Georgia Institute of Technology College of Sciences

We consider Klein-Gordon equations with an external potential and cubic nonlinearity in three spatial dimensions. It is assumed that the linear operator has internal modes, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein treated the case when the linear operator has one large eigenvalue and proved the decay of the solution. In 2022, we solved the general one eigenvalue case. In our recent work, we solved the multiple internal modes case: the operator can has multiple and possibly degenerate eigenvalues. Indeed, we determined the sharp decay rate of the overall solution, as well as distinct decay rates for different modes of the solution. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu.

This series of talks will introduce the immersed curve perspective of knot Floer homology and give applications to to 3-manifold topology.

Margulis inequalities and Margulis functions (a.k.a Foster-Lyapunov stability) have played a major role in modern dynamics, in particular in the fields of homogeneous dynamics and Teichmuller dynamics.
Moreover recent exciting developments in the field of random walks over manifolds give rise to related notions and questions in a much larger geometrical content, largely motivated by recent work of Brown-Eskin-Filip-Rodriguez Hertz.

I will explain what are Margulis functions and Margulis inequalities and describe the main lemma due to Eskin-Margulis (“uniform expansion”) that allows one to prove such an inequality. I will also try to sketch some interesting applications.

No prior knowledge is needed, the talk will be self-contained and accessible.

Zoom Link: https://gatech.zoom.us/j/94689623118?pwd=Ie8Ir2bExulIP4joQbcmZiwsxpIq75.1 Meeting ID: 946 8962 3118 Passcode: 910355

Three fifteen-minute talks by local speakers.

Logan Post: Almost all even-parity binary words are shuffle squares

Jasper Seabold: Using Grid Graphs to Study Hypergraph Ramsey Questions

Adri Wessels: