Geometric Analysis Reading Seminar

Fall 2024:
  • Main Topic: Cheeger-Colding theory
  • Time: 3pm on Thursday
  • Location: 2-255

    • Summer 2024:
  • Topic: Geometry measure theory
  • Time: 1pm on Tuesday
  • Location: zoom
  • Schedule: link.


    • Spring 2024:
  • Time: 11am on Wednesday
  • Location: 2-255
    • Feb 14: Xinrui Monotonicity and Ricci curvature [Col12], [CM14]
    • Feb 21: Yang SLag and the Donaldson-Scaduto conjecture [EL24]
    • Feb 28: Shrey Index bounds for shrinkers [IRS19]
    • Mar 6: Giada Index bounds for minimal hypersurfaces [Son23]
    • Mar 13: Mike Green's Function and Positive Mass Theorem [AMO24]
    • Mar 20: Keaton Multiplicity one for MCF of surfaces I [BK24]
    • Mar 27: Spring break
    • Apr 3: Jingze Multiplicity one for MCF of surfaces II [BK24]
    • Apr 10: Tang-Kai Genus-zero shrinkers [Bre16]
    • Apr 17: Benjy Donaldson-Sun theory [DS14]
    • Apr 24: Max
    • May 1: Dain Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces [EM24]
    • May 8: Yiqi Milnor's conjecture in dimension three [Liu13]
    Fall 2023:
  • Topic: Stable Bernstein problem
  • Time: 4pm on Thursday
  • Location: 2-255

    • Sep 7: Giada Introduction [Cho21]
    • Sep 14: Tang-Kai Green's function on Stable Minimal Hypersurfaces [CL21, Sec. 1-2.0]
    • Sep 21: Dain/Keaton Integration over level sets of the Green's function [CL21, Sec. 2.1-3]
    • Sep 28: Dain Stern's Bochner formula [CL21, Sec. 3]
    • Oct 5: Keaton Munteanu-Wang's Monotonicity Formula [CL21, Sec. 4]
    • Oct 12: Yang Overview of [CMR22]
    • Oct 19: Aaron Completeness of conformal metric [CMR22]
    • Oct 16: Jingze Overview of [CL22]
    • Nov 12: Aaron mu-bubble [CL22]
    Spring 2023:
  • Topic: Generic MCF
  • Time: 4pm on Monday
  • Location: 2-429
    • Feb 20: Jingze Introduction [CCMS20, Sec. 1.1-1.4]
    • Feb 27: Xinrui Preliminaries [CCMS20, Sec. 1.5-2.6]
    • Mar 6: Irvine conference
    • Mar 13: Keaton Uniqueness I (Merle-Zaag) [CCMS20, Sec. 3-5]
    • Mar 20: Keaton Uniqueness II [CCMS20, Sec. 3-5]
    • Mar 27: Spring break
    • Apr 3: Giada Constructions for compact one-sided flows [CCMS20, Sec. 6]
    • Apr 10: Marathon
    • Apr 17: Tang-Kai General constructions I [CCMS20, Sec. 7]
    • Apr 24: Tang-Kai General constructions II [CCMS20, Sec. 7]
    • May 1: Break
    • May 8: Jingze Regularity for negative time [CCMS20, Sec. 8]
    • May 15: Xinrui Low-entropy case [CCMS20, Sec. 10]
    Fall 2022:
  • Topic: Bamler's Ricci flow theory
  • Time: 4:30pm on Monday
  • Location: 2-361

    • Sep 26: Yiqi Brief Introduction to Ricci flow [Sun, Sec. 1-5]
    • Oct 3: Keaton Brief review of Perelman's theory and Nash Entropy [Sun, Sec. 6]
    • Oct 12: Tang-Kai Overview of Bamler's theory [Bam21]
    • Oct 17: Shrey Heat kernels and inequalities [Sun, Sec. 7.1-7.4], [Bam20a]
    • Oct 24: Mike Hn-Center, Wasserstein distance, and a gradient estimate [Sun, Sec. 7.4-7.6], [Bam20a]
    • Oct 31: Xinrui Nash Entropy and Heat kernel upper bound I [Sun, Sec. 7.7-7.8], [Bam20a]
    • Nov 7: Giada Heat kernel upper bound II and other heat kernel bounds [Sun, Sec. 7.8-7.11], [Bam20a]
    • Nov 14: Jingze P*-parabolic neighborhoods, and epsilon-regularity [Sun, Sec. 7.12-7.14], [Bam20a]
    • Nov 21: Tang-Kai Synthetic RF preliminaries and introduction to metric flows [Sun, Sec. 8.1-8.2], [Bam20b]
    • Nov 28: Keaton Metric flows and H-concentration [Sun, Sec. 8.2-8.3], [Bam20b]
    • Dec 5: Feng Distribution bound for metric flows and time continuity [Sun, Sec. 9.1-9.2], [Bam20b]
    • Dec 16: Dain Future completion, space of metric flow pairs and compactness [Sun, Sec. 9.3-9.6], [Bam20b]
    • Dec 19: Julius Limits of Ricci flows [Sun, Sec. 10], [Bam20c]
    Summer 2022:
  • Topic: Strong rigidity of cylinders as Ricci shrinkers
  • Time: 11am on Tuesday
  • Location: zoom

    • Jun 28: Xinrui Introduction and discussion [CM21, Sec. 0], [LW21]
    • Jul 5: Xinrui Some commutator relations and estimates [CM21, Sec. 1]
    • Jul 12: Jingze Properties of \mathcal{P}, I, and D [CM21, Sec. 2-3.2]
    • Jul19: Aaron Growth estimates of eigenfunctions [CM21, Sec. 3.3-4.2]
    • Jul 26: Tang-Kai Jacobi fields and extensions [CM21, Sec. 5-6]
    • Aug 2: Feng Variations of geometric quantities [CM21, Sec. 7]
    • Aug 9: Keaton Local rigidity [CM21, Sec. 8]
    • Aug 16: Tang-Kai Improvement and the proof of the main theorems [CM21, Sec. 9]
    Spring 2022:
  • Topic: Tensor maximum principle and applications
  • Time: 10:30am on Wednesday
  • Location: 2-429
  • Schedule: link.


    • Fall 2021:
  • Topic: Multiplicity one of min-max hypersurfaces
  • Time: 5pm on Thursday
  • Location: 2-361

    • Sep 23: Feng GMT review and Major developments in min-max theory [MN20, Sec. 2.1]
    • Oct 7: Tang-Kai Min-max construction of minimal hypersurfaces [MN20, Sec. 2.2]
    • Oct 14: Xinrui Gromov's volume spectrum, Weyl law [MN20, Sec. 2.3], [LMN18]
    • Oct 21: Feng Min-max theory for PMC hypersurfaces [ZZ19], [ZZ20], [Zho20, Sec. 1]
    • Oct 28: Feng Min-max theory for PMC hypersurfaces II
    • Nov 4: Keaton Morse index upper bound [MN16], [Zho20, Sec. 3]
    • Nov 11: Xinrui Compactness for hypersurfaces with bounded Morse index [Sha17], [Zho20, Sec. 2]
    • Nov 18: Feng Multiplicity one convergence [Zho20, Sec. 4]
    • Dec 2: Tang-Kai Applications to volume spectrum and the proof of multiplicity one conjecture [Zho20, Sec. 5]
    • Dec 9: Keaton Morse index lower bound [MN18]

    References:
    [AMO24] V. Agostiniani, L. Mazzieri, and F. Oronzio, A Green's Function Proof of the Positive Mass Theorem.
    [Bam16] R. Bamler, Cheeger-Colding-Naber theory.
    [Bam20a] R. Bamler. Entropy and heat kernel bounds on a Ricci flow background.
    [Bam20b] R. Bamler. Compactness theory of the space of super Ricci flows.
    [Bam20c] R. Bamler. Structure theory of non-collapsed limits of Ricci flows.
    [Bam21] R. Bamler. Recent Developments in Ricci Flows.
    [BK24] R. Bamler and B. Kleiner, On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces.
    [Bre16] S. Brendle, Embedded self-similar shrinkers of genus 0.
    [CMR22] G. Catino, P. Mastrolia, and A. Roncoroni. Two rigidity results for stable minimal hypersurfaces.
    [Che01] J. Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds.
    [CC96] J. Cheeger and T. H. Colding, Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products.
    [CC97] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I.
    [Cho21] O. Chodosh, Stable minimal surfaces and positive scalar curvature, class notes.
    [CCMS20] O. Chodosh, K. Choi, C. Mantoulidis, and F. Schulze. Mean curvature flow with generic initial data.
    [CL21] O. Chodosh and C. Li. Stable minimal hypersurfaces in R^4.
    [CL22] O. Chodosh and C. Li. Stable anisotropic minimal hypersurfaces in R^4.
    [Col96a] T. H. Colding, Shape of manifolds with positive Ricci curvature.
    [Col96b] T. H. Colding, Large manifolds with positive Ricci curvature.
    [Col12] T. H. Colding, New monotonicity formulas for Ricci curvature and applications. I.
    [CM08] T. H. Colding and W. P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces.
    [CM14] T. H. Colding and W. P. Minicozzi II, Ricci curvature and monotonicity for harmonic functions.
    [CM21] T. H. Colding and W. P. Minicozzi II. Singularities of Ricci flow and diffeomorphisms.
    [DS14] S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry.
    [EM24] N. Edelen and P. Minter, Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces.
    [EL24] S. H. Esfahani and Yang Li, On the Donaldson-Scaduto conjecture.
    [IRS19] D. Impera, M. Rimoldi, and A. Savo, Index and first Betti number of f-minimal hypersurfaces and self-shrinkers.
    [LW21] Y. Li and B. Wang. Rigidity of the round cylinders in Ricci shrinkers.
    [LMN18] Y. Liokumovich, F. C. Marques, and A. Neves. Weyl Law for the Volume Spectrum.
    [Liu13] G. Liu, 3-manifolds with nonnegative Ricci curvature.
    [MN16] F. C. Marques and A. Neves. Morse index and multiplicity of min-max minimal hypersurfaces.
    [MN18] F. C. Marques and A. Neves. Morse index of multiplicity one min-max minimal hypersurfaces.
    [MN20] F. C. Marques and A. Neves. Applications of Min-Max Methods to Geometry.
    [Sha17] B. Sharp. Compactness of minimal hypersurfaces with bounded index.
    [Son23] A. Song, Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces.
    [Sun] A. Sun's notes from Bamler's 2020 Class.
    [Zho20] X. Zhou. On the Multiplicity One Conjecture in Min-Max Theory.
    [ZZ19] X. Zhou and J. Zhu. Min-max theory for constant mean curvature hypersurfaces.
    [ZZ20] X. Zhou and J. Zhu. Existence of hypersurfaces with prescribed mean curvature I - generic min-max.

    These seminars were organized by a few people among the speakers.

    Older seminars: link 1, link 2, link 3.


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