Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 2024130
Optimization and Normal Matrices
Mar 18, 2024380
RMHS
Mar 06, 2024350
Geometric Approaches to Frame Theory
Nov 10, 20231100
Introduction to Knot Theory
Oct 02, 20231160
Frames as Loops
Sep 27, 20231330
Applications of Grassmannians and Flag Manifolds
Jul 06, 20232020
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20232620
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20224300
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20224310
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20226580
Geometric Approaches to Frame Theory
Mar 16, 20226560
Geometric Approaches to Frame Theory
Feb 24, 20226430
Animations
Dec 08, 20215890
Some Applications of Symplectic Geometry
Nov 16, 20217780
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20217240
Random Graph Embeddings with General Edge Potentials
Sep 28, 20218280
What is a Random Knot? And Why Do We Care?
Sep 23, 20217600
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,0790
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20219500
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,2970
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,2700
What is (Applied) Symplectic Geometry?
Sep 27, 20201,1960
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20201,8240
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,2690
Modeling Topological Polymers
Sep 26, 20191,4010
Symplectic Geometry and Frame Theory
Jul 03, 20191,5370
Stiefel Manifolds and Polygons
Jul 01, 20191,8870
Visualizing Higher Dimensions
Jun 14, 20191,9710
Tensors in Differential Geometry
Jun 02, 20192,2112
Symplectic Geometry and Frame Theory
Jan 21, 20191,4830
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20191,8540
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,2000
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,1170
Modeling Topological Polymers
Nov 04, 20181,4700
Symplectic Geometry and Frame Theory
Oct 31, 20181,3140
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,0510
Symplectic Geometry and Frame Theory
Sep 24, 20181,3540
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,4900
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20181,6050
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, 20181,7120
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension
Nov 04, 20171,9030
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, 20171,8130
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Sep 07, 20171,7190
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Aug 03, 20171,9550
Polytopes and Polygons
Polyhedra, sampling algorithms for random polygons, and applications to ring polymer models
Jul 27, 20171,7630
Random Quadrilaterals
Concavity, a question of Sylvester, and how to generate random quadrilaterals
Jan 13, 20172,0220
The Geometry of Random Polygons
From obtuse triangles to DNA models and synthetic polymers
Nov 21, 20161,4260
Random Triangles
What's the probability that a random triangle is obtuse? Or, what the heck is a random triangle, anyway?