a riemannian metric gives rise to geodesics. as unparameterised curves, at each point there is one geodesic in every direction. does this arrangement of curves determine the metric and will any such arrangement of curves determine a metric? these are classical questions considered, for example, by roger liouville in 1889 who made considerable progress in two dimensions. this talk will discuss these questions in two and three dimensions, mostly based on joint work with robert bryant and maciej dunajski.

there are many well-known concentration results for sums of independent random matrices, e.g. those of rudelson, ahlswede-winter, tropp, and oliveira. we move beyond the independent setting, and prove a chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an reversible markov chain, confirming a conjecture due to wigderson and xiao. our proof is based on a new multi-matrix extension of the golden-thompson inequality which follows from complex interpolation methods.
joint work with a. garg, y. lee, and z. song.

the motion of a flying saucer is restricted by the three-dimensional geometry of the space in which it moves. i shall discuss three possible cases, one of which may be interpreted as an explanation of engel's 1893 construction of the exceptional lie algebra g2. this is joint work with pawel nurowski.

donaldson-thomas (abbreviated as dt) theory is a sheaf theoretic technique of enumerating curves on a calabi-yau threefold. classical dt invariants give a virtual count of gieseker stable sheaves provided that no strictly semistable sheaves exist. this assumption was later lifted by the work of joyce and song who defined generalized dt invariants using hall algebras and the behrend function, their method being motivic in nature. in this talk, we will present a new approach towards generalized dt theory, obtaining an invariant as the degree of a virtual cycle inside a deligne-mumford stack. the main components are an adaptation of kirwan’s partial desingularization procedure and recent results on the structure of moduli of sheaves on calabi-yau threefolds. based on joint work with young-hoon kiem and jun li.

biological systems make use of feedback in an extraordinary number of ways, on scales
ranging from molecules to cells to organisms to ecosystems. in this talk i will discuss the use
of concepts from feedback control theory in the design of feedback circuits at the molecular,
cellular, and multicellular level. after a brief survey of relevant concepts from control theory
and synthetic biology, i will present some recent results that demonstrate how feedback
controllers can be implemented in cells, including ratiometric control of protein
concentrations, reference tracking of cell concentration, and fractional control of cell
populations. using these examples and others from the literature, i will discuss some of the
common features and possible architectures for implementation of biomolecular feedback
control systems, some limits on performance predicted by the theory, and some
implementation challenges that require more effort and new approaches.