Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 2024530
Optimization and Normal Matrices
Mar 18, 2024690
RMHS
Mar 06, 2024550
Geometric Approaches to Frame Theory
Nov 10, 20231370
Introduction to Knot Theory
Oct 02, 20231380
Frames as Loops
Sep 27, 20231510
Applications of Grassmannians and Flag Manifolds
Jul 06, 20232260
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20232870
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20224530
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20224540
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20226850
Geometric Approaches to Frame Theory
Mar 16, 20226760
Geometric Approaches to Frame Theory
Feb 24, 20226690
Animations
Dec 08, 20216070
Some Applications of Symplectic Geometry
Nov 16, 20218060
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20217490
Random Graph Embeddings with General Edge Potentials
Sep 28, 20218560
What is a Random Knot? And Why Do We Care?
Sep 23, 20217900
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,1050
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20219730
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,3200
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,2990
What is (Applied) Symplectic Geometry?
Sep 27, 20201,2280
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20201,8500
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,2990
Modeling Topological Polymers
Sep 26, 20191,4240
Symplectic Geometry and Frame Theory
Jul 03, 20191,5660
Stiefel Manifolds and Polygons
Jul 01, 20191,9250
Visualizing Higher Dimensions
Jun 14, 20192,0110
Tensors in Differential Geometry
Jun 02, 20192,2422
Symplectic Geometry and Frame Theory
Jan 21, 20191,5060
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20191,8910
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,2200
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,1380
Modeling Topological Polymers
Nov 04, 20181,5000
Symplectic Geometry and Frame Theory
Oct 31, 20181,3400
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,0780
Symplectic Geometry and Frame Theory
Sep 24, 20181,3870
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,5190
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20181,6350
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, 20181,7400
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension
Nov 04, 20171,9290
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.
Oct 18, 20171,8480
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Sep 07, 20171,7440
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Aug 03, 20171,9900
Polytopes and Polygons
Polyhedra, sampling algorithms for random polygons, and applications to ring polymer models
Jul 27, 20171,7960
Random Quadrilaterals
Concavity, a question of Sylvester, and how to generate random quadrilaterals
Jan 13, 20172,0510
The Geometry of Random Polygons
From obtuse triangles to DNA models and synthetic polymers