Geometry-Research-S20
Research in Geometric Analysis: Manifold Learning
at Lehman College, CUNY
with Prof Chen-Yun Lin and Prof Chris Sormani
funded by Prof Chen-Yun Lin’s PSC-CUNY and Startup Grants.
Students interested in joining our 2022 team should see our new webpage.
2020-2021 Teams all completed Differential Geometry before starting:
Summer 2020 Team: Esteban Alcantara, Maziar Farahzad, Julinda Pillati Mujo
Fall 2020 Team: Esteban Alcantara, Maziar Farahzad, Julinda Pillati Mujo
Spring 2021 Team: Esteban Alcantara, Maziar Farahzad, Julinda Pillati Mujo
Summer 2021 Team: Dahkota Debold, Abdelali Hourmati,
Fall 2021 Team: Dahkota Debold, Abdelali Hourmati,
Spring-Summer 2022 Team has a new webpage.
Funded team members are paid $25 per hour.
Students who wish to work with Prof Chen-Yun Lin must learn MATLAB:
CUNY has MATLAB available here
Sign into CUNY MATLAB and
do the beginning (1-3) of the 2 hour MATLAB onramp course,
and try out Easy Plotting of Functions in 2D and 3D adjusting the examples.
Note that we will be using MATLAB to find the eigenfunctions and eigenvalues of very large matrices. MATLAB is finding these using the Jacobi method. Here's a talk by Dr. Urschel about this algorithm.
All Students Interested in Differential Geometry Research can start by going over Prof Sormani’s course:
Differential Geometry (including the lessons after the final)
and by viewing these playlists:
Continuity and Limits in Metric Spaces
Derivatives in Higher Dimensions (very important)
Diffeomorphisms and Inverse Function Theorem (very important)
Implicit Function Theorem (very important)
Tangent Vectors and Covariant Differentiation (very important)
Areas of Parametrized Surfaces and Wormholes - Dr. Jorge Basilio
Curvature (only if interested)
Hessians (very important)
Laplacians and Eigenfunctions (very important)
Fourier Series: Part 1 and Part 2 and MATLAB and Gibbs Phenomenon - Dr. Steve Brunton
Eigenmaps and Diffusion Distances (very important)
MatLab Creating Data Sets of Points (very important)
Matlab and Diffusion Maps (very important)
Gromov-Hausdorff and Intrinsic Flat Convergence (for fun)
Videos about manifold learning:
A video about the research Prof Chen-Yun Lin has done directly related to this project. All the necesary terminology will be presented by video in June before the project begins.
Selected Papers:
Mikhail Belkin and Partha Niyogi
1. Laplacian eigenmaps and spectral techniques for embedding and clustering (paper)
2. Laplacian eigenmaps for dimensionality reduction and data representation Neural Computation, Vol 15, (2003) 1373-1396. (reprint) (preprint)
3. Towards a theoretical foundation for laplacian-based manifold methods. (paper)
Coifman and Lafon, “Diffusion Maps”, Appl Comput Harmon Analysis, Vol 21 (2006) 5-30. (reprint) (document with excerpts and explanations for students)
See also Varadhan’s Diffusion processes in a small time interval. Comm. Pure Appl. Math., 20(4):659–685, 1967 and others cited in Portegies’ paper.
More about Manifold Learning.
Page maintained by Prof Sormani sormanic@gmail.com