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Mathematics Department

Colloquium Series

Spring 2021

All talks are 3:40-4:40 p.m. online at this Google Meet link

  • Apr
    28
  • Output-Weighted Active Sampling for Bayesian Uncertainty Quantification and Prediction of Rare Events
    Location: Virtual Talk
    Time: 03:40 PM

    View Abstract

    We introduce a class of acquisition functions for sample selection that leads to faster convergence in applications related to Bayesian uncertainty quantification and Bayesian optimization of/about rare extreme events. The approach follows the paradigm of active learning, whereby existing samples of a black-box function are utilized to optimize the next most informative sample. The proposed method aims to take advantage of the fact that some input directions of the black-box function have a larger impact on the output than others, which is important especially for systems exhibiting rare and extreme events. The acquisition functions introduced in this work leverage the properties of the likelihood ratio, a quantity that acts as a probabilistic sampling weight and guides the active-learning algorithm towards regions of the input space that are deemed most relevant. We demonstrate the advantages of the proposed approach in the probabilistic quantification of rare events in dynamical systems and the identification of their precursors, as well as the quantification of extreme event statistics for pitch motions and associated vertical bending moments in ship dynamics. We also discuss connections and implications for Bayesian optimization and present applications related to path planning for anomaly (rare event) detection in environment exploration, as well as optimization of an experimental turbulent jet for maximum mixing.
  • Apr
    14
  • The mathematics of taffy pulling
    Jean-Luc Thiffeault
    University of Wisconsin
    Location: Virtual Talk
    Time: 03:40 PM

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    Taffy is a type of candy made by repeated 'pulling' (stretching and folding) a mass of heated sugar. The purpose of pulling is to get air bubbles into the taffy, which gives it a nicer texture. Until the late 19th century, taffy was pulled by hand -- an arduous task. The early 20th century saw an avalanche of new devices to mechanize the process. These devices have fascinating connections to the topological dynamics of surfaces, in particular with pseudo-Anosov maps. The motion of the pins of the taffy puller cab be related to orbits of singularities on closed surfaces of genus one and higher. Special algebraic integers such as the Golden ratio and the lesser-known Silver ratio make an appearance, as well as more exotic numbers. We examine different designs from a mathematical perspective, and discuss their efficiency.
  • Mar
    31
  • The sedimentation of flexible filaments
    Saverio Spagnolie
    University of Wisconsin
    Location: Virtual Talk

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    The deformation and transport of elastic filaments in viscous fluids play central roles in many biological and technological processes. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance may cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. Preliminary results for suspensions of flexible filaments will also be discussed.
  • Mar
    10
  • Numerical Analysis meets Algebraic Topology
    Hal Schenck
    Auburn University
    Location: Virtual Talk
    Time: 03:40 PM

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    One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM). A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, there are many open questions about splines: for a triangular mesh T and smoothness order one, the dimension of the vector space C^1_3(T) of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a formula for the dimension of the space C^1_2(T) in terms of the combinatorics and geometry of the mesh T, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology.
  • Mar
    02
  • Informing Anti-Human Trafficking Efforts with Operations Research Models
    Kayse Lee Maass
    Northeastern University
    Location: Virtual Talk
    Time: 03:40 PM

    View Abstract

    Human trafficking is a prevalent and malicious global human rights issue, with an estimated 24 million victims currently being exploited worldwide. A major challenge to its disruption is the fact that human trafficking is a complex system interwoven with other illegal and legal networks, both cyber and physical. Efforts to disrupt human trafficking must understand these complexities and the ways in which a disruption to one portion of the network affects other network components. As such, operations research models are uniquely positioned to address the challenges facing anti-human trafficking efforts. This presentation will discuss ongoing interdisciplinary anti-human trafficking efforts focusing on prevention, network disruption, and survivor empowerment. Specifically, we will discuss 1) the adaptions to current network interdiction models that are necessary for adequately representing human trafficking contexts and 2) a budget-constrained optimization model that maximizes the societal value of locating additional shelters for human trafficking survivors.
  • Feb
    24
  • Singularities, Hyperplane Arrangements, and Bernstein--Sato Polynomials
    Daniel Bath
    Purdue University
    Location: Virtual Talk

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    We consider the singular locus of a complex valued function, that is, the points where all the partial derivatives vanish. There are several invariants attached to the singular structure that attempt to measure how bad (or good) the singularities are. In particular, the roots of the classical Bernstein--Sato polynomial--a polynomial coming from a very general differential equation--all encode subtle information about the singularities, but the importance of some roots is better understood than others. For hyperplane arrangements (products of linear polynomials) we identify a novel measurement for how bad the singularities are. Namely, very small roots of the Bernstein--Sato polynomial force the vector fields tangent to our arrangement to be complicated (specifically not free).
  • Feb
    10
  • Speeding Up Real Algebraic Geometry via Number Theory
    Maurice Rojas
    Texas A&M
    Location: Virtual Talk
    Time: 03:40 PM

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    The common problem underpinning a myriad of applications, ranging from complexity theory to optimization and engineering, is approximating the real solutions of systems of polynomial equations. Speed is especially important because real-world applications yield systems of equations taking many hours of distributed computing time. Apropos of this, we reveal a dramatic new speed-up for counting real roots of certain sparse polynomial systems: The first (deterministic) algorithm with polynomial dependence on the logarithm of the degree. We attain this speed-up by an application of diophantine approximation. We explain how counting real roots is related to number theory, and discuss how these results point to average-case speed-ups for arbitrary sparse polynomial systems.
  • Feb
    03
  • The Wave Equation: from water waves to laser beams
    Reza Malek-Madani
    United States Naval Academy
    Location: Virtual Talk
    Time: 03:40 PM

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    The wave equation, u_tt - u_xx =0, and its many linear and nonlinear variations, has a rich history, going back at least to the 1750s and the remarkable formula of Jean D'Alembert. In this talk I will start with a short review of this equation and then propose how and why to apply a Galerkin-Spectral method to a version of the wave equation that appears in the study of laser beams.
  • Jan
    20
  • Bispectral mode decomposition of nonlinear flows
    Location: Virtual Talk
    Time: 03:40 PM

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    Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. This talk introduces bispectral mode decomposition as a direct means of deducing flow structures that are associated with triadic interactions from experimental or numerical data. Triadic interactions are characterized by quadratic phase coupling which can be detected by the bispectrum. The new method maximizes an integral measure of this third-order statistic to compute modes associated with frequency triads, as well as a mode bispectrum that identifies resonant three-wave interactions. Unlike the classical bispectrum, the decomposition establishes a causal relationship between the three frequency components of a triad. This permits the distinction of sum and difference interactions, and the computation of interaction maps that indicate regions of strong nonlinear coupling. The proposed method is demonstrated on high-fidelity numerical data and experimental data obtained from particle image velocimetry.
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