Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
The talk is postponed to a later date (TBD) as the speaker contracted Covid recently.
Translating solitons and self shrinkers are solitons of the mean curvature flow. They are not only special solutions of the MCF but blow-up models of singularities of MCF. In this talk, we firstly introduce a half-space type theorem of translating solitons. More precisely, we prove that complete translating solitons can lie on the upper part of a hyperplane and cannot lie on the lower part of it. Secondly, we introduce the rigidity of a self-shrinker with free boundary in a ball. We prove that any graphical self-shrinker with free boundary in a ball is a flat disk passing through the center of the ball.
Watch online via this Zoom link.In this talk we will present some connections between different random graph models. The classical Erdos-Renyi and Chung-Lu models with independent edges are well-known and easy to work with, whereas more complicated models such as regular random graph, scale-free random graph and preferential attachment random graph are more difficult to be studied. However there are similar properties which can be found on many different models. The connections between models allow us to prove a result on one model using a similar result on another model. We will also talk about some examples of properties which can be proved using this method such as convergence of limiting ditribution of eigenvalues, power law of eigenvalues, average distance, etc.
Keywords: Random graph; Preferential attachment; Erdos-Renyi; Chung-Lu; regular random graph.
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We consider the asymptotic limits of two dimensional incompressible stochastic Navier Stokes equation and one dimensional stochastic Schrodinger equation. These limits include large and moderate deviations, central limit theorem, and the law of the iterated logarithm. For large and moderate deviations, we will discuss both the Azencott method and the weak convergence approach and show how they can be used to derive the Strassen's compact law of the iterated logarithm. The exit problem will also be given as an application.
Watch online via this Zoom link.A Riemannian metric on a closed manifold is called Zoll when all of its geodesics are closed and have the same period. Zoll metrics on the two-sphere were constructed by Zoll in the beginning of the nineteen hundreds, but many questions about them are still open. In this talk, I will explain my motivation to look for higher dimensional analogues of Zoll metrics, where closed geodesics are replaced by closed embedded minimal hypersurfaces. Then, I will discuss some recent results about the construction and geometric understanding of these new geometries. This is a joint project with F. Marques (Princeton) and A. Neves (UChicago).
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This is joint work with Eyal Neuman.
Polymer models give rise to some of the most challenging problems in
probability and statistical physics. We typically model a polymer using a
random walk, where the time parameter n of the walk represents distance
along the polymer starting from one end. That is, we imagine that the
polymer is built up by adding molecules one by one at random angles. We
usually include a self-avoidance term, reflecting the idea that different parts
of the polymer cannot be in the same place at the same time. A difficult
problem, unsolved in the most important physical cases, is to predict the
end-to-end distance or radius of the polymer.
In this talk, I will discuss two extensions of the random polymer model.
1) Moving polymers can be modeled by stochastic partial differential equations. If the polymer takes values in one-dimensional Euclidean space,
we give fairly sharp upper and lower bounds for its radius. We find
that there is more stretching than in typical one-dimensional polymer
models that do not have time dependence.
2) Random surfaces can be modeled by elastic manifolds, also called discrete Gaussian free fields. These models originate in quantum field
theory. If the dimensions of the parameter space and the range are
the same, we can derive bounds on the radius of the polymer. These
bounds are fairly sharp in two dimensions.
We will explain the models mentioned above and give an outline of our
proof techniques.
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Fifty years ago Blaine Lawson constructed the first examples, beyond geodesic spheres and Clifford tori, of closed minimal surfaces embedded in the round 3-sphere. I will review some natural questions (and a few answers) concerning the space of such surfaces, with a focus on results recently obtained for the Lawsons in collaboration with Nicos Kapouleas: namely a characterization by their topology and symmetries and, for an infinite subfamily, the computation of their Morse index and nullity.
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Nematic liquid crystals consist of aggregates of rodlike molecules with orientational ordering, that is described through a unit vector n, called the director. In statics, the typical problem for nematics is to determine the vector field n in a spatial domain, which can be affected by either external actions or boundary conditions. Within the variational approach an energy functional, dependent on n and on its gradient, is minimized according to the boundary conditions. When nematics are confined on curved surfaces, the director field is influenced by both geometrical and topological constraints. To determine the nematic alignment on a surface it is more convenient to deal with an effective functional defined on the surface itself rather than in a spatial domain.
In this talk we will discuss how, naively (but not too naively), this effective functional can be obtained. We will compare the classical two-dimensional model with our revised one and show how the nematic texture is affected by both intrinsic and extrinsic surface curvature. We will illustrate both examples where the surface is fixed and others where the surface is deformable.
With reference to deformable surfaces, we examine a generalization of the classical Plateau problem to an axisymmetric nematic film bounded by two coaxial parallel rings. At equilibrium, the shape of the nematic film results from the competition between surface tension, which favors the minimization of the area, and the nematic elasticity, which instead promotes the alignment of the molecules along a common direction. Depending on two dimensionless parameters, one related to the geometry of the film and the other to the constitutive moduli, the Gaussian curvature of the equilibrium shape may be everywhere negative, vanishing, or positive.
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For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^n, n = 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter. This property distinguishes hyperbolic spaces from Euclidean and spherical ones, where the quantity is bounded below by $3 \pi^2$. We finish by talking about horoconvex domains.
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