Eigenmaps and Manifold Learning

Research in Geometric Analysis:

at Lehman College of the City University of New York

with Prof Chen-Yun Lin and Prof Christina Sormani

funded by various sources including Prof Lin's PSC-CUNY grant

Fall 2022 Meetings:

Fall 2022 Meetings with the professors are online 7:45-9:45pm on Tuesdays.

Students who need to leave early can present first and watch the video later.

Team members work alone or together during the week.

Fall 2022 Team:

Karla Hernandez (Lehman Student)

Abdelali Hourmati (Lehman Alum)

Alexandra Kirlan (CCNY Alum)

Tabitha Ramirez (CCNY Student)

Dahkota DeBold (CCNY Alum)

Alex Cernei (CCNY Alum)

Summer 2022:

Summer 2022 Meetings with the professors were 6:30-8:30pm on Tuesdays.

Team members worked alone or together during the week for 10 hours per week.

Summer 2022 Team:

Karla Hernandez (Lehman Student)

Abdelali Hourmati (Lehman Alum)

Tabitha Ramirez (CCNY Student)

Alex Cernei (CCNY Alum)

Alexandra Kirlan (CCNY Alum)

as well as some prior members who join for fun as they wish:

Dahkota DeBold (CCNY Alum)

Julinda Pillati Mujo (Lehman Alum)

Prior Research Team Members: (see website and 2020-21 team photos)

Esteban Alcantara (Lehman Alum, masters student at City College)

Dahkota Debold (CCNY Alum, math doctoral student at U Tennessee)

Maziar Farahzad (SUNYSB Alum, math doctoral student at U Toronto)

Abdelali Hourmati (Lehman Alum, currently employed in NYC)

and Julinda Pillati Mujo (Lehman Alum, physics doctoral student at the CUNYGC)

We are currently recruiting students at all levels:

Masters students with Differential Geometry, Analysis or Physics training

Advanced Math Majors with Differential Geometry, Analysis or Physics training

Undergraduates who have completed Linear Algebra with an A/A-

Our Annual Recruitment Event was held on Tuesday April 5 6-8pm.

Here's a link to the video of the event (in case you missed it) and a followup video. Ten top students recommended by faculty teaching linear algebra and differential geometry from Lehman College and City College attended our event. Prof Lin gave a presentation and five current/past team members showed students their work. Here's a link to photos of the event. Be sure to email Professor Sormani sormanic@gmail.com if you wish to join the team so we can discuss funding.

Preparation to join our team: (must complete before June 19 to join)

Linear Algebra Preparation: (all team members should complete before June 19)

Students are expected to have completed a course in linear algebra. Those who wish to review some topics as needed may consult Prof Sormani's Linear Algebra course with videos particularly Lessons 21-28 which cover eigenvectors, eigenvalues, and eigenfunctions. The matrices we will be using are very large, so we will be using MATLAB to find the eigenvectos and eigenvalues and MATLAB is using the Jacobi Iteration Algorithm. See Lesson 23 and the following lecture:

Finding Eigenvectors and Eigenvalues using the Jacobi Algorithm by Dr. Urschel (before June 20)

It is also useful to know one of the most basic methods of data analysis using Linear Algebra:

Principal Component Analysis of Data Sets using Linear Algebra by Siraj Raval (before June 27)

We will not be using Principal Component Analysis but instead a much more advanced technique, Manifold Learning (Dimension Reduction via Eigenmaps) that captures the geometry of the data set.

MATLAB Preparation: (all team members should complete before June 19)

All CUNY students have access to MATLAB here.

All students should learn the basics of MATLAB

MATLAB onramp course (2 hours) (complete before June 20)

MATLAB Fundamentals course (16.5 hours) (complete before June 27)

MATLAB for Data Processing and Visualization Course (8 hours) (complete in July)

Differential Geometry Preparation: (for advanced team members)

Advanced team members are expected to have completed a course in Differential Geometry similar to this course by Professor Sormani Differential Geometry (including lessons after the final). Direct link to videos about key topics are in these playlists:

Metric Spaces and Open Sets

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)

Gromov-Hausdorff and Intrinsic Flat Convergence (for fun)

Manifold Learning Preparation: (for advanced team members)

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.

More about Manifold Learning.

Schedule:

June 7-June 20 Start Linear Algebra and MATLAB preparation

as described above (see dates in red). Please work 10 hours per week.

Take screenshots of your work and put it in a googledoc entitled:

EMAPS22-prep-lastname-firstname

with your last and first names.

Advanced students (who have completed a course in Differential Geometry) should also start working on Differential Geometry preparation as described above.

Funding:

Students earning a salary for summer research must show up 6:30-8:30 pm every Tuesday and must upload 8 hours worth of work to their weekly googledoc to be paid for that week. The salary is $25 an hour. The initial hire is for four weeks starting mid July. We will seek additional funding and hope to extend the appointments to the end of August for those students that are actively engaged in the project. Please email Prof Lin and Prof Sormani to apply for this funding including a link to your prep googledoc so we can determine who we should hire. The googledoc should include the hours you plan to work and photos of your MATLAB completion certificates and photos of the lessons you watched or completed.

Further Information about our meetings (including notes):

is here on the team's private page.

Final Work:

Poster by Undergrads

Paper by Lin and Sormani including some images by students.

Page maintained by Prof Sormani sormanic@gmail.com