Geometry and Analysis

TBA

TBA

TBA

I will discuss the geometry of CMC surfaces embedded in Euclidean space with finite genus and, among other things, prove a local area estimate for such surfaces. This is joint work with Bill Meeks.

Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs, especially in relation to physical models. Since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, it is well known that the optimal regularity of a Riemannian structure is governed by that of the Ricci tensor in harmonic coordinates. In this talk, we will discuss the conformal analogue problem: we consider a closed 3-dimensional Riemannian manifold (M, g) with g in the Sobolev class W^{2, q} with q > 3 and show that the optimal regularity of the conformal structure is governed by that of the conformally invariant Cotton tensor.

The proof requires the resolution of the renowned Yamabe problem for W^{2, q} metrics, which is interesting on its own. In turn, the resolution of the Yamabe problem relies on new results on the existence, regularity and expansion of the Conformal Green's function for W^{2, q} metrics, which we provide in any dimension n \geq 3 and for q>n/2.

This is based on joint work with R. Avalos and A. Royo Abrego.

The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for metric 1-currents as superposition of 1-rectifiable sets in complete and separable metric spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents and a more refined analysis in the Banach space setting. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).

In this talk, I will discuss inverse source problems for nonlinear evolution equations, particularly for parabolic and pseudoparabolic equations perturbed by p-Laplacian diffusion and power-law reaction terms. We also consider inverse problems for these equations with time-fractional derivatives. Such parabolic and pseudoparabolic equations, modified by p-Laplacian diffusion and power-law reaction terms, have various applications—for instance, in modeling non-Newtonian fluids, filtration processes, and more.
For all these problems, we establish sufficient conditions on the data that guarantee the existence and uniqueness of weak solutions to the given problems. In addition, we obtain some qualitative properties of the solutions, such as extinction in finite time, large time behavior, etc.

Skyrmions, topological solitons arising in nonlinear sigma models, have long been proposed as effective models for atomic nuclei. While exact solutions remain unknown, a variety of approximations, often inspired by better understood solitons such as monopoles and instantons, have been developed. In particular, Sutcliffe recently introduced a novel approximation using an ultra-discrete computation of the holonomy of a JNR instanton. In this talk, I will outline the mathematical framework behind this construction, before focusing particularly on the constituent locations of the resulting Skyrmions. I will present new theoretical and computational tools for analysing these locations, reveal their connections to classical geometry and monopole theory, and discuss their potential implications for future research. Based up joint work with Josh Cork.

In this talk I will focus on simply-connected holomorphic symplectic manifolds - both Kähler (hyperkähler manifolds) and non-Kähler. The only known example of the latter was constructed by Bogomolov-Guan. First, I will remind the construction of known families which are mostly based on the same idea. Then I will outline some geometry of some particular examples with its connection to G_2-manifolds and birational geometry. Later, I will speak about the subvarieties of such manifolds with the main ideas rely on the deformation theory, calibrated manifolds and Lagrangian fibrations. In particular, I will show the nonexistence of some tori in hyperkahler manifolds.