I will discuss generalized global symmetries and higher p-form symmetries from the perspective of spontaneous symmetry breaking. I will mostly focus on continuous (non-discrete) symmetries, and discuss the generalizations of Goldstone's theorem, the Goldstone modes, and the Coleman-Mermin-Wagner theorem to generalized global and higher form symmetries.

Manolescu and Piccirillo recently initiated a program of constructing potential counterexamples to the smooth 4-dimensional Poincare conjecture via a homeomorphism of the zero surgeries of two knots. They mainly focus on examples of pairs where one knot is not slice in and the other is unknown if slice. We instead consider pairs where one knot is slice and study the resulting homotopy spheres.

It has been known since their definition due to Labourie in 2004 that Anosov representations are stable- any sufficiently small perturbation of an Anosov representation is still Anosov. In 2014, Kapovich, Leeb and Porti characterized Anosov representations as those sending geodesics in the word hyperbolic group \Gamma to “Morse quasigeodesics” in the associated symmetric space G/K. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. Supplementing their proof with explicit estimates in the symmetric space allows us to describe explicit neighborhoods of Anosov representations. This is a thesis defense. https://utexas.zoom.us/j/93336434967?pwd=eDZEUi9EdEhLcHJXQjk4eHBuNThMQT09 Meeting ID: 933 3643 4967 Passcode: morse

The study of hyperbolic and relatively hyperbolic groups acting on CAT(0) cube complexes has produced exciting recent results in geometric group theory. I will talk about a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex called a relatively geometric action. In joint work with Daniel Groves, we develop analogues of tools used to construct and study geometric actions of hyperbolic and relatively hyperbolic groups on CAT(0) cube complexes, including a relatively geometric version of Agol's Theorem. I will also discuss some of the structural theorems we hope to prove and a potential application to the Relative Cannon Conjecture.

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

Leray in 1934 and Hopf in 1950 have proved the existence of weak solutions of Navier stokes. To have the result, we can actually make use of eigenfunctions of Stokes operator and also the Bogovskii operator. This is now a classical process to find existence on a bounded domain and can inspire a lot of other existence results.