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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
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