Optimization Informed by Geometric Invariant Theory and Symplectic Geometry
May 22, 2024260
Math 105
Apr 12, 2024670
Complex Analysis
Mar 27, 2024830
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Mar 26, 2024860
Optimization and Normal Matrices
Mar 18, 20241040
RMHS
Mar 06, 2024830
Geometric Approaches to Frame Theory
Nov 10, 20231600
Introduction to Knot Theory
Oct 02, 20231580
Frames as Loops
Sep 27, 20231760
Applications of Grassmannians and Flag Manifolds
Jul 06, 20232520
Frames, Optimization, and Geometric Invariant Theory
Apr 10, 20233261
Hodge and Gelfand Theory in Clifford Analysis and Tomography
Aug 22, 20224860
Topological Polymers and Random Embeddings of Graphs
Aug 16, 20224800
Finding Good Coordinates for Sampling Configuration Spaces
Mar 17, 20227110
Geometric Approaches to Frame Theory
Mar 16, 20227000
Geometric Approaches to Frame Theory
Feb 24, 20227050
Animations
Dec 08, 20216270
Some Applications of Symplectic Geometry
Nov 16, 20218440
A Lie Algebraic Perspective on Frame Theory
Oct 05, 20217890
Random Graph Embeddings with General Edge Potentials
Sep 28, 20218970
What is a Random Knot? And Why Do We Care?
Sep 23, 20218220
The (Symplectic) Geometry of Spaces of Frames
Feb 19, 20211,1370
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
Jan 10, 20211,0030
An Introduction to Symplectic Geometry and Some Applications
Oct 17, 20201,3650
New Stick Number Bounds from Random Sampling of Confined Polygons
Oct 03, 20201,3330
What is (Applied) Symplectic Geometry?
Sep 27, 20201,2590
New Stick Number Bounds from Random Sampling of Confined Polygons
Mar 08, 20201,8960
Hamiltonian Group Actions on Frame Spaces
Jan 09, 20201,3340
Modeling Topological Polymers
Sep 26, 20191,4490
Symplectic Geometry and Frame Theory
Jul 03, 20191,5890
Stiefel Manifolds and Polygons
Jul 01, 20191,9760
Visualizing Higher Dimensions
Jun 14, 20192,0630
Tensors in Differential Geometry
Jun 02, 20192,2862
Symplectic Geometry and Frame Theory
Jan 21, 20191,5310
Using Differential Geometry to Model Complex Biopolymers
Jan 14, 20191,9220
What’s the Probability a Random Triangle is Obtuse?
For MATH 161
Dec 05, 20181,2470
What’s the Probability a Random Triangle is Obtuse?
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
Nov 19, 20182,1650
Modeling Topological Polymers
Nov 04, 20181,5380
Symplectic Geometry and Frame Theory
Oct 31, 20181,3580
The Geometry of Topologically Constrained Random Walks
Oct 14, 20182,1220
Symplectic Geometry and Frame Theory
Sep 24, 20181,4090
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Jul 12, 20181,5390
Random Walks are Almost Closed
Loop closure is surprisingly non-destructive
Apr 19, 20181,6730
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, 20181,7650
From random walks to closed polygons
A natural map from random walks to equilateral polygons in any dimension
Nov 04, 20171,9600
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, 20171,8810
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Sep 07, 20171,7720
The Geometry of Polygon Space
Acute triangles, convex quadrilaterals, flag means, and more
Aug 03, 20172,0310
Polytopes and Polygons
Polyhedra, sampling algorithms for random polygons, and applications to ring polymer models
Jul 27, 20171,8260
Random Quadrilaterals
Concavity, a question of Sylvester, and how to generate random quadrilaterals