Video Lectures

We show that two generic, open, convex or concave toric domains in R4 are symplectomorphic if and only if they agree up to reflection. The proof uses barcodes in positive S1-equivariant symplectic homology, or equivalently in cylindrical contact...

In the 1940s Mahler initiated the program of determining the bass note spectrum Spec(P):={infx⎯⎯∈Λ∖0⎯⎯∣∣P(x⎯⎯)∣∣,Λ⊂ℝk a unimodular lattice} for some homogeneous form P. Understanding this spectrum is central in the geometry of numbers and offers a...

Galaxies are at the core of nearly all modern astrophysical studies. They serve as essential cosmological probes, tracing the structure of the universe, while also providing the stage on which stars form and black holes grow. Despite their...

For an embedded stable curve over the real numbers we introduce a hyperplane arrangement in the tangent space of the Hilbert scheme. The connected components of its complement are labeled by embeddings of the graph of the stable curve to a compact...

A sequence of nonnegative real numbers a1,a2,…,an, is log-concave if a2i≥ai−1ai+1 for all i ranging from 2 to n−1. Examples of log-concave inequalities range from inequalities that are readily provable, such as the binomial coefficients ai=(ni), to...

Teichmuller dynamics give us a nonhomogeneous example of an action of SL_2(R) on a space H_g preserving a finite measure. This space is related to the moduli space of genus g curves. The SL_2(R) action on H_g has a complicated behavior: McMullen...

In this talk, we will explore recent developments in the study of coherent structures evolving by incompressible flows. Our focus will be on the behavior of fluid interfaces and vortex filaments. We include the dynamics of gravity Stokes interfaces...

Stellar streams form as dwarf galaxies and globular clusters tidally disrupt around their host galaxy. To date, more than 100 streams have been detected around our Galaxy. In this talk, I will begin with an overview of how streams form, how they can...

Algebraic torsion is a means of understanding the topological complexity of certain homomorphic curves counted in some Floer theories of contact manifolds.  This talk focuses on algebraic torsion and the contact invariant in embedded contact...

Topology of the Hitchin system has been studied for decades, and interesting connections were found to orbital integrals, non-abelian Hodge theory, mirror symmetry etc. I will explain that a large part of the symmetries in these geometries above are...