basis of local fields is needed. Through the study of correlation functions on the lattice, there emerged the existence of a `fermionic' basis with good properties. In this talk we give a survey of these developments.
Since the 2008 financial crisis, the main focuses of mathematical finance deeply evolved. Before the crisis, research on risks and securities pricing was dominated by arbitrage pricing theory. Price innovations where considered as exogenous and the broad framework was to relate, through a Green function, solutions of a parabolic PDE (or of an evolution semi-group), expectation of processes and financial outcome of dynamic hedging strategies. Market dynamics were "erased" by the logic of "risk neutral" diffusion. Concomitantly, risk measurement was focused on Value-at-Risk estimation, in other words, profit and loss distribution quantiles. Since the crisis a range of questions arose, that previously didn't get the traction they deserved, and relate to the dynamic behavior of markets during crises, the anticipation of dynamic instabilities putting markets in a "metastable" state, credit contagion, systemic risk and the impact - positive or negative - of financial regulations. In this presentation, we shall try to provide a bird's-eye view of these questions, pre- and post-crisis, and provide fuel for thought in a field which becomes richer and more complex every day.
Jennifer Tour Chayes (Microsoft Research)
Age of Networks
Everywhere we turn these days, we find that networks can be used to describe relevant interactions. In the high tech world, we see the Internet, the World Wide Web, mobile phone networks, and a variety of online social networks. In economics, we are increasingly experiencing both the positive and negative effects of a global networked economy. In epidemiology, we find disease spreading over our ever growing social networks, complicated by mutation of the disease agents. In biomedical research, we are beginning to understand the structure of gene regulatory networks, with the prospect of using this understanding to manage many human diseases. In this talk, I look quite generally at some of the models we are using to describe these networks, processes we are studying on the networks, algorithms we have devised for the networks, and finally, methods we are developing to indirectly infer network structure from measured data. I'll discuss in some detail particular applications to cancer genomics, applying network algorithms to suggest possible drug targets for certain kinds of cancer.