Analysis Seminar
Fall 2021 (Thursdays 2 pm)
Time: Thursday September 2, 2021
Place: Manchester 124
Speaker: Dr. Leandro Lichtenfelz
Title: The Euler Equations from a Geometric Perspective: Part 1
Abstract: This series of talks will be an introduction to Arnold’s idea of describing the Euler equations of hydrodynamics, classically a system of PDEs, as an ordinary differential equation (in fact, a geodesic equation) on an infinite-dimensional group. Several other PDEs of interest in hydrodynamics and optimal transport were later found to admit a similar description; we will look at some examples. For these equations, there is a rich interplay between the geometry of the group and the behavior of solutions of the original equation. I will give a survey of recent results and discuss some open problems.
Time: Thursday September 9, 2021
Place: Manchester 124
Speaker: Dr. Leandro Lichtenfelz
Title: The Euler Equations from a Geometric Perspective: Part 2
Abstract: This series of talks will be an introduction to Arnold’s idea of describing the Euler equations of hydrodynamics, classically a system of PDEs, as an ordinary differential equation (in fact, a geodesic equation) on an infinite-dimensional group. Several other PDEs of interest in hydrodynamics and optimal transport were later found to admit a similar description; we will look at some examples. For these equations, there is a rich interplay between the geometry of the group and the behavior of solutions of the original equation. I will give a survey of recent results and discuss some open problems.
Time: Thursday September 16, 2021 (Canceled)
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Time: Thursday September 23, 2021
Place: Manchester 124
Speaker: Dr. Leandro Lichtenfelz
Title: The Euler Equations from a Geometric Perspective: Part 3
Abstract: This series of talks will be an introduction to Arnold’s idea of describing the Euler equations of hydrodynamics, classically a system of PDEs, as an ordinary differential equation (in fact, a geodesic equation) on an infinite-dimensional group. Several other PDEs of interest in hydrodynamics and optimal transport were later found to admit a similar description; we will look at some examples. For these equations, there is a rich interplay between the geometry of the group and the behavior of solutions of the original equation. I will give a survey of recent results and discuss some open problems.
Time: Thursday September 30, 2021
Place: Manchester 124
Speaker: Dr. Miaohua Jiang
Title: Standing on the Shoulders of Giants
Abstract: I will tell four stories that happened in my mathematical research career. The stories share a common and interesting theme: the research topics all came to me unexpectedly and they essentially determined the main directions of my research. The talk is accessible to undergraduates.
Time: Thursday October 7, 2021
Place: Manchester 124
Speaker: Dr. Mauricio Alexander Rivas
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Time: Thursday October 14, 2021
- Meeting ID: 921 6908 5500 & Passcode: 255273
Speaker: Dr. Eric Autry
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Time: Thursday October 21, 2021
- Meeting ID: 921 6908 5500 & Passcode: 255273
Speaker: Dr. Marty Golubitsky
Title: Hopf Bifurcation in Different Contexts
Abstract: Hopf bifurcation is the basic tool for finding small amplitude periodic solutions in systems of ordinary differential equations. The periodic solutions appear when a complex conjugate pair of simple eigenvalues of the Jacobian at a stable equilibrium cross the imaginary axis with nonzero speed. The kinds of periodic solutions that appear can depend on whether the system of ODE has symmetry or whether the system is a coupled system associated to a network of ODE. This talk describes some of the different types of periodic solutions that can be expected to appear in these contexts and some of the applications that these periodic solutions can help explain.
Time: Thursday October 28, 2021
- Meeting ID: 921 6908 5500 & Passcode: 255273
Speaker: Dr. Feride Tiglay
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Time: Thursday November 4, 2021
Place: Manchester 124
Speaker: Dr. Miaohua Jiang
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Time: Thursday November 11, 2021
Place: Manchester 124
Speaker: Dr. Mauricio Alexander Rivas
Title: The Main Eigencurves for the Laplacian, and $L^2(\Omega)$-orthogonality of Steklov Eigenfunctions
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In this talk, three classes of eigencurves for the Laplacian on a cube $\Omega$ in $\mathbb{R}^3$ are constructed. Explicit formulae for points on the eigencurves are provided, and then a local picture for the eigencurves crossing those points are presented. Theoretical or applicable motivations for studying each class of eigencurves are discussed. Part of the construction of eigencurves leads to formulae showing (asymptotical) interior $L^2$-orthogonality of Steklov eigenfunctions; the functions are known to be $L^2$-orthogonal on the boundary. This is joint work with Manki Cho at the University of Houston-Clear Lake.
Time: Thursday November 18, 2021
Place: Manchester 124
Speaker: Dr. Kaitlin Hill
Title: Noise-induced tipping in piecewise-smooth systems
Abstract: In this talk, I will give an update on our progress to extend the Freidlin-Wentzell theory of large deviations to piecewise-smooth systems, for the particular case where the system has a single switch in the dynamics. We’ll first look at the example of a 2-dimensional piecewise-smooth system with a switching manifold at x=0 and discuss the most-probable path followed in a noise-induced transition across the switch. Then we’ll generalize this to n dimensions, and introduce the most probable path as the minimum of a rate functional that we can derive using the concept of gamma-convergence.
A bit about Prof.Hill: Professor Hill earned her Ph.D. from Northwestern University in 2017. She then worked as a postdoctoral scholar at the University of Minnesota, before accepting a position at Wake Forest University as a Teacher-Scholar postdoctoral fellow. More information about Prof. Hill and her research can be found on her webstie here: https://kaitlinhill.weebly.com
Time: Thursday December 2, 2021
Place: Manchester 124
Speaker: Dr. Stephen Robinson
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Spring 2021 (Thursdays 3 pm through Zoom)
Date: Thursday February 11
Speaker: Dr. John Holmes
Title: New results on weakly dispersive PDEs
Abstract: We will discuss two models for water waves. The first models waves near the equator, where the coriolis effect is important. The second equation is the FORQ equation, which is related to the celebrated KdV equation. Together with Feride Tiglay and Ryan Thompson, we study these equations in Besov spaces and show that the data-to-solution map is not better than continuous. In this talk we will motivate and introduce Besov spaces based on the Littlewood-Paley decomposition. We will also outline the proofs of nonuniform dependence and present some new estimates in Besov spaces.
Date: Thursday February 18
Speaker: Dr. Jeremy Marzuola
Title: Some recent progress on nodal domains
Abstract: We will first discuss some recent work on characterizing the number of nodal sets for an eigenfunction that was initiated with Graham Cox and Chris Jones, then has been further developed with Greg Berkolaiko and Graham Cox. The work of Berkolaiko-Cox-M gives an especially nice means of quantifying nodal sets for separable problems. Then, time permitting we will discuss more closely properties of nodal sets for low energy eigenfunctions on nearly rectangular domains that we looked at in joint with with Tom Beck and Yaiza Canzani.
Date: Thursday February 25
Speaker: Dr. John Gemmer
Title: Gamma-convergence for novices: Part 1
Abstract: Gamma convergence provides an asymptotic description for the behavior of a family of minimization problems and has been used to provide a rigorous framework for studying singular perturbation problems that arise in homogenization theory, phase transitions, free discontinuity problems, and elasticity theory to name a few. In 2002 Andrea Braides wrote a book entitled “Gamma convergence for beginners” which requires a solid foundation in measure theory and functional analysis to read in depth which is not my definition of a beginner. My goal in these three lectures is to introduce the topic of gamma convergence to a broad audience somewhere at the novice level, i.e. a Wake Forest student taking MST 711. Part 1 of the lecture will serve as an introduction to weak convergence and the direct method of the calculus of variations. Part 2 will focus on gamma convergence with respect to the weak topology and contain several examples of computing gamma limits. Finally part 3 will focus on a collaboration with Kaitlin Hill in which we apply gamma-convergence techniques to understand noise induced tipping in piecewise-smooth dynamical systems.
Date: Thursday March 4
Speaker: Dr. John Gemmer
Title: Gamma-convergence for novices: Part 2
Abstract: Gamma convergence provides an asymptotic description for the behavior of a family of minimization problems and has been used to provide a rigorous framework for studying singular perturbation problems that arise in homogenization theory, phase transitions, free discontinuity problems, and elasticity theory to name a few. In 2002 Andrea Braides wrote a book entitled “Gamma convergence for beginners” which requires a solid foundation in measure theory and functional analysis to read in depth which is not my definition of a beginner. My goal in these three lectures is to introduce the topic of gamma convergence to a broad audience somewhere at the novice level, i.e. a Wake Forest student taking MST 711. Part 1 of the lecture will serve as an introduction to weak convergence and the direct method of the calculus of variations. Part 2 will focus on gamma convergence with respect to the weak topology and contain several examples of computing gamma limits. Finally part 3 will focus on a collaboration with Kaitlin Hill in which we apply gamma-convergence techniques to understand noise induced tipping in piecewise-smooth dynamical systems.
Date: Thursday March 11
Speaker: Dr. John Gemmer
Title: Gamma-convergence for novices: Part 3
Abstract: Gamma convergence provides an asymptotic description for the behavior of a family of minimization problems and has been used to provide a rigorous framework for studying singular perturbation problems that arise in homogenization theory, phase transitions, free discontinuity problems, and elasticity theory to name a few. In 2002 Andrea Braides wrote a book entitled “Gamma convergence for beginners” which requires a solid foundation in measure theory and functional analysis to read in depth which is not my definition of a beginner. My goal in these three lectures is to introduce the topic of gamma convergence to a broad audience somewhere at the novice level, i.e. a Wake Forest student taking MST 711. Part 1 of the lecture will serve as an introduction to weak convergence and the direct method of the calculus of variations. Part 2 will focus on gamma convergence with respect to the weak topology and contain several examples of computing gamma limits. Finally part 3 will focus on a collaboration with Kaitlin Hill in which we apply gamma-convergence techniques to understand noise induced tipping in piecewise-smooth dynamical systems.
Date: Thursday March 18
Speaker: Dr. Irfan Glogic
Title: Self-similar blowup for nonlinear wave equations
Abstract: One of the remarkable features of time-dependent nonlinear partial differential equations is the possibility of spontaneous onset of singularities (also called blowup). Namely, a smooth and localized initial profile evolves after a finite amount of time into a singular form, where either the profile itself or some derivative becomes infinite (hence the name blowup). This phenomenon has both physical and mathematical significance, and determining whether a given nonlinear model admits blowup is one of the central questions of the modern analysis of PDEs. In this talk we concentrate on nonlinear wave equations and we discuss a type of blowup that appears to be generic, namely the self-similar one. We furthermore outline a general framework for studying stability of self-similar blowup, we discuss the new mathematical problems this approach generates, and we mention some results we obtained in recent years.
Date: Thursday March 25
Speaker: Dr. Marco Lopez
Title: Invariant measures of Markov transformations and their entropy
Abstract: In this talk we will give a review of metric and topological entropies for various classes of discrete dynamical systems, with especial emphasis on Markov transformations.
Date: Thursday April 1
Speaker: Dr. Nsoki Mavinga
Title: Delay Differential Equations with Applications to the Analysis of the Spread of Vector-Borne Diseases
Abstract: Many problems in applied sciences give rise to delay differential equations. These are differential equations in which the current rate of change of the system depends not only on the current state but also on the history of the system; i.e. the system has memory. In this talk, we will discuss the stability of equilibrium solutions for a two-lag delay differential equation which models the spread of infectious diseases; namely, vector-borne diseases where the lags are incubation periods in humans and vectors. We show that there are some values of the transmission and recovery rates for which either the disease dies out or it spreads into an endemic state. The approach is based on the linearization method and the analysis of roots of transcendental equations.
Date: Thursday April 8
Speaker: Dr. Mostafa Rezapour
Title: A new approach in derivative-free optimization (part 1)
Abstract: Optimization is one of the most important components of machine learning and deep learning. However, in modern computational science, machine learning is not simply a consumer of optimization. The interaction between optimization and machine learning is one of the most important recent developments in computational science. In the first part of the presentation, we review the classic optimization algorithms for solving unconstrained optimization problems. Moreover, we discuss how to apply deep neural networks to solve ODEs and PDEs. Finally, in the second part of the presentation, we discuss how to use deep neural-networks and their well-known capability as universal function approximator in derivative-free unconstrained optimization problems, where the objective function is possibly nonsmooth. This approach may provide improved practical performance in cases where the objective function is extremely noisy or stochastic.
Date: Thursday April 15
Speaker: Dr. Mostafa Rezapour
Title: A new approach in derivative-free optimization (part 2)
Abstract: Optimization is one of the most important components of machine learning and deep learning. However, in modern computational science, machine learning is not simply a consumer of optimization. The interaction between optimization and machine learning is one of the most important recent developments in computational science. In the first part of the presentation, we review the classic optimization algorithms for solving unconstrained optimization problems. Moreover, we discuss how to apply deep neural networks to solve ODEs and PDEs. Finally, in the second part of the presentation, we discuss how to use deep neural-networks and their well-known capability as universal function approximator in derivative-free unconstrained optimization problems, where the objective function is possibly nonsmooth. This approach may provide improved practical performance in cases where the objective function is extremely noisy or stochastic.
Date: Thursday April 22
Speaker: Dr. Kate Meyer
Title: Dynamics of flow-kick disturbance models
Abstract: To incorporate repeated disturbances into a differential equation (DE) model of ecological processes, one might embed the disturbance continuously in the DE or resolve the disturbance discretely. For example, do harvests from a logistic population appear continuously as x’ = x(1 − x) − h(x) or do individual harvests periodically kick the state x as it flows according to x’ = x(1 − x)? Are fires always smoldering in a model savannah, or do they burn at discrete timepoints? In this talk we’ll explore the flow-kick approach to modeling repeated, discrete disturbances and examine the dynamic implications of this modeling choice. We’ll position continuous disturbances as limits of repeated, discrete ones, and discuss how flow-kick systems both mimic and depart from their continuous analogs.
