We describe how one can recover the Mori--Mukai
classification of smooth 3-dimensional Fano manifolds using mirror
symmetry, and indicate how the same ideas might apply to the
classification of smooth 4-dimensional Fano manifolds. This is joint
work in progress with Corti, Galkin, Golyshev, and Kasprzyk.
I will consider the physical background, and general thinking behind, the recent programme aimed at applying topos theory to the foundations of physics.
The central axis of the famous DNA double helix can become knotted
or linked as a result of numerous biochemical processes, most notably
site-specific recombination. Site-specific recombinases are naturally
occurring enzymes that cleave and reseal DNA molecules in very precise ways.
As a by product of their main purpose, they manipulate cellular DNA in
topologically interesting and non-trivial ways. So if the axis of the DNA
double helix is circular, these cut-and-seal mechanisms can be tracked by
corresponding changes in the knot type of the DNA axis. In this talk, I'll
explain several topological strategies to investigate these biological
situations. As a concrete example, I will disscuss my recent work, which
predics what types of DNA knots and links can arise from site-specific
recombination on DNA twist knots.
In this talk we will present some recent results on the long time
asymptotics of the generalized (non-Markovian) Langevin equation (gLE). In particular,
we will discuss about the ergodic properties of the gLE and present estimates on the rate of convergence to equilibrium, we will present
a homogenization result (invariance principle) and we will discuss
about the convergence of the gLE dynamics to the (Markovian) Langevin
dynamics, in some appropriate asymptotic limit. The analysis is based on the approximation of the gLE by a
high (and possibly infinite) dimensional degenerate Markovian system,
and on the analysis of the spectrum of the generator of this Markov
process. This is joint work with M. Ottobre and K. Pravda-Starov.