CIMS Weekly Bulletin | About this seminar | Contact Info |

Speaker: | Ben Blum-Smith | |
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Title: | Invariant polynomials and quotients of spheres | |

Abstract
Classical invariant theory is about trying to describe the collection
("ring") of multivariate polynomials over Q or C that are unaffected
by ("invariant under") some specific group of coordinate
transformations. The theory is mature, and there are many powerful
general theorems that describe this ring's good properties.
Surprisingly, a number of these theorems fail if the polynomials are
restricted to integer coefficients.
In this talk, we investigate one such failure. In the classical case,
all the invariant polynomials can be written uniquely in terms of a
preselected few. Over the integers, this works for some transformation
groups but not others. Which groups?
In another twist, this question turns out to be closely related to a
question in pure topology: if a group of transformations acts on a
sphere, is the quotient a nice topological space like a sphere or a
ball? Or is it something more exotic? |

Speaker: | Eric Vanden-Eijnden | |
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Title: | Small Noise, Big Impact: The Inexorable Effect of Random Perturbations on Dynamical Systems | |

Abstract
Small random perturbations may have a dramatic impact on the evolution of dynamical systems, and large deviation theory (LDT) is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. In this talk I will review some of the theoretical and computational aspects behind these calculations, with illustrations from applications in material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative or degenerate noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable. |

Speaker: | Afonso Bandeira | |
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Title: | On Phase Transitions for Spiked Random Matrix and Tensor Models | |

Abstract
A central problem of random matrix theory is to understand
the eigenvalues of spiked random matrix models, in which a prominent
eigenvector (or low rank structure) is planted into a random matrix.
These distributions form natural statistical models for principal
component analysis (PCA) problems throughout the sciences, where the
goal is often to recover or detect the planted low rank structured. In
this talk we discuss fundamental limitations of statistical methods to
perform these tasks and methods that outperform PCA at it. Emphasis
will be given to low rank structures arising in Synchronization
problems.
Time permitting, analogous results for spiked tensor models will also be discussed.
Joint work with: Amelia Perry, Alex Wein, and Ankur Moitra. |

Speaker: | Thomas Leblé | |
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Title: | 1d/2d log-gases: searching for transitions | |

Abstract
Large systems of particles with logarithmic interactions are interesting models of statistical mechanics in dimension 1 or 2, with connections to random matrix theory or the quantum Hall effect. As the temperature T varies, so does their microscopic behaviour. Is anything remarkable occurring for some particular value of T? What about the low/high-temperature limits? I will present a few things that we know and describe some open questions. |

Speaker: | Shafer Smith | |
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Title: | Baroclinic turbulence in Earth's oceans | |

Abstract
I'll discuss the nonlinear equilibrated states of a two-parameter family of idealized local mean flows that capture the primary mesoscale baroclinic instability types found in the world's oceans. The model flow yields a mean potential vorticity (PV) gradient that consists of baroclinic part, equal to the stretching term from the baroclinic shear, and a depth-averaged part, equal to the sum of beta and the stretching associated with lateral buoyancy gradients at the ocean's surface. The model may be susceptible to Phillips-type baroclinic instability, arising from sign changes in the baroclinic PV gradient, Charney-type instability, which occurs when the surface buoyancy gradient has the opposite sign of the barotropic PV gradient, or both (and may be stable for a narrow slice of parameter space). The two instability types (which may occur alone or in the same flow) lead to distinct turbulent states and eddy fluxes. In some instances, the equilibrated flow exhibits a transition scale below which energetic submesoscale flows develop, displaying features that bear resemblance to Surface Quasigeostrohphic (SQG) turbulence. When such flows result from a Charney-type instability, however, the submesoscale range is non-inertial, being forced at all scales by submesoscale instabilities (akin to, but weaker than Mixed Layer Instabilities). These flows also exhibit a turbulent 'boundary layer,' with a thickness given roughly by the Charney depth scale from the associated linear instability problem. The detailed structure of the turbulence resulting from this family of mean flows provides a useful simplified model for the regional variation of eddy structure in the global ocean. |