about 50 billion cells in your body go through mitosis each day, a process that requires a mother cell replicating and splitting its dna among two daughter cells. we know that dna is super-coiled and is very knotted in the nucleus for space purposes. yet, with overwhelming probability, dna manages to split evenly even in this highly tangled state. as it turns out, there is an enzyme called topoisomerase ii which cuts and glues dna to let other strands of dna pass through. no one is quite sure how this enzyme works or makes decisions on when to cut. in this talk, we'll explore a model that assumes topo ii makes strand cuts based off local topological properties. we'll also look at some results of numerical simulations to see how well this model mimics the true behavior of topo ii.

initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. a natural problem is to find a parameterization of all possible initial data. a standard method for this is called the conformal method. in this talk, we'll discuss the successes and failures of this method, and future directions for research.

compressive sensing in its most practical form aims to recover a function that exhibits sparsity in a given basis from as few function samples as possible. one of the fundamental results of compressive sensing tells us that $o(s \log^4 n)$ samples suffice in order to robustly and efficiently recover any function that is a linear combination of $s$ arbitrary elements from a given bounded orthonormal set of size $n > s$. furthermore, the associated recovery algorithms (e.g., basis pursuit via convex optimization methods) are efficient in practice, running in just polynomial-in-$n$ time. however, when $n$ is very large (e.g., if the domain of the given function is high-dimensional), even these runtimes may become infeasible.

if the orthonormal basis above is fourier, then the sparse recovery problem above can also be solved using sparse fourier transform (sft) techniques. though these methods aim to solve the same problem, they have a different focus. principally, they aim to reduce the runtime of the recovery algorithm as much as absolutely possible, and are willing to sample the function a bit more often than a compressive sensing method might in order to achieve that objective. by doing so, one can indeed achieve similar recovery guarantees to basis pursuit, but with radically reduced runtimes that depend only logarithmically on $n$. however, sfts are highly adapted to the special properties of the fourier basis, making their extension to other orthonormal bases difficult.

in this talk we will present a general framework that can be used in order to construct a highly efficient sft algorithm. the framework abstracts many of the components required for sft design in an attempt to simplify the application of sft ideas to other basis choices. extension of arbitrary sfts to the chebyshev and legendre polynomial bases will also be discussed.

we will discuss, using explicit examples, how dynamical systems can be used to study certain problems in number theory and geometry.

it is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. if we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; friedlander-iwaniec showed $x^2+y^4$ is prime infinitely often, and heath-brown showed the same for $x^3+2y^3$. we will demonstrate a family of multivariate sparse polynomials all of which take infinitely many prime values.

a planar set that contains a unit segment in every direction is called a kakeya set. these sets have been studied intensively in geometric measure theory and harmonic analysis since the work of besicovich (1919); we find a new connection to game theory and probability. a hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. at each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. thus they are engaged in a zero sum game, where the payoff is the capture time. we show that every rabbit strategy yields a kakeya set; the optimal rabbit strategy is based on a discretized cauchy random walk, and it yields a kakeya set k consisting of 4n triangles, that has minimal area among such kakeya sets. (talk based on joint work with y. babichenko, r. peretz, p. sousi and p. winkler).

holomorphic symplectic manifolds are the higher-dimensional
analogs of k3 surfaces and their local and global deformation theories
enjoy many of the same nice properties. by work of namikawa, some aspects
of the story generalize to singular symplectic varieties, but the lack of
a well-defined period map means the moduli theory is badly behaved. in
joint work with c. lehn, we consider locally trivial
deformations---deformations along which the singularities don't
change---and show that in this context most of the results from the smooth
case extend. in particular, we prove a version of the global torelli
theorem and derive some applications to the geometry of birational
contractions of moduli spaces of vector bundles on k3 surfaces.