Geometry & Analysis Day is a one day event at the Department of Mathematics at CU Boulder. The event consists of talks by invited speakers and informal discussion followed by a dinner.
10:00-11:00 Sema Salur Deformation Theory of Calibrated Submanifolds
Abstract: Let $N^{2n}$ be a smooth symplectic manifold equipped with a closed, nondegenerate differential 2-form $\omega$. An $n$-dimensional submanifold $L^n$ of $N^{2n}$ is called Lagrangian if the restriction of $\omega$ to $L$ is zero.
Lagrangian submanifolds and their deformations have important applications in symplectic geometry and mathematical physics. In partciular, they play a role in establishying the correspondence between “Calabi-Yau mirror pairs” in string theory.
In this talk we first study the deformations of (special) Lagrangian submanifolds and then extend the theory to “Lagrangian Type” submanifolds inside $G_2$ manifolds. If time permits, we will also discuss relations between $G_2$ and contact structures.
11:00-12:00 Coffee & Discussion
12:00-1:00 Catherine Searle Torus actions, maximality, and non-negative curvature
Abstract: The classification of compact Riemannian manifolds with positive or non-negative sectional curvature is a long-standing problem in Riemannian geometry. One successful approach has been the introduction of symmetries, and an important first case to understand is that of continuous abelian symmetries. In recent work with Escher, we obtained an equivariant diffeomorphism classification of closed, simply-connected non-negatively curved Riemannian manifolds admitting an isotropy-maximal torus action, with implications for the Maximal Symmetry Rank Conjecture for non-negatively curved manifolds. I will discuss joint work with Escher and Dong, that builds on this work to extend the classification to those manifolds admitting an almost isotropy-maximal action.
1:00-3:00 Lunch
3:00-4:00 James P. Kelliher Striated regularity for a class of active transport equations
Abstract: We say that the regularity of a function in $\R^d$ is striated if it has higher regularity along certain hypersurfaces than it does in directions normal to those hypersurfaces. In the context of active transport equations, we say that striated regularity is propagated if such regularity is maintained as both the scalar and the hypersurfaces are transported by the flow map for the velocity field. (Being active scalars, the velocity field at time $t$ is a function of the scalar at time $t$, making the underlying PDE nonlinear.)
We first discuss briefly the transport of vorticity by the 2D Euler equations. Two simplifying properties of these equations are that the transport is by a divergence-free vector field, and the transport equations related to the hypersurfaces are pure transport. We outline how the propagation of striated regularity was first established for the 2D Euler equations in a landmark 1991 result of Jean-Yves Chemin. We place Chemin's approach and subsequent approaches by later authors in a larger framework that we then apply to a seemingly very similar active transport equation: the aggregation equation with Newtonian potential (the inviscid form of a limiting case of the Keller-Segel equation). Such equations lack, however, the two simplifying properties of the 2D Euler equations. We explain how we overcome these difficulties to obtain the propagation of the regularity of level sets of the active scalar, a restricted form of propagation of striated regularity. This extends a recent result of Bertozzi, Garnett, Laurent, and Verdera for patch boundary regularity for the aggregation equation.
This is joint work with Hantaek Bae of Ulsan National Institute of Science and Technology (UNIST).
There will be a dinner at the Boulder Dushanbe Teahouse at 6pm. Click here for a map. The address is: 1770 13th St, Boulder.
Geometry & Analysis Day is supported by the Simons Foundation and the Meyer Fund.