Direct Sampling of Confined Polygons in Linear Time
Jan 03, 2025720
Balancing Graphs Using Geometric Invariant Theory
Oct 04, 20242220
Finding Good Coordinates for Sampling: The Importance of Geometry
Sep 06, 20242130
Matrix Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
Jul 15, 20242480
Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 20243580
Math 105
Apr 12, 20242720
Complex Analysis
Mar 27, 20242840
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 20243330
Optimization and Normal Matrices
Mar 18, 20243340
RMHS
Mar 06, 20242600
Geometric Approaches to Frame Theory
Nov 10, 20233900
Introduction to Knot Theory
Oct 02, 20233350
Frames as Loops
Sep 27, 20233960
Applications of Grassmannians and Flag Manifolds
Jul 06, 20234930
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20235651
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20227140
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20227110
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20229540
Geometric Approaches to Frame Theory
Mar 16, 20229420
Geometric Approaches to Frame Theory
Feb 24, 20229450
Animations
Dec 08, 20218030
Some Applications of Symplectic Geometry
Nov 16, 20211,1040
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20211,0540
Random Graph Embeddings with General Edge Potentials
Sep 28, 20211,1310
What is a Random Knot? And Why Do We Care?
Sep 23, 20211,0670
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,4100
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,2470
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,6420
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,6180
What is (Applied) Symplectic Geometry?
Sep 27, 20201,5100
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20202,2040
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,5780
Modeling Topological Polymers
Sep 26, 20191,7150
Symplectic Geometry and Frame Theory
Jul 03, 20191,8600
Stiefel Manifolds and Polygons
Jul 01, 20192,2980
Visualizing Higher Dimensions
Jun 14, 20192,4980
Tensors in Differential Geometry
Jun 02, 20192,6182
Symplectic Geometry and Frame Theory
Jan 21, 20191,7790
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20192,1900
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,4530
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,4440
Modeling Topological Polymers
Nov 04, 20181,7760
Symplectic Geometry and Frame Theory
Oct 31, 20181,5950
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,3640
Symplectic Geometry and Frame Theory
Sep 24, 20181,6530
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,8130
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20181,9430
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, 20182,0050
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension
Nov 04, 20172,2590
A Geometric Approach to Sampling Loop Random Flights
In statistical physics, the basic (and highly idealized) model of a ring polymer like bacterial DNA is a closed random flight in 3-space with equal-length steps, often called an equilateral random polygon. While random flights without the closure condition are easy to simulate and analyze, the fact that the steps in a random polygon are not independent has made it challenging to develop practical yet provably correct sampling and numerical integration techniques for polygons. In this talk I will describe a geometric approach to the study of random polygons which overcomes these challenges. The symplectic geometry of the space of polygon conformations can be exploited to produce both Markov chain and direct sampling algorithms; in fact, this approach can be generalized to produce a sampling theory for arbitrary toric symplectic manifolds. This is joint work with Jason Cantarella, Bertrand Duplantier, and Erica Uehara.