LSAMP-F16

Metric Geometry Research Group

Fall 2016-Spring 2017 with Professor Sormani

We will explore a variety of metric spaces and measure

the Gromov-Hausdorff distances between these metric

spaces by finding explicit distance preserving maps into

larger metric spaces. Compact, connected and discrete

metric spaces will be studied.

Amanda Rodriguez, Fifonsi Lantonkpode and Benjamin Arthur

with Professor Christina Sormani

Reference:

[BBI] Burago-Burago-Ivanov's A Course in Metric Geometry

[M] Marsden's Elementary Classical Analysis

Fall 2017:

We meet Mondays 8-9 pm Wednesdays 4:15-5:15 pm

Initial Meetings for the above trio of leaders to prepare their initial presentations:

Wed 9/28 4:15-5:30 pm: See 1.1-1.2 of [BBI]

Metric Spaces: Euclidean, Taxi, Spherical and Networks

Distance Preserving Maps: Euclidean Plane into Euclidean Space

but not sphere into Euclidean Plane, HW on these notions

Mon 10/3 9-10 pm: See 7.3 of [BBI] which we will come back to later

Balls and Tubular Neighborhoods,

Hausdorff Distance (only a quick introduction)

Gromov-Hausdorff Distance (only a quick introduction)

Wed 10/5 4:15-5:15 pm: Gromov-Hausdorff Distance using Hausdorff Distances

Th 10/6 8-9 pm: Gromov Hausdorff Convergence

No meetings 10/10 and 10/12 Practice Using Tex to Design a Presentation

Mon 10/17 8-9pm: Proving Lattices GH converge to Taxi Square

Wed 10/19 4:15-5:15pm Complete Metric Spaces

Mon 10/24 8-9pm Compact Metric Spaces (defined using sequences)

Wed 10/26 4:15-5:15pm Compactness

Mon 10/31 no meeting (students who wish to do work together can come)

Meetings with the Full Team starting 11/2:

New Members: Shanell George, Vanessa Ortiz, Ulysses Hernandez

Wed 11/2 4:15-5:15pm Compactness

Mon 11/7 8-9pm Continuous Functions and Compactness

Wed 11/9 4:15-5:15pm discussed students concerns

Mon 11/14 8-9pm GH convergence and Epsilon Correspondences

Wed 11/16 3:30-5pm Pre-Graduate School Workshop Carman Hall B08

Mon 11/21 8-9pm GH Limits of Triangles

Wed 11/23 4:15-5:15pm GH Limits of Triangles

Mon 11/28 8-9pm GH Limits of Tetrahedra

Wed 11/30 4:15-5:15pm GH Limits of Tetrahedra

Mon 12/5 8-9pm GH Limits of Tetrahedra

Wed 12/7 4:15-5:15pm Gromov's Compactness Theorem for Finite Metric Spaces

Mon 12/12 8-9pm Prepare paper to hand in to LSAMP

Spring 2017:

In addition to continuing Research on GH Convergence we will learn more Metric Space Topics: Metric Spaces, Balls, Open Sets, Compact Sets, Continuity Properties, Hausdorff Measure, Lipschitz functions, Differentiation in Euclidean Spaces, Inverse and Implicit Function Theorems, Lebesgue Measure...

Ulysses is not at Lehman this Spring. Fifonsi is now a masters student and will attend when he can.

Mon 1/30 4:15-5:15 Lebesgue Integration of indicator functions and simple functions

Wed 2/1 4:15-5:15 Lebesgue and Riemann Integral agree when Riemann Integrable

Mon 2/6 4:15-5:15 Rectifiable Lengths of Curves in Metric Spaces

Wed 2/8 4:15-5:15 Continuity: Proof that C(t)=(cos(t), sin(t)) is a continuous curve in the plane. So is C(t) = v + tw.

Wed 2/15 4:15-5:15 Rectifiable lengths of differentiable curves

Wed 2/22 4:15-5:15 Lebesgue Measure

Mon 2/27 4:15-5:15 Lebesgue Integration

Wed 3/1 4:15-5:15 Reviewing our Research from Last Fall

Mon 3/6 4:15-5:15 Working on the Write up (final day for most team members)

Wed 3/8 4:15-5:15 Continuous Images of Sequentially Compact Sets

Mon 3/13 4:15-5:15 Pre-Images of Open Sets

Wed 3/15 4:15-5:15 Lipschitz Functions

Wed 3/22 4:15-5:15 Review of Gromov-Hausdorff Convergence

Mon 3/27 4:15-5:15 Review of Hausdorff Distances

Wed 3/29 4:15-5:15 Working Together

Mon 4/3 4:15-5:15 Gromov-Hausdorff Convergence to a Circle

Wed 4/5 4:15-5:15 Gromov-Hausdorff Convergence to a Point (Fifonsi here today)

Wed 4/19 4:15-5:15 Hausdorff Measure

Th 4/20 4:15-5:15 Hausdorff Dimension

Wed 5/1 4:15-5:15 Gromov-Hausdorff Convergence to a Circle (Fifonsi here today)

Mon 5/8 4:15-5:15 Hausdorff Measure in Taxicab space