Selected Presentations & Appearances
In this talk, reporting on joint work with Sergey Cherkis, I discuss the
recent proof that at large distances Yang Mills instantons on ALF spaces decompose into a sum of U(1) instantons.
In this talk, I describe my recent proof of a large part of a conjecture of Gromov's on the L_p cohomology of symmetric spaces.
The Nahm transform provides a fundamental tool for analyzing Yang-Mills instanton moduli spaces. In this talk, after reviewing the transform's origins in representation theory, I discuss recent progress (based on joint work with S. Cherkis and A. Larrain-Hubach) in extending the transform to ALF gravitational instantons.
In this talk I will discuss the geometry of p-harmonic forms
and basic vanishing and stability theorems in L^p-cohomology. I will
briefly
sketch how to apply these techniques to prove a conjecture of Gromov on
the reduced L^p cohomology of negatively curved symmetric spaces.
I will the Nahm transform for multi-centered Taub-NUT spaces, with an emphasis on the invertibility of the transform and the induced isometry of moduli spaces. This is joint work with Sergey Cherkis and Andres Larrain-Hubach.
In this talk I will lay the foundations of the geometry of p-harmonic forms and L^p-Hodge theory. As an application, I will give strong evidence for (half of) a conjecture of Gromov on the L^p cohomology of negatively curved symmetric spaces.
Monotonicity techniques developed to study the small scale structure of singularities of nonlinear pdes have been sharpened to give large scale information about eigenfunctions with small eigenvalues.
These improved monotonicity estimates can then return to nonlinear pde
to provide better understanding of their singularities.
In this talk, I discuss various applications of monotonicity estimates to eigenvalue problems arising in topology and number theory and conclude with an application to a toy problem concerning singularities arising in Yang Mills theory.
In this talk we survey the application of monotonicity methods to bound Betti numbers and the dimensions of Laplace Beltrami eigenspaces with small eigenvalue on negatively curved spaces.
The goal of this talk is to explain aspects of the isometry between the moduli space of Yang-Mills instantons on ALF gravitational instantons of type A_{k-1} and the space of solutions to a system of odes on line segments.
This is joint work with Sergey Cherkis and Andres Larrain Hubach.
In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach
In this talk, we introduce many new variations of Nahm transforms, including transforms from Sp(k) connections to Sp(j) connections.
I discuss the effect of negative curvature and large girth on singularity formation in Yang Mills and harmonic maps.
Abstract: "Motivated by questions about the spectral geometry of
hyperbolic manifolds arising in number theory, we seek a lower bound for
the first nonzero eigenvalue of the Laplace Beltrami operator on 1-forms
on hyperbolic manifolds. We prove an analog of Cheeger's inequality.
Joint work with Michael Lipnowski."
A talk about my joint work with Luca DiCerbo on new estimates of Betti numbers
This talk was a report on my continuing joint work with Sergey Cherkis on the Nahm transform for instantons on ALF spaces
In this talk I discuss my recent work with Michael Lipnowski on estimates for eigenvalues of Laplace Beltrami operators motivated by conjectures of alegari et al.
This talk was written for an audience of mathematics students from underdeveloped countries.
Assuming only an undergraduate mathematics background, I introduced major questions and ideas of Yang Mills theory.
I discuss recent progress in collaboration with Sergey Cherkis and Andres Larrain-Hubach
on establishing Cherkis's conjectured Nahm transform for Yang Mills instantons on ALF gravitational instantons.
I discuss recent progress in collaboration with Sergey Cherkis and Andres Larrain-Hubach
on establishing Cherkis's conjectured Nahm transform for Yang Mills instantons on ALF gravitational instantons.