Events
Department of Mathematics and Statistics
Texas Tech University
This talk is concerned with the Uniformization problem: which intrinsic qualities of a metric space allow a good parameterization by a
Euclidean space? In the first part of the talk, we consider geometrically good (conformal, quasiconformal) parameterizations. While
including a wide range of fractal examples, spaces as such enjoy geometric and analytic properties and a great amount of first-order
calculus can be performed on them. In the second part of the talk, we discuss measure-theoretically good (Lipschitz, Holder)
parameterizations. The problem of classifying spaces admitting such parameterizations is is one of the most important problems
in geometric measure theory and it is associated to the famous Traveling Salesman Problem.
Geometric PDEs are concerned with utilizing PDE techniques to study geometric problems. A general theme in my research is the investigation of equilibrium configurations with respect to natural quantities modelling the energy or entropy of geometric objects. Those equilibria enjoy several extremal properties that are usually described by elliptic PDEs. Consequently, understanding these equations would advance our knowledge about the associated geometries. In this talk, I'll describe my contribution to fundamental conjectures in the following concrete directions. First, we'll discuss PDEs on manifolds, exploiting elliptic equations that arise in special manifolds particularly in dimension four. The second direction focuses on geometric eigenvalue problems. Here the geometry of an object is examined through studying appropriate elliptic operators and their eigenvalues. Third, we'll talk about geometric flows and applications in which PDEs arise as a mechanism to change the shape of a manifold.
This talk is concerned with the Uniformization problem: which intrinsic qualities of a metric space allow a good parameterization by a
Euclidean space? In the first part of the talk, we consider geometrically good (conformal, quasiconformal) parameterizations. While
including a wide range of fractal examples, spaces as such enjoy geometric and analytic properties and a great amount of first-order
calculus can be performed on them. In the second part of the talk, we discuss measure-theoretically good (Lipschitz, Holder)
parameterizations. The problem of classifying spaces admitting such parameterizations is is one of the most important problems
in geometric measure theory and it is associated to the famous Traveling Salesman Problem.
Geometric PDEs are concerned with utilizing PDE techniques to study geometric problems. A general theme in my research is the investigation of equilibrium configurations with respect to natural quantities modelling the energy or entropy of geometric objects. Those equilibria enjoy several extremal properties that are usually described by elliptic PDEs. Consequently, understanding these equations would advance our knowledge about the associated geometries. In this talk, I'll describe my contribution to fundamental conjectures in the following concrete directions. First, we'll discuss PDEs on manifolds, exploiting elliptic equations that arise in special manifolds particularly in dimension four. The second direction focuses on geometric eigenvalue problems. Here the geometry of an object is examined through studying appropriate elliptic operators and their eigenvalues. Third, we'll talk about geometric flows and applications in which PDEs arise as a mechanism to change the shape of a manifold.
TBA