Talks

In this talk I will discuss one of the frontiers of both theory and data analysis in gravitational wave astronomy - understanding the ringing of black holes and probing them from real data. I will review past efforts started from Chandrasekhar, Detweiler, et al in analyzing modes of black holes and explain what we currently understand in both linear and nonlinear wave properties, as well as the corresponding detection aspect. At last I will show a few pressing problems and where we will be heading.

I will discuss BBN and a new python-based tool (PRyMordial) which allows one to easily simulate it both in the context of a standard cosmological model as well as in various scenarios of physics beyond the Standard Model. I’ll discuss how BBN provides a unique probe of physics relevant for the ~ MeV scale, and how it constrains or hints at modifications to the standard picture.

Complete minimal surfaces in R^3 have been much studied but much less is known about R^4. I will recall the main tools in R^4 and give a couple of examples of minimal embeddings of the complex plane in R^4. Then I will focus on complete minimal tori of curvature -8π and one end: in R^3 there is a unique example (the Chen-Gackstetter square torus) but in R^4 we can construct examples on all the rectangular tori. I will discuss the strategy for the construction and indicate the many questions which remain open. Joint work with Marc Soret.

We discuss the topological realization problem for minimally embedding compact surfaces in round spheres and balls. In 1970, using the solution to the Plateau problem, Lawson constructed orientable minimal surfaces of each genus embedded in $S^3$. In recent work with Karpukhin, McGrath and Stern, using equivariant eigenvalue optimization methods and a priori eigenspace dimension bounds, we constructed orientable free boundary minimal surfaces in $B^3$ of any genus and (positive) number of boundary, components. Now we have extended our methods to handle the nonorientable case, constructing embedded minimal surfaces in $S^4$ diffeomorphic to the connect-sum of any number of real projective planes: these all have area (so Willmore bending energy) under $8\pi$ and enjoy other interesting geometric features. The analogous construction for nonorientable free boundary minimal surfaces embedded in $B^4$ looms on the horizon. Open problems and speculation about potential discretizations may be offered if time permits....

We will talk of interpolation problems of two types.

First type of interpolation we talk of is that given two real analytic curves can one interpolate them with a minimal or maximal surface or a CMC surface? -- a version of Plateau's problem. For minimal surfaces this problem was solved by Douglas and Rado in great generality. We show that indeed, if the curves are "close" enough in a certain sense, then interpolation is possible. We will also talk of existence of a maximal surface containing a given real analytic curve and a special singularity, under certain conditions.

The second type of interpolation we will talk about is given a array of surfaces placed at some periodic intervals, can one interpolate them by a minimal/maximal surface, in the sense that the height functions of surfaces at these arrays sum up to a height function of a minimal/maximal surface.This work uses some Euler-Ramanujan identities.

A number of theories predict heavy dark matter, including WIMPs and high mass composite states formed in the early universe. Discovering the heaviest dark matter candidates, with a unit mass in excess of a microgram, requires methods beyond traditional underground experiments. For high mass dark matter searches in general (including WIMPS) new search methods will be necessary in the coming decades. I will discuss the most trenchant past searches for high mass dark matter along with future prospects.