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Geometry and Topology Seminar: Joonas Ilmavirta, University of Jyväskylä, Finland, Geophysics and algebraic geometry

SAS 4201

Many areas of interest within the Earth are anisotropic, meaning that the speed of sound is different in different directions. It turns out that pressure waves are far better behaved than shear waves, but fortunately the different polarizations are coupled together through algebraic geometry. I will explain the surprising power of algebraic geometry in the…

Geometry and Topology Seminar:  Igor Zelenko, Texas A&M University, Gromov’s h-principle for corank two distribution of odd rank with maximal first Kronecker index

SAS 3282

The natural question is: do structures satisfying given open relations (called the genuine solutions of the differential relation) exist on a given manifold? Replacing all derivatives appearing in a differential relation by the additional independent variables one obtains an open subset of the corresponding jet bundle. A formal solution of the differential relation is a…

Geometry and Topology Seminar: Arunima Bhattacharya, UNC Chapel Hill, Lagrangian Mean Curvature Equations

SAS 1216

In this talk, we will introduce the special Lagrangian and Lagrangian mean curvature type equations. We will derive a priori interior estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian angle. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data…

Geometry and Topology Seminar: Mathew Kushelman, NC State, On Liouville’s Theorem for Conformal Maps

SAS 1216

A theorem of Liouville asserts that the simplest conformal transformations on Euclidean space---translations, dilations, reflections, and inversions---generate all conformal transformations when the dimension is at least 3.  I will describe a new proof of this theorem which is shorter and more elementary than the argument, due to Nevanlinna, found in most modern textbooks.

Geometry and Topology Seminar: Ruzica Mijic, Technical University of Vienna, An Introduction to Laguerre Geometry

SAS 1216

In traditional Euclidean geometry, points serve as the foundational elements for constructing and analyzing space. In contrast, Laguerre geometry, a non-Euclidean geometry, uses oriented circles (or hyperspheres, in the context of higher dimensions) and oriented lines (or hyperplanes), as fundamental objects. Here, a “point” is simply a circle with radius zero, i.e. having no special…

Geometry and Topology Seminar: Marithania Silvero Casanova, Universidad de Sevilla, Positivity, fiberedness and link homology

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Khovanov homology is a link invariant which categorifies Jones polynomial. In this talk we present several results concerning Khovanov homology of fibered positive links; in particular, we extend the result by Stosic stating that braid positive links have vanishing Khovanov homology in homological grading 1. We also explore Khovanov homology of certain cable links and…

Geometry and Topology Seminar: Daniel Weser, UNC, A Heintze-Karcher inequality with free boundaries and applications to capillarity theory

SAS 1216

 In volume-constrained capillarity problems, minimizers may have free boundaries adhering to the container. Recent work in the study of capillarity problems has utilized stability theory for the volume-constrained isoperimetric problem to classify the shape of global minimizers and (in the case without free boundary) critical points. In this talk, I will discuss joint work with…

Geometry and Topology Seminar: Kai-Wei Zhao, University of Notre Dame, On the blowup of regularized solutions to the Jang equation and constant expansion surfaces

SAS 1216

Schoen-Yau proved the spacetime positive energy theorem by reducing it to the time-symmetric (Riemannian) case using the Jang equation. To acquire solutions to the Jang equation, they introduced a family of regularized equations and took the limit of regularized solutions, whereas a sequence of regularized solutions could blow up in some bounded regions enclosed by apparent horizons. They analyzed the blowup behavior near and outside the apparent horizons, but what happens inside…

Geometry and Topology Seminar: Andrew Shedlock, NC State, Recovery of a complete Riemannian Manifold using the local source-to-solution operator for the Electro-Magnetic Wave Operator

SAS 1216

In this talk we consider an Electro-Magnetic Wave operator on a complete Riemannian manifold, which generalizes the standard wave equation to include first order and zeroth order terms.  The Cauchy Problem for the Electro-Magnetic Wave operator with zero initial values and a smooth compactly supported forcing function has a unique smooth solution. We study the…

Geometry and Topology Seminar: Lili Yan, University of Minnesota, Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation

SAS 1216

In this talk, we consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions 2 and higher. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on…

Geometry Topology Seminar: Paweł Dłotko, DIOSCURI Centre for Topological Data Analysis, Warsaw, Poland, From Euer number to statistics and back. Tangential approach to Topological Data Analysis.

SAS 4201

In this talk, I will give a brief and intuitive overview of the methods of Topological Data Analysis we developed in my Discuri Centre. Starting from Euler numbers and its generalization we will get to the regime of statistical goodness of fit test. We will explore alternative methods of constructing graph based visualization of data.…

Geometry and Topology Seminar: Ao Sun, Lehigh University, Interpolation method in mean curvature flow

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The interpolation method is a very powerful tool to construct special solutions in geometric analysis. I will present two applications in mean curvature flow: one is constructing a new genus one self-shrinking mean curvature flow, and another one is constructing immortal mean curvature flow with higher multiplicity convergence. The talk is based on joint work…

Geometry and Topology: Peter McGrath, NC State, New Minimal Surfaces in B^3

SAS 4201

A fundamental question in Differential Geometry which has spurred much recent work is whether every orientable compact surface with boundary can be minimally embedded in the Euclidean 3-ball with free boundary.  I will discuss work (joint with M. Karpukhin, R. Kusner, and D. Stern)  which resolves this problem.

Geometry and Topology Seminar: Tony Liimatainen, University of Helsinki, Finland, Geometric inverse problems and inverse problems for the minimal surface equation

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We start by giving a short introduction to geometric inverse problems. Then we present our recent results regarding inverse problems for the minimal surface equation. Minimal surfaces are solutions to a quasilinear elliptic equation. We determine a minimal surface up to an isometry from the corresponding Dirichlet-to-Neumann map in dimension 2. Applications of the results…

Geometry and Topology Seminar: Peter J. Olver, University of Minnesota, Structure and Generators of Differential Invariant Algebras

SAS 1216

The structure of algebras of differential invariants, particularly their generators, is based on the symbolic invariant calculus provided by the method of equivariant moving frames.  I will discuss a new computational algorithm that will, in many cases, determine whether a given set of differential invariants is generating.  As an example, we establish a new result…

Geometry and Topology Seminar: Adam Lowrance, Vassar College, The average value of invariants of 2-bridge knots.

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We show how to use continued fraction representations of 2-bridge knots to compute the average value of different invariants of the set of 2-bridge knots with fixed crossing number c. Examples include the Seifert genus, braid index, and the absolute value of the signature. We also mention other properties of the probability distributions of these…