The knot concordance group consists of knots modulo the equivalence relation of concordance, with the operation inducted by connected sum. We define an infinite family of integer-valued concordance homomorphisms coming from the knot Floer complex. We will compare these invariants to other concordance homomorphisms and discuss applications. This is joint work with I. Dai, M. Stoffregen, and L. Truong

One generally considers 3-manifolds up to homeomorphism. We consider 3-manifolds up to a weaker notion of equivalence, called homology cobordism. Under this equivalence relation, the set of 3-manifolds forms a group, with the operation induced by connected sum. We use Heegaard Floer homology to give new results about the structure of this group. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

Determining the maximal number of linearly independent vector fields on the n-sphere is a classical question in topology generalizing the hairy ball theorem. Radon and Hurwitz produced a lower bound using Clifford algebras, and then Adams showed it’s also an upper bound. Today I’ll present Adams’ calculation, reducing the problem to a question about K-theory, then answering it with spectral sequences.

In this talk, I will discuss recent results -- obtained in collaboration with C. Henderson and A. Tarfulea -- on the local well-posedness of the inhomogeneous Landau equation. We construct a unique classical solution on some time interval, for initial data that is Holder continuous and polynomially decaying in the velocity variable. Our construction also leads to an improved continuation criterion for solutions.

I will discuss some of the theory of convex projective manifolds: quotients of open subsets of real projective space by the action of discrete subgroups of PGL(n,R). These give an interesting generalization of hyperbolic manifolds. For instance, the fundamental groups of such manifolds are often word-hyperbolic, and there are parallels to the theory of quasifuchsian groups. I'll explain some of these concepts in the case of a compact manifold, and then discuss what can be said in the case where we no longer assume that the fundamental group of the manifold is word-hyperbolic.

I will discuss stacks classifying families of broken lines and broken cycles. These stacks give new ways to organize algebraic structures, and have enticing applications to symplectic geometry. For example, in joint work with Jacob Lurie, we've shown that families of broken lines classify non-unital Aoo structures while giving a clear pathway to enrich Lagrangian Floer theory over spectra (which are more powerful invariants than chain complexes). The stack of broken cycles, on the other hand, is a universal recipient for (co)sheaves of Fukaya categories on 1-dimensional Lagrangian skeleta.

Abstract: In this talk, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three- dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

We will show how minimal dynamical systems and etale groupoids can be used to construct finitely generated simple groups with prescribed properties. For example, one can show that there are uncountably many different growth types (in particular quasi-isometry classes) among finitely generated simple groups, or embed the Grigorchuk group into a simple torsion group of intermediate growth. Other properties like torsion and amenability will be also discussed.

Weak (variational) solutions to the Navier-Stokes equations in R^3 are plagued by many pathologies even for relatively nice initial data. One way around this problem is to work with a fundamentally different notion of solution based on the theory of semigroups. I will introduce these mild solutions and sketch the classic well-posedness theorem due to Kato.