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Tue Oct 09

Math Physics Seminar

12:20pm - Vincent Hall 313
Signal Processing and Classification
Jimmy Broomfield

This talk will center around signal classification and it will include a demonstration in python. We will focus on how exploring three major
feature domains: The original time sampled data, the frequency domain, and the use of wavelet decompositions. A range of real world applications
will be discussed

Tue Nov 06

Math Physics Seminar

12:20pm - Vincent Hall 313
Contact Point Forces Acting on a Ball, Rolling on a Horizontal Plane and Actuated by Internal Point Masses
Stuart Rogers

This talk derives the contact point forces, namely the normal force and static friction, that act on a ball rolling without slipping on a horizontal plane. It is assumed that the ball is actuated by n point masses, which are free to move inside the ball. The dynamics of a disk and ball actuated by internal point masses are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed

Tue Nov 20

Math Physics Seminar

12:20pm - Vincent Hall 313
Application of Invariant Signatures: Solving Jigsaw Puzzles
Rob Thompson


Tue Dec 04

Math Physics Seminar

12:20pm - Vincent Hall 313
Revivals and Fractalization in the Linear Free Space Schrödinger Equation with Pseudoperiodic Boundary Conditions”
Natalie Sheils

We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific ("rational'') times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other ("irrational'') times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogeneous linear boundary conditions in terms of numerically computable eigenfunctions. (Joint work with Peter Olver and David Smith)

Tue Dec 11

Math Physics Seminar

12:20pm - Vincent Hall 313
Application of Invariant Signatures: Solving Jigsaw Puzzles
Rob Thompson
Tue Jan 29

Math Physics Seminar

12:20pm - Vincent Hall 209
Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
Ian Marqutte, University of Queensland

I will review results on classification of quantum superintegrable systems on two-dimensional Euclidean space allowing separation of variables in Cartesian coordinates and possessing an extra integral of third or fourth order. The exotic quantum potential satisfy a nonlinear ODE and have been shown to exhibit the Painleve property. I will also present different constructions of higher order superintegrable Hamiltonians involving Painleve transcendents using four types of building blocks which consist of 1D Hamiltonians allowing operators of the type Abelian, Heisenberg, Conformal or Ladder. Their integrals generate finitely generated polynomial algebras and representations can be exploited to calculate the energy spectrum. I will point out that for certain cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. I will describe how other sets of integrals can be build and used to provide a complete solution.

Tue Feb 05

Math Physics Seminar

12:20pm - Vincent Hall 209
Choreography in 3-body Classical Mechanics
Sasha Turbiner, ICN-UNAM, Mexico

By definition the choreography (dancing curve) is the trajectory
on which n classical bodies move chasing each other without collisions
with equal time delay. The first choreography (the remarkable Figure Eight)
at zero angular momentum was discovered unexpectedly by C Moore
(Santa Fe Institute) in 1993 for 3 equal masses in R^3 Newtonian gravity, numerically
and rigorously confirmed by Chenciner-Montgomery (2000). At the moment
about 6,000 choreographies are known, all numerically, in Newtonian gravity.
Possibly all known dancing curves are non-algebraic.

Does there exist (non)-Newtonian gravity for which the dancing curve is known
analytically? - Yes, a single example is known - it is the algebraic
lemniscate by Jacob Bernoulli (1694) - and it will be a concrete subject of the talk. Astonishingly, Newtonian Figure Eight coincides with algebraic lemniscate with
\chi^2 deviation 10^{-7}. Both choreographies admit any odd numbers of bodies
on them. 3-body choreography on algebraic lemniscate defines maximally
superintegrable trajectory with seven constants of motion and, possible,
polynomial algebra of integrals.

Tue Feb 19

Math Physics Seminar

12:20pm - Vincent Hall 209
Invariants of Finite and Discrete Group Actions Via Moving Frames
Peter Olver

The equivariant moving frame method is adapted to algorithmically construct complete, minimal sets of generating invariants for finite or, more generally, discrete group actions, both linear and nonlinear. The resulting invariants are piecewise analytic and admit a rewrite rule that allows one to immediately express any other invariant (polynomial, rational, smooth, analytic, etc.) as a function of the generating invariants. The talk will be elementary and no a priori knowledge of invariant theory or moving frames will be assumed.

Tue May 07

Math Physics Seminar

12:20pm - Vincent Hall 209
Polytopes and elliptopes in the foundations of quantum mechanics
Michael Janas and Michel Janssen, School of Physics and Astronomy, University of Minnesota

Bell's inequality for classical correlations is perhaps the most famous result in quantum foundations. Less familiar is the so-called Tsirelson bound, for the corresponding quantum correlations. We first review Bell's inequality in its original form, where one attempts to classically simulate the following quantum experiment: Alice and Bob independently measure a spin component of one of a pair of spin-1/2 particles entangled in the singlet state. We show that the resulting set of classical and quantum correlations allowed in this setup may be visualized as a tetrahedron and an elliptope, respectively, in three-dimensional space. We then consider how to generalize these results to the case of either more settings or more outcomes (i.e., higher spin). In the case of more than two outcomes, the restrictions on the classical simulations lead to high-dimensional polytopes and highly-faceted polyhedra. In the case of four settings we arrive at the Clauser-Horne-Shimony-Holt (CHSH) inequality, a form of the Bell inequality commonly used in experimental tests. In this context, we briefly discuss connections to semi-definite programming and the topic of spectrahedral shadows.

Tue May 14

Math Physics Seminar

12:30pm - Vincent Hall 209
3rd order symmetries for systems on the 2-sphere and the 2 two-sheet 2-hyperboloid -- Superintegrability and cubic algebras
Willard Miller, University of Minnesota (Joint with Ian Marquette, University of Queensland, and Bjorn Berntson, KTH Royal Institute of Techn

There are many papers concerning 3rd and higher order superintegrable systems on 2D Euclidean space, particularly by Pavel Winternitz, his students and collaborators. However, we are not aware of similar studies on nonflat spaces. We decided to study the 2-sphere (with separation in polar coordinates) and the 2-hyperboloid (with separation in horospherical coordinates). The computations were very complex for these nonflat systems and the results less rich than for flat space, which is to be expected. In each case we have a Hamiltonian operator H a 2nd order symmetry operator A and compute a 3rd order symmetry operator B.
We classified the potentials which yielded

a nonzero B. They fall into 4 classes: 1) Systems that are 2nd order superintegrable, 2) Systems that are 3rd order superintegrable but do not generate a cubic algebra, 3) Systems with algebraically dependent generators A and B, 4) Systems that are 3rd order superintegrable with a cubic algebra.

We pay special attention to the class 3 cases which also occur for Euclidian space with little change. We call these functionally dependent superintegrable systems, and show that they always permit a computation of the A eigenfunctions by simple quadrature. For the hyperboloid we show that the TTW method applied to a 2nd order
superintegrable system in horospherical coordinates yields a 3rd order superintegrable system.

Tue May 21

Math Physics Seminar

12:20pm - Vincent Hall 209
Second quantization of elliptic Calogero-Sutherland models
Bjorn Berntson

Calogero-Sutherland models are a class of completely integrable (quantum) many-body systems. In the most general case, particles interact pairwise through an elliptic potential. We discuss the second quantization of several elliptic Calogero-Sutherland models. In degenerate (trigonometric, rational) cases, there is a well-studied correspondence between Calogero-Sutherland systems and Benjamin-Ono equations, a nonlinear, integrable integro-differential equation. We discuss the correspondence in the elliptic regime, where second quantization of a particular system leads to a new two-component elliptic Benjamin-Ono equation. We construct a Lax pair for this equation via a Riemann-Hilbert problem on the torus. This is joint work with Edwin Langmann and Jonatan Lenells.