Direct Sampling of Confined Polygons in Linear Time
Jan 03, 20251490
Balancing Graphs Using Geometric Invariant Theory
Oct 04, 20243180
Finding Good Coordinates for Sampling: The Importance of Geometry
Sep 06, 20242970
Matrix Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
Jul 15, 20243210
Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 20244280
Math 105
Apr 12, 20243150
Complex Analysis
Mar 27, 20243360
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 20243940
Optimization and Normal Matrices
Mar 18, 20244010
RMHS
Mar 06, 20243150
Geometric Approaches to Frame Theory
Nov 10, 20234440
Introduction to Knot Theory
Oct 02, 20234000
Frames as Loops
Sep 27, 20234630
Applications of Grassmannians and Flag Manifolds
Jul 06, 20235550
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20236351
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20227700
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20227730
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20221,0060
Geometric Approaches to Frame Theory
Mar 16, 20221,0090
Geometric Approaches to Frame Theory
Feb 24, 20221,0120
Animations
Dec 08, 20218610
Some Applications of Symplectic Geometry
Nov 16, 20211,1730
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20211,1170
Random Graph Embeddings with General Edge Potentials
Sep 28, 20211,2020
What is a Random Knot? And Why Do We Care?
Sep 23, 20211,1290
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,4740
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,3050
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,7240
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,6930
What is (Applied) Symplectic Geometry?
Sep 27, 20201,6000
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20202,2750
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,6480
Modeling Topological Polymers
Sep 26, 20191,7790
Symplectic Geometry and Frame Theory
Jul 03, 20191,9320
Stiefel Manifolds and Polygons
Jul 01, 20192,3750
Visualizing Higher Dimensions
Jun 14, 20192,6590
Tensors in Differential Geometry
Jun 02, 20192,6912
Symplectic Geometry and Frame Theory
Jan 21, 20191,8390
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20192,2670
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,5110
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,5180
Modeling Topological Polymers
Nov 04, 20181,8390
Symplectic Geometry and Frame Theory
Oct 31, 20181,6550
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,4250
Symplectic Geometry and Frame Theory
Sep 24, 20181,7250
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,8850
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20182,0020
Equilateral Polygons, Shapes of Ring Polymers, and an Application to Frame Theory
Random flights in 3-space forming a closed loop, or random polygons, are a standard simplified model of so-called ring polymers like bacterial DNA. Equilateral random polygons, where all steps are the same length, are particularly interesting (and challenging) in this context. In this talk I will describe an (almost) toric Kähler structure on the moduli space of equilateral polygons and show how to exploit this structure to get a fast algorithm to directly sample the space. Using work of Hausmann–Knutson, the Kähler structure on the space of equilateral polygons can be realized as the Kähler reduction of the standard Kähler structure on the Grassmannian of 2-planes in complex n-space. This means that equilateral polygons in 3-space can be lifted to Finite Unit-Norm Tight Frames (FUNTFs) in complex 2-space. I will describe how to modify the polygon sampler to produce a FUNTF sampler and show that optimal packings in the 2-sphere lift to FUNTFs with low coherence.
Mar 12, 20182,0800
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension