Harvard

Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

It has been recently pointed out that maximal gauged supergravities in four dimensions often come in one-parameter families. The parameter measures the combination of electric and magnetic vectors that participate in the gauging. I will discuss the higher-dimensional origin of these dyonic gaugings, when the gauge group is chosen to be ISO(7). This gauged supergravity arises from consistent truncation of massive type IIA on the six-sphere, with its dyonically-gauging parameter identified with the Romans mass. The (AdS) vacua of the 4D supergravity give rise to new explicit AdS4 backgrounds of massive type IIA. I will also show that the 3D field theories dual to these AdS4 solutions are Chern-Simons-matter theories with a simple gauge group and level k also given by the Romans mass.

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections. After summarizing the evidence for these conclusions, I will delve into our efforts to create mathematical models that connect these patterns to the developmental mechanism leading to a beak. It turns out that there are simple (but precise) constraints on any mathematical model that leads to the observed phenomenology, leading to explicit predictions for the time dynamics of beak development in song birds. Experiments testing these predictions for the development of zebra finch beaks will be presented.

Based on the following papers:

http://www.pnas.org/content/107/8/3356.short

http://www.nature.com/ncomms/2014/140416/ncomms4700/full/ncomms4700.html

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.

I will discuss a few problems  that involve randomness , chosen randomly  (?) from the following : (i) the probability of a coin landing on a side  (ii) optimal strategies for throwing accurately, (iii)  the statistical mechanics of a ribbon, (iv) the intermittent dynamics of a growing polymeric assembly (v) fat tails from feedback.



In Clarendon Lab