Posted July 13, 2024
Last modified September 16, 2024
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3:30 pm Lockett 232
Ian Tobasco, Rutgers University
Homogenization of Kirigami and Origami-Based Mechanical Metamaterials
Mechanical metamaterials are many-body elastic systems that deform in unusual ways, due to the interactions of nearly rigid building blocks. Examples include origami patterns with many folds, or kirigami patterns made by cutting material from an elastic sheet. In either case, the local deformations of the pattern involve internal degrees of freedom which must be matched with the usual global Euclidean invariances --- e.g., groups of origami panels move by rotations and translations while the whole pattern bends into a curved shape. This talk will introduce the homogenization problem for kirigami and origami metamaterials to a broad audience, and describe our recent results. Our goal is to explain the link between the design of the individual cuts/folds and the bulk deformations they produce. This is joint work with Paul Plucinsky (U. Southern California, Aerospace and Mechanical Engineering) and Paolo Celli (Stony Brook U., Civil Engineering). This talk will be mathematically self-contained, not assuming a background in elasticity.
Posted November 11, 2024
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3:30 pm Lockett 232
Benjamin Dodson, Johns Hopkins University
Global well-posedness and scattering for the radial, conformal wave equation
In this talk we prove global well-posedness and scattering for the radially symmetric nonlinear wave equation with conformally invariant nonlinearity. We prove this result for sharp initial data.
Posted January 13, 2025
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3:30 pm – 4:30 pm Lockett 232
Trinh Tien Nguyen, University of Wisconsin Madison
Boundary Layers in Fluid Dynamics and Kinetic Theory
Abstract: In this talk, I will discuss recent results on Prandtl boundary layer theory in fluid dynamics. We demonstrate that the Prandtl expansion holds for initial data that is analytic near the boundary under the no-slip boundary condition. I will then present a recent result on the validity of the Prandtl expansion from Boltzmann theory, marking an important step toward justifying other types of approximate solutions (arising from fluid dynamics) as macroscopic limits of the kinetic Boltzmann equations.
Posted January 10, 2025
Last modified January 17, 2025
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3:30 pm – 4:30 pm Lockett 232
Suhan Zhong, Texas A&M University
Polynomial Optimization in Data Science
Abstract: Optimization plays a pivotal role in data science. Recent advances in polynomial optimization have introduced innovative methods to solve many challenging problems in this field. In this talk, I will showcase the application of polynomial optimization through the lens of two-stage stochastic models. Additionally, I will provide a brief overview of the underlying theory and discuss potential future research directions.
Posted January 15, 2025
Last modified January 21, 2025
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3:30 pm – 4:30 pm Lockett 232
Saber Jafarpour, University of Colorado Boulder
Safety and Resilience of Learning-enabled Autonomous Systems: A Monotone Contracting System Perspective.
Abstract: Learning-enabled autonomous systems are increasingly deployed for decision-making in safety-critical environments. Despite their substantial computational advantages, ensuring the safety and reliability of these systems remains a significant challenge due to their high dimensionality and inherent nonlinearity. In this talk, we leverage tools and techniques from control theory to develop theoretical and algorithmic methods for certifying the safety and robustness of learning-enabled autonomous systems. Our approach investigates safety and resilience from a reachability perspective. We employ contraction and monotone systems theories to develop computationally efficient frameworks for approximating reachable sets of autonomous systems. We demonstrate how these frameworks can be applied to verify and train robust standalone neural networks and to provide run-time safety assurance in systems with learning-based controllers.
Posted December 5, 2024
Last modified January 22, 2025
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3:30 pm Lockett 232
Ken Ono, University of Virginia
Partitions detect primes
This talk presents “partition theoretic” analogs of the classical work of Matiyasevich that resolved Hilbert’s Tenth Problem in the negative. The Diophantine equations we consider involve equations of MacMahon’s partition functions and their natural generalizations. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are precisely the prime numbers. To this end, we produce explicit additive bases of all graded weights of quasimodular forms, which is of independent interest with many further applications. This is joint work with Will Craig and Jan-Willem van Ittersum.
Posted January 28, 2025
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3:30 pm – 4:30 pm Lockett 232
Federico Glaudo, Institute for Advanced Study, Princeton
A Journey through PDEs and Geometry
This talk will explore a range of intriguing questions that lie at the crossroads of partial differential equations and geometry. Topics include the stability of near-solutions to PDEs, the isoperimetric inequalities on curved spaces, as well as the random matching problem. The aim is to make the ideas accessible and engaging for a broad mathematical audience.
Posted February 5, 2025
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2:30 pm – 3:30 pm Zoom
Ajay Chandra, Imperial College London
An Invitation to Singular Stochastic Partial Differential Equations
Abstract: In this talk I will start by motivating the fundamental importance of singular stochastic partial differential equations in (i) our understanding of the large-scale behaviour of dynamic random systems and (ii) developing a rigorous approach to quantum field theory. I will describe the key mathematical difficulties these equations pose, and sketch how combining analytic, probabilistic, and algebraic arguments have allowed mathematicians to overcome these difficulties and develop a powerful new PDE theory. I’ll also discuss some more recent developments in this area, namely applications to gauge theory and non-commutative probability theory.
Posted February 19, 2025
Last modified March 10, 2025
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3:30 pm – 4:30 pm Lockett 232
Justin Holmer, Brown University
Dynamics of Solitary Waves
Solitary waves arise as exact coherent structures in a range of nonlinear wave equations, including the nonlinear Schrödinger, Korteweg–de Vries, and Benjamin–Ono equations. These equations have broad applications in areas such as water wave theory, plasma physics, and condensed matter physics. When certain types of perturbations are introduced, the solitary wave retains its overall form while its shape and position adjust to accommodate the new conditions. In this talk, I will present some theoretical results on the modulation of solitary wave profiles under such perturbations, supported by numerical simulations that illustrate and validate these findings.