In this talk, I will give an outline of our recent proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated microlocal analysis used in earlier proofs by Bryant-Griffiths-Yang and Nakamura-Maeda; instead, it is based on a new local existence theorem for a class of nonlinear, first-order PDE systems that we call "strongly symmetric positive." These are a subclass of the symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not fall under one of the standard types (elliptic, hyperbolic, and parabolic).
As in earlier proofs, we construct solutions via the Nash-Moser implicit function theorem, which requires showing that the linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies "smooth tame estimates." We accomplish this in two steps: (1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive after a carefully chosen change of variables. (2) Show that any such system has local solutions that satisfy smooth tame estimates.
The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE systems used in prior proofs, while step (1) requires only linear algebra.
The talk will focus on the main ideas of the proof; technical details will be kept to a minimum.
This is joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang.
Isometric embedding via strongly symmetric positive systems
Feb. 15, 2018 11am (Math 220)
Mike Fried (University of California at Irvine)
X
Reminder: This week we will be meeting in Room 220, instead of Room 350. Also, next week the seminar will meet on Tuesday, in the Number Theory time at 11AM. I will give an exposition on how the construction this week gives tools tying together three sets of famous problems.
Last week we used the phrase "extreme" extensions of a group . This name for Frattini covers, was to contrast them with split extensions of the group. We then produced the universal object which covers any particular case of such covers. It is projective in the category of profinite groups. Also, it decomposes as the fiber product over of covers
, with kernel a pro-free -adic group, one for each prime dividing the order of .
a seminar participant asked how we characterize these extensions cohomologically. That answer appears as the 4th among Propositions labeled "-pieces, Parts 1, 2, 3 and 4" for the properties we need for the groups .
Part 1 is above. We do Part 2 this week: The explicit construction of when the -Sylow in is normal. We will also construct the quotients from these groups that will bind the spaces that turn the Regular Inverse Galois problem into a pure diophantine problem.
The Main Conjecture, when is a dihedral group, is equivalent to an unsolved problem on certain torsion points on hyperelliptic Jacobians of a fixed dimension . This is known only for , where the spaces of hyperelliptic jacobians are modular curves. The Main Conjecture has been shown for the case corresponding to for any group .
Sequences of Frattini covers of any finite group
Feb. 15, 2018 2pm (MATH 350)
Functional Analysis
Sooran Kang (Chung-Ang University, Korea)
X
We first briefly discuss the (quantum) Yang-Mills theory on noncommutative spaces (in particular noncommutative tori) developed by Connes and Rieffel, and then we review the Yang-Mills functional and Laplace’s equation on the quantum Heisenberg manifolds .
In particular, we review the Grassmannian connection and the minimizing connection on the finitely generated projective bimodule over and , where is a generalized fixed-point algebra of a certain crossed product -algebra that also can be realized as a quantum Heisenberg manifold with different parameters.
Then we give various examples of compatible linear connections on , on its submodule associated to the projection whose trace is , and on the tensor product , and we discuss how the Yang-Mills theory of is different from that of the noncommutative tori.
Linear Connections on Quantum Heisenberg Manifolds
Isometric embedding via strongly symmetric positive systems