Finding Good Coordinates for Sampling: The Importance of Geometry
Sep 06, 2024820
Matrix Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
Jul 15, 20241300
Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 20242300
Math 105
Apr 12, 20241930
Complex Analysis
Mar 27, 20242040
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 20242250
Optimization and Normal Matrices
Mar 18, 20242310
RMHS
Mar 06, 20241780
Geometric Approaches to Frame Theory
Nov 10, 20232820
Introduction to Knot Theory
Oct 02, 20232540
Frames as Loops
Sep 27, 20232910
Applications of Grassmannians and Flag Manifolds
Jul 06, 20233820
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20234531
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20226060
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20225950
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20228420
Geometric Approaches to Frame Theory
Mar 16, 20228240
Geometric Approaches to Frame Theory
Feb 24, 20228200
Animations
Dec 08, 20217180
Some Applications of Symplectic Geometry
Nov 16, 20211,0010
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20219320
Random Graph Embeddings with General Edge Potentials
Sep 28, 20211,0190
What is a Random Knot? And Why Do We Care?
Sep 23, 20219510
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,2830
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,1440
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,5130
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,4870
What is (Applied) Symplectic Geometry?
Sep 27, 20201,4030
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20202,0630
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,4660
Modeling Topological Polymers
Sep 26, 20191,5960
Symplectic Geometry and Frame Theory
Jul 03, 20191,7300
Stiefel Manifolds and Polygons
Jul 01, 20192,1460
Visualizing Higher Dimensions
Jun 14, 20192,2740
Tensors in Differential Geometry
Jun 02, 20192,4882
Symplectic Geometry and Frame Theory
Jan 21, 20191,6500
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20192,0530
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,3700
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,3120
Modeling Topological Polymers
Nov 04, 20181,6660
Symplectic Geometry and Frame Theory
Oct 31, 20181,4770
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,2530
Symplectic Geometry and Frame Theory
Sep 24, 20181,5370
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,6950
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20181,8130
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, 20181,8960
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension
Nov 04, 20172,1360
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, 20172,0370
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more