Direct Sampling of Confined Polygons in Linear Time
Jan 03, 20251980
Balancing Graphs Using Geometric Invariant Theory
Oct 04, 20243730
Finding Good Coordinates for Sampling: The Importance of Geometry
Sep 06, 20243470
Matrix Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
Jul 15, 20243540
Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 20244700
Math 105
Apr 12, 20243480
Complex Analysis
Mar 27, 20243700
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 20244350
Optimization and Normal Matrices
Mar 18, 20244430
RMHS
Mar 06, 20243410
Geometric Approaches to Frame Theory
Nov 10, 20234900
Introduction to Knot Theory
Oct 02, 20234410
Frames as Loops
Sep 27, 20235060
Applications of Grassmannians and Flag Manifolds
Jul 06, 20235940
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20236741
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20227980
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20228130
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20221,0340
Geometric Approaches to Frame Theory
Mar 16, 20221,0360
Geometric Approaches to Frame Theory
Feb 24, 20221,0410
Animations
Dec 08, 20218910
Some Applications of Symplectic Geometry
Nov 16, 20211,2030
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20211,1570
Random Graph Embeddings with General Edge Potentials
Sep 28, 20211,2290
What is a Random Knot? And Why Do We Care?
Sep 23, 20211,1520
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,5090
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,3390
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,7680
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,7290
What is (Applied) Symplectic Geometry?
Sep 27, 20201,6330
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20202,3000
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,6810
Modeling Topological Polymers
Sep 26, 20191,8030
Symplectic Geometry and Frame Theory
Jul 03, 20191,9600
Stiefel Manifolds and Polygons
Jul 01, 20192,4090
Visualizing Higher Dimensions
Jun 14, 20192,7530
Tensors in Differential Geometry
Jun 02, 20192,7232
Symplectic Geometry and Frame Theory
Jan 21, 20191,8680
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20192,2940
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,5430
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,5610
Modeling Topological Polymers
Nov 04, 20181,8720
Symplectic Geometry and Frame Theory
Oct 31, 20181,6830
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,4530
Symplectic Geometry and Frame Theory
Sep 24, 20181,7580
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,9220
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20182,0390
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, 20182,1310
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension