Talks

This talk presents a study of biconservative hypersurfaces M4r(r = 0, 1, 2, 3, 4) in the pseudo-Euclidean space E5s(s = 0, 1, 2, 3, 4, 5) with constant norm of the second fundamental form. We find that every such hypersurface in E5s with a diagonal shape operator has constant mean curvature and constant scalar curvature. This is a joint work with Ram Shankar Gupta, Andreas Arvanitoyeorgos, and Marina Statha.

There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.

CMC (constant mean curvature) surfaces are critical with respect to area as long as we restrict to competitor surfaces which 'enclose the same volume'. The index of a CMC surface is the number of ways we can locally deform it to reduce its area whilst enclosing the same amount of volume. We will show that the index of a CMC surface is bounded linearly from above in terms of its genus and a Willmore-type energy i.e. the more unstable a surface is the more genus and/or ‘Willmore energy’ it must have. Joint work with Luca Seemungal. 

An unavoidable part of studying astrophysics based on catalogs of detected events is quantifying the probability of detecting different types of events. I will briefly discuss the types of design considerations that go into constructing such estimates and how they will scale with larger catalog sizes. I will also introduce the wide variety of uses for such data products, including uncovering unexpected features within the data caused by the fact that humans build and operate the detectors.

Binary neutron star mergers are critical for understanding the dynamics of dense matter, the origin of gravitational waves, and the formation channels of the heaviest elements through the r-process. I will review how long-lived remnants can act as central engines for multimessenger observations. I will then discuss how we can identify phase transitions within neutron stars or their remnants using such observations. Phase transitions alter the system’s dynamics and can produce distinct observable signatures, potentially detectable with next-generation facilities and observatories. These signatures can be used to probe matter at supranuclear densities and to test fundamental physics.

Heavy right-handed neutrinos are highly motivated due to their connection with the origin of neutrino masses via the seesaw mechanism. If the right-handed neutrino Majorana mass is at or below the weak scale, direct experimental discovery of these states is possible in laboratory experiments. However, there is no a priori basis to expect right-handed neutrinos to be so light since the Majorana mass is a technically natural parameter and could comfortably reside at any scale, including at scales far above the weak scale. Here we explore the possibility that the right-handed neutrino Majorana mass originates from electroweak symmetry breaking. Working within an effective theory with two Higgs doublets, nonzero lepton number is assigned to the bilinear operator built from the two Higgs fields, which is then coupled to the right-handed neutrino mass operator. In tandem with the neutrino Yukawa coupling, following electroweak symmetry breaking a seesaw mechanism operates, generating the light SM neutrino masses along with right-handed neutrinos with masses below the electroweak scale. This scenario leads to novel phenomenology in the Higgs sector, which may be probed at the LHC and at future colliders. There are also interesting prospects for neutrinoless double beta decay and lepton flavor violation. We also explore some theoretical aspects of the scenario, including the technical naturalness of the effective field theory and ultraviolet completions of the right-handed neutrino Majorana mass.

The Dark Energy Spectroscopic Instrument (DESI) is the first of a new generation of Dark Energy experiments and probes evolution in the universe using galaxy clustering. Within the galaxy clustering signal, the projected location of the Baryon Acoustic Oscillations (BAO) acts as a standard ruler to map cosmic evolution. I will present the latest BAO results from the DESI Data Release 2 (DR2) sample, which contains 3 years of data, and their impact on our understanding of dark energy and neutrino masses. Finally, I will consider how the amplitude of the BAO signal can help us measure the Hubble constant, potentially helping to solve the Hubble tension.

For a closed Riemann surface X, and an irreducible representation R from the fundamental group of X to PSL(2,C), a seminal theorem of Donaldson proves the existence of an R-equivariant harmonic map from the universal cover of X into hyperbolic 3-space. We shall discuss a generalization of this result to the case when X is a punctured Riemann surface arising from an element of the enhanced Teichmüller space, and R is a framed PSL(2,C)-representation. Our technique involves the harmonic map heat flow, and the relation between the asymptotic behaviour of a harmonic map and the singular-flat geometry of its Hopf differential. This is joint work with Gobinda Sau.