Direct Sampling of Confined Polygons in Linear Time
Jan 03, 20252570
Balancing Graphs Using Geometric Invariant Theory
Oct 04, 20244550
Finding Good Coordinates for Sampling: The Importance of Geometry
Sep 06, 20244200
Matrix Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
Jul 15, 20244220
Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 20245330
Math 105
Apr 12, 20243860
Complex Analysis
Mar 27, 20244030
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 20244960
Optimization and Normal Matrices
Mar 18, 20245070
RMHS
Mar 06, 20243890
Geometric Approaches to Frame Theory
Nov 10, 20235600
Introduction to Knot Theory
Oct 02, 20234920
Frames as Loops
Sep 27, 20235710
Applications of Grassmannians and Flag Manifolds
Jul 06, 20236660
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20237331
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20228930
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20228760
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20221,0950
Geometric Approaches to Frame Theory
Mar 16, 20221,1100
Geometric Approaches to Frame Theory
Feb 24, 20221,1120
Animations
Dec 08, 20219220
Some Applications of Symplectic Geometry
Nov 16, 20211,2730
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20211,2180
Random Graph Embeddings with General Edge Potentials
Sep 28, 20211,2870
What is a Random Knot? And Why Do We Care?
Sep 23, 20211,2150
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,5930
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,4040
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,8840
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,8040
What is (Applied) Symplectic Geometry?
Sep 27, 20201,7090
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20202,3710
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,7400
Modeling Topological Polymers
Sep 26, 20191,8710
Symplectic Geometry and Frame Theory
Jul 03, 20192,0320
Stiefel Manifolds and Polygons
Jul 01, 20192,5040
Visualizing Higher Dimensions
Jun 14, 20193,0020
Tensors in Differential Geometry
Jun 02, 20192,8042
Symplectic Geometry and Frame Theory
Jan 21, 20191,9370
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20192,3660
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,5870
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,6330
Modeling Topological Polymers
Nov 04, 20181,9310
Symplectic Geometry and Frame Theory
Oct 31, 20181,7420
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,5170
Symplectic Geometry and Frame Theory
Sep 24, 20181,8300
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20182,0090
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20182,1270
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.