Talks

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.

Dark matter (DM) remains one of the most enduring mysteries in fundamental physics, motivating a wide array of theoretical models and experimental searches. Axions and axion-like particles (ALPs) are theoretically well-motivated, as they naturally arise from theories with broken global symmetries. These particles may serve as viable DM candidates themselves or act as mediators between DM and the Standard Model. In this talk, I will present an overview of axion and axion-mediated DM models, including both classic QCD axions and broader classes of ALPs. Emphasis will be placed on novel mechanisms shaping the cosmic DM abundance, and on innovative strategies for detecting these elusive particles. I will highlight how forthcoming laboratory efforts, such as fixed-target and direct detection experiments, alongside astrophysical observations, are poised to explore uncharted regions of parameter space and deepen our understanding of the connections between axion physics and dark matter.

In this talk, I will give an overview of why the PN formalism is still relevant to model gravitational waves, focusing on recent synergies with other techniques and research topics. Taking the example of EFT-inspired higher curvature gravity theories, I will present a way towards building better gravitational wave tests to be used by next generation detectors.

How did the universe evolve prior to the creation of the cosmic microwave background? There are no direct observational probes of the universe’s expansion history prior to the onset of Big Bang nucleosynthesis (BBN), and numerous theories predict deviations from radiation domination during the universe’s first second. Meanwhile, a persistent discrepancy between local and cosmological measurements of the Hubble constant has prompted us to reconsider the evolution of the universe between BBN and recombination. Since the growth of dark matter density perturbations depends on the expansion rate, deviations from the standard expansion history leave imprints on the matter power spectrum. I will discuss how adding decaying massive particles or fast-rolling scalar fields to the standard cosmological model impacts the abundance and structure of dark matter halos. Both cases illustrate how small-scale structure provides a powerful probe of the evolution of the universe prior to recombinatio

In slow-roll inflationary models, the inflaton can undergo excursions on the order of the Planck scale, leading to significant changes in the properties of fields coupled to the inflaton, referred to as spectator fields. These changes may result in transitions between weakly and strongly interacting regimes, or even alterations in mass squared within the spectator field sector during inflation. Such dynamics can induce phase transitions, which have profound implications for the early Universe. In this talk, I will explore the phenomenological consequences of these phase transitions, focusing on the production of gravitational waves, curvature perturbations, non-Gaussianities, dark matter, and baryon number. I will also demonstrate how gravitational waves generated by scalar perturbations induced by phase transitions may potentially explain the alleged gravitational wave signals observed in recent pulsar timing array studies.