Joint Princeton/Institute for Advanced Study Geometric Measure Theory Reading Seminar Fall 2018 Organized by Otis Chodosh, Christina Sormani, and Ruobing Zhang We will meet weekly on Wednesdays 1:15-2:50 pm in Fine Hall 1201 at Princeton (this is right before the Princeton Differential Geometry and Geometric Analysis Seminar). The shuttle from IAS leaves from Fuld Hall at 12:40 and arrives at Palmer Square at 12:48 then walk 15 minutes to Fine Hall. We will start when the group from IAS arrives. Or use bikeshare. The first three weeks will be reviews by Prof Sormani preparing everyone for the IAS Emerging Topics Workshop and then after that different participants will present different papers in detail choosing papers most closely related to their research interests. Postdocs and grad students from all nearby institutions are invited to participate. Schedule: Sept 26: Federer-Flemming's Geometric Measure Theory Topics: integer rectifiable and Integral currents on Euclidean space, weak and flat convergence Source: Frank Morgan's Geometric Measure Theory Presentation by Christina Sormani Key ideas: By generalizing the notion of an oriented submanifold of Euclidean Space, considering it as a current, T, acting on differential forms, Federer-Fleming were able to produce a solution to the Plateau problem in this generalized class. They proved a compactness theorem for integral currents: if the T have support in a compact region of Euclidean space and have a uniform upper bound on their mass and the mass of their boundaries, then a subsequence converges in the flat and weak sense to an integral current. The limit of the boundaries of the currents is the boundary of the limit. The mass (which is a weighted volume) is lower semicontinuous under flat and weak convergence. An integral current has a set of positive density which is rectifiable (covered almost everywhere by biLipschitz charts) although the support may have higher dimension. We did not have time to go over the coarea formula nor the slicing theorem, which are also beautiful parts of Federer-Fleming's work. Next week we turn to Ambrosio-Kirchheim's Currents on Metric spaces, where all of this is successfully generalized with new proofs. If anyone would like to present the original Federer-Flemming proof of their compactness theorem and present it, there is a beautiful exposition in Frank Morgan's book. If anyone would like to present the more modern theory of flat chains mentioned by Otis Chodosh during the talk, he can provide you with notes from a course he took at Stanford with White. Oct 3: Ambrosio-Kirchheim's Currents on Metric Spaces Topics: integer rectifiable and integral currents on complete metric spaces, weak convergence Source: Ambrosio-Kirchheim's Acta paper Presentation by Christina Sormani Key ideas: Using DeGiogi’s tuples of Lipschitz functions to replace differential forms, Ambrosio-Kirchheim extend all the work of Federer-Fleming to the setting of complete metric spaces including slicing and boundary rectifiability and the compactness theorem. They do not prove their compactness theorem using a deformation theorem but instead prove it by induction using the slicing theorem. Their notion of mass is defined using a mass measure which is not just an integer weighted multiple of a Hausdorff measure but also involves an area factor. It is lower semicontinuous with respect to weak convergence. It is defined using a mass measure which is defined as the smallest measure satisfying a natural integral inequality. If anyone would like to present the proof of their compactness theorem, you may find it in their paper. Oct 10: Intrinsic Flat Convergence and Integral Current Spaces Topics: intrinsic flat convergence of Riemannian manifolds and integral current spaces Necessary Background for Emerging Topics Workshop next week Source: Sormani-Wenger's JDG paper Presentation by Christina Sormani Defn of Integral Current spaces and intrinsic flat convergence as is Sormani-Wenger JDG, and some basic properties like lower semicontinuity of mass and convergence of boundaries, and examples, plus the statement of Wenger’s Compactness Theorem (from Wenger CVPDE). Week of October 15-19 IAS Emerging Topics Workshop organized by Gromov and Sormani will have many talks on intrinsic flat convergence and scalar curvature Oct 24: Antoine Song: Sweepouts and Widths Topics: Defn of continuous and discrete sweepouts, width and Gromov-Guth Continuous 1 sweepouts, Almgren map, discrete sweepoutsn fineness, width, Almgren Pitts Theoren, Multiparameter sweepouts, Almgren, Guth, Marques-Neves, including weak homotopy eqiv of infinite real projective space, p width and Gromov-Guth estimate with rough idea of proof of the upper bound using skeletons of cubes Oct 31: Ben Lowe: Gromov’s Filling Inequality and Systolic Estimate Sources: Notes on Gromov's Systolic Estimate by Larry Guth and A Short Proof of Gromov's Filling Inequality by Stefan Wenger. Nov 7: no meeting Workshop on Mean Curvature and Regularity at IAS Nov 14: no meeting Geometric Analysis Workshop at Rutgers (must register) Nov 21: no meeting Thanksgiving Nov 27: Aaron Naber: Some new proofs of old results about Ricci curvature 1:15-2:50 pm in Fine Hall 1201 Abstract: Consider a noncollapsed limit M_i->X of manifolds with lower Ricci curvature bounds. In a couple weeks I will discuss how the singular set S^k(X) of X is k-rectifiable. In an effort to give a (vastly) simplified introduction to a couple of the ideas needed in the proof I will discuss some new approaches to some classical results in Ricci curvature in this informal lecture. Specifically I will show that singular set of X is codimension 2 and that X is homeomorphic to a manifold away from a codimension 2 subset. Among other things, we will replace the Reifenberg construction of Cheeger-Colding with a purely analysis proof based on the transformation lemma. See Naber’s notes on Reifenberg. Nov 27 at 6pm at IAS Neumeyer speaks about isoperimetric inequalities) Dec 5: Zihui Zhao: Boundaries of domains in Euclidean space of low regularity 1:15-2:50 pm in Fine Hall 1201 See her homepage for links to related articles. Participants: Shaoyun Bai (Princeton) Alice Chang (Princeton) Eric Chen (Princeton) Gao Chen (IAS) Otis Chodosh (IAS and Princeton) Sanghoon Lee (Princeton) Wenbo Li (CUNY) Yangyang Li (Princeton) Ben Lowe (Princeton) Elena Maeder-Baumdicker (Princeton) Antoine Song (Princeton) Christina Sormani (IAS, SCGP, and CUNY) Daniel Stern (Princeton) Ruobing Zhang (Princeton and Stony Brook) Siyi Zhang (Princeton) Zihui Zhao (IAS) If you would like to join, contact Prof Sormani at sormanic@gmail.com |
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