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Graduate Analysis Seminar : Past Events

Past Seminars

The following seminars have already happened, you may instead view upcoming seminars in this series.

Jan. 28, 2013

An Introduction to Semigroup Theory

Karen Zaya : 3 p.m. in SEO 512

Feb. 4, 2013

An Introduction to Semigroup Theory

Karen Zaya : 3 p.m. in SEO 512

Feb. 11, 2013

Dissipative Operators and Adjoint Semigroups

Landon Kavlie : 3 p.m. in SEO 512
Abstract A continuation of the Introduction to Semigroups Series.

Feb. 18, 2013

Analytic Semigroups

Landon Kavlie : 3 p.m. in SEO 512
Abstract A continuation of the Introduction to Semigroups Series. We will introduce the concept of an analytic semigroup of type $(\alpha,M)$. Moreover, we will describe what it means for an operator $A$ to generate an analytic semigroup and show that any $A\in\mathcal{G}\mathcal{A}_b(\theta,M)$ generates an analytic semigroup of type $(\theta-\pi/2)$.

Feb. 25, 2013

Semigroups in Fluid Dynamics

Roman Shvydkoy : 3 p.m. in SEO 512
Abstract We will discuss how semigroups introduced in the previous talks enter into the questions of stability of incompressible fluids. We will review perturbation theorems for generators, and prove that linearizations of the Euler or Navier-Stokes equations generate C_0-semigroups. We will argue that the Navier-Stokes semigroup is in fact analytic.

March 11, 2013

Conditioning, Martingales, and the Markov Property

Geoff Lindsell : 3 p.m. in SEO 512
Abstract This talk will explore the basic properties of conditional expectation and martingales in the discrete time setting. We will discuss the fundamental inequalities, convergence concepts, and theorems (e.g. Uniformly Integrable Martingales, Doob's Optional Sampling Theorem) which may be generalized to the continuous time setting. The only prerequisites are a basic understanding of measure theory.

March 18, 2013

Conditioning, Martingales, and the Markov Property (continued)

Geoff Lindsell : 3 p.m. in SEO 512
Abstract We continue to explore Doob's theory of martingales in the discrete time setting (rigorously enough to get an idea of how these concepts work in the continuous time setting). We will discuss two major theorems from the theory of uniformly integrable martingales: Doob's L^p inequality, and the Optional Sampling Theorem.

April 1, 2013

Analysis of Black-Scholes equation

Venu Tammali : 3 p.m. in SEO 512
Abstract We will first develop the tools needed for deriving the Black-Scholes equation, such as Brownian process, Ito calculus, etc. Then we will derive the equation. Finally we will give a risk neutral measure for option pricing as an alternative to the Black-Scholes equation.

April 8, 2013

Null space and Restricted Isometric Property Applications in Compressive Sensing

Venu Tammali : 3 p.m. in SEO 512
Abstract We will present a seminar on Null Space Property and Restricted Isometry Property of Matrices, which have huge consequences in Compressive Sensing Techniques used in Image Processing, MP3 conversion, and other applications.

April 15, 2013

An Introduction To Sobolev Spaces

Matthew Kaplan : 3 p.m. in SEO 512
Abstract An introductory level talk in which we motivate and prove results leading to the Sobolev imbedding theorems.

April 22, 2013

Introduction to Sobolev Spaces

Matthew Kaplan : 3 p.m. in SEO 512
Abstract A continuation of last week's discussion.

April 29, 2013

On the incompressible Euler equations

Anne Bronzi : 3 p.m. in SEO 512
Abstract Well-known results concerning the incompressible Euler equations will be discussed, namely the local well-posedness, the Beale-Kato-Majda continuation criterion and the global well-posedness in dimension two.

Aug. 26, 2013

Organizational Meeting

: 3 p.m. in SEO 512

Sept. 4, 2013

An Introduction to Littlewood-Paley Theory

Landon Kavlie : 3 p.m. in SEO 512
Abstract We will begin with a brief introduction to harmonic analysis and Fourier multipliers. After that, we will look at dyadic decompositions and the Littlewood-Paley theorem.

Sept. 11, 2013

An Introduction to Littlewood-Paley Theory

Landon Kavlie : 3 p.m. in SEO 512
Abstract A continuation of last week's talk. We will finish the proof of the Littlewood-Paley theorem and present some applications.

Sept. 18, 2013

Measuring Smoothness

Karen Zaya : 3 p.m. in SEO 512
Abstract We discuss and compare a few different ways to measure the smoothness of a function.

Oct. 2, 2013

Introduction to Unbounded Operators

Matthew Kaplan : 3 p.m. in SEO 512

Oct. 7, 2014

TBD

Jack Arbunich : 4 p.m. in SEO 512
Abstract We will continue our crash-course in basic harmonic analysis. Topics to be covered include criteria for convergence of Fourier series, Weyl's equidistribution theorem, and the isoperimetric inequality.

Oct. 21, 2014

Applications of Fourier Series

Keaton Quinn : 4 p.m. in SEO 512
Abstract We'll continue our crash course in harmonic analysis today with an overview of some applications of Fourier series, such as the heat equation on a circle, the isoperimetric inequality and Weyl's equidistribution theorem.

Oct. 28, 2014

Derivation of Nonlinear Schrödinger equation from many-body systems

Zhihui Xie : 4 p.m. in SEO 1227
Abstract The nonlinear Schrödinger equations have attracted a lot of attention from the mathematical community, which emphasizes the importance of making their derivation rigorous. In this talk, we look at the derivation of a certain type of NLS from many-body interactions of bosonic particles with many-body interactions.

Nov. 4, 2014

Dynamics of the Navier-Stokes Equations

Landon Kavlie : 4 p.m. in SEO 512
Abstract In this talk, I introduce the Navier-Stokes equations. I then describe how these equations produce an infinite-dimensional dynamical system.

Nov. 18, 2014

Basics of the Heat Equation on $\mathbb{R}^n$

Trevor Leslie : 4 p.m. in SEO 512
Abstract We will introduce the Fourier transform and use it to solve the heat equation on $\mathbb{R}^n$. We will also give sufficient conditions for uniqueness of solutions.

Nov. 25, 2014

Time Independent Schrödinger Equation via a Variational Method, Part 1

Jack Arbunich : 4 p.m. in SEO 512
Abstract Part 1: (11/25) We define an energy functional associated to the time independent Schrödinger equation and prove the existence of a minimizer. Part 2: (12/2) We show that the minimizer is a solution, and we prove uniqueness and regularity results.

Dec. 2, 2014

Time Independent Schrödinger Equation via a Variational Method, Part 2

Trevor Leslie : 4 p.m. in SEO 512
Abstract Part 1: (11/25) We define an energy functional associated to the time independent Schrödinger equation and prove the existence of a minimizer. Part 2: (12/2) We show that the minimizer is a solution, and we prove uniqueness and regularity results.

Feb. 18, 2015

Gaining two derivatives on a singular force in the 2D Navier-Stokes equations.

Landon Kavlie : 2 p.m. in SEO 427
Abstract It is well known that linear parabolic equations like the heat equation gain two derivatives on the force, globally. An analogous result is known for the 2D Navier-Stokes equations with a force that is "smooth enough." In this talk, we use classical and harmonic analysis techniques to extend these results.

Feb. 25, 2015

Chaotic dynamics of Perturbed Quadratic Maps

Robert Kozma : 2 p.m. in SEO 427
Abstract Even the simplest nonlinear function $z^2+c$ gives rise to chaotic dynamics resulting in very intricate mathematical objects such as the Mandelbrot set, and Julia sets. In this talk we first go over some basic properties of the Mandelbrot set, and Julia sets. We then go on to describe novel results on the behavior of Julia sets of certain families of rational maps arising from $z^2+c$ by adding a small perturbation term $\lambda/z^2$ and taking the limit as $\lambda \to 0$. We will see that for certain $c$ values the resulting Julia sets have some astonishing geometric and topological properties. Using symbolic dynamics and Cantor necklaces, we prove that as $\lambda\to 0$, the Julia for this family set evolves into a space-filling fractal curve.

March 11, 2015

Chernoff's Proof of the Tychonoff Theorem Using Nets

Trevor Leslie : 2 p.m. in SEO 427
Abstract In this talk, we will define nets and use them to give a short proof of Tychonoff's theorem.

April 8, 2015

Unbounded Operators and Stone's Theorem

Jack Arbunich : 2 p.m. in SEO 427
Abstract I will introduce some basic concepts/definitions used in studying unbounded operators. With bounded self-adjoint operators the exponential of an operator makes sense via its Taylor series as the series converges in norm. This method is not possible if our operator is unbounded. Stone's Theorem states that if we have a strongly continuous one parameter unitary group, then there is a self-adjoint operator $A$ on our Hilbert Space such that the one parameter group is the exponential of the operator $A$.

April 15, 2015

Stone's Theorem (part 2)

Jack Arbunich : 2 p.m. in SEO 427
Abstract With bounded self-adjoint operators the exponential of an operator makes sense via its Taylor series as the series converges in norm. This method is not possible if our operator is unbounded. Stone's Theorem states that if we have a strongly continuous one parameter unitary group, then there is a self-adjoint operator $A$ on our Hilbert Space such that the one parameter group is the exponential of the operator $A$.

Aug. 31, 2015

Organizational Meeting

Trevor Leslie : 3 p.m. in SEO 612

Sept. 14, 2015

Semiclassical approximations to quantum mechanical expectation values (Introduction for Graduate Students)

Wolfgang Gaim : 3 p.m. in SEO 612
Abstract The main goal of this talk will be to prepare the students for my talk in the analysis and applied mathematics seminar. Therefore I will make an introduction into semiclassical quantum mechanics where I will explain the basic framework and discuss some important applications and theorems. I will then use these techniques in a more general setting and give an introduction into space-adiabatic perturbation theory.

Sept. 28, 2015

Bloch's theorem and the magnetic Bloch-Floquet transform

Wolfgang Gaim : 3 p.m. in SEO 612
Abstract In my talk I will make a short introduction into solid state physics, where I will show a proof and some implications of Bloch's theorem for Hamiltonians with periodic potentials. In addition I will introduce the magnetic Bloch-Floquet transform.

Oct. 12, 2015

Elements of a theory of pseudo-differential operators

Han Liu : 3 p.m. in SEO 612

Oct. 26, 2015

Parametrix of a Pseudodifferential Operator

Jack Arbunich : 3 p.m. in SEO 612
Abstract We will examine what it means to invert a pseudo-differential operator. In doing so, we will verify the uniqueness and existence of such a creature, and show the existence of a parametrix for semi-elliptic operators.

Nov. 2, 2015

Banach Algebras and Gelfand Theory

Keaton Quinn : 3 p.m. in SEO 612
Abstract We give the introductory theory for commutative unital Banach algebras. We start with the basic definitions, introduce some spectral theory, provide a characterization of the spectrum of an operator, and introduce the Gelfand Transform.

Nov. 16, 2015

Introduction to Modular Forms I

Charles Alley : 3 p.m. in SEO 612
Abstract Modular forms are functions on the upper half plane which obey a certain transformation law under the action of $SL_2(\mathbb{Z})$. In this talk I will present basic definitions and mention a few applications. The study of modular forms is a vast and rich area of research. Questions and interests of the audience will be used to decide on a related topic for the second talk.

Jan. 25, 2016

Graduate Analysis Seminar Organizational Meeting

Trevor Leslie : 2 p.m. in SEO 612

Feb. 8, 2016

The Schwarzian Derivative

Charles Alley : 2 p.m. in SEO 612
Abstract In this talk I will introduce the Schwarzian derivative and give a survey of some of its properties and related results. The Schwarzian derivative is a quadratic differential operator on the space of locally injective holomorphic functions which is distinguished by the property $S(f)=0$ if and only if $f$ is a linear fractional transformation.

Feb. 22, 2016

A PDE on a compact manifold

Keaton Quinn : 2 p.m. in SEO 612
Abstract We turn a geometric question about curvatures of compact Riemannian 2-manifolds into a existence question for a PDE. We describe how one defines a PDE on a manifolds and how to extend the analytic tools used to study them to this general setting.

Feb. 29, 2016

Topological Vector Spaces

Wesley Townsend : 2 p.m. in SEO 612
Abstract We survey different methods of putting a topology on vector spaces. In particular, we study the topology generated by a family of seminorms and the operator topologies that arise in the study of differential equations.

March 7, 2016

Homogenization for Elliptic PDEs via Two-Scale Convergence

Trevor Leslie : 2 p.m. in SEO 612
Abstract We study the elliptic PDE $-\nabla \cdot (A^\varepsilon \nabla u^\varepsilon) = f$ on a bounded domain with Dirichlet boundary conditions, where $A^\varepsilon(x) = A(x/\varepsilon)$ and $A(y)$ is a periodic matrix with bounded coefficients. We derive an approximation of this equation of the form $-\nabla \cdot (\overline{A} \nabla u) = f$ and prove that the solution $u$ approximates the solution $u^\varepsilon$ of the original equation. The proof uses the method of two-scale convergence; basic properties of two-scale convergence will be stated (without proof) for completeness.

March 14, 2016

Multiscale Expansion in Periodic Structures

Jack Arbunich : 2 p.m. in SEO 612
Abstract Building off some of the ideas last week, we shall explore an alternative method in multiscale analysis on periodic structures via a formal two scale asymptotic expansion. We will motivate this by considering a model for an electron moving in a periodic potential created by atoms in a crystal lattice. If we search for high frequency solutions, where the wavelength is comparable to the period of the lattice, then two natural length scales arise, a microscopic and macroscopic length scale. Starting from a microscopic description of a problem, we convert to a semiclassical scaling in time and space, and seek an approximate macroscopic description through multiscale expansion. Our aim will be to highlight the formal expansion and to show the framework for the stability result which verifies indeed that our approximation has an agreeable degree of accuracy. Along the way we will definitely encounter the Bloch-Floquet eigenvalue problem and possibly some snacks which for now is left to the whim of the speaker.

March 28, 2016

Some basic properties of Radon measures

Trevor Leslie : 2 p.m. in SEO 612
Abstract With the aim of understanding the dual space of $C_0(X)$, the continuous functions on a (locally compact Hausdorff) space $X$ which vanish at infinity, we introduce Radon measures and prove that there is a natural bijection between the set of positive linear functionals on $C_c(X)$ (the space of compactly supported continuous functions on $X$) and the space of Radon measures on $X$.

April 11, 2016

Modeling the Multilayer Grow-Or-Go Model for Glioblastoma Multiforme

Miko Vesović : 2 p.m. in SEO 612
Abstract An existing model for understanding the growth of glioblastoma multiforme (GBM) takes into account its multilayer structure which consists of proliferative, invasive, and necrotic cells. It expresses the growth of a tumor in a host environment during ideal conditions in an attempt to find methods of suppressing the growth. During numerical simulations of the system, certain adjustments were made which could potentially improve the model's ability to represent the growth while still leaving the model solvable.

April 18, 2016

Curvature Functions for Surfaces and Elliptic PDEs.

Keaton Quinn : 2 p.m. in SEO 612
Abstract We investigate which curvatures compact surfaces can carry by turning the question into one of solutions to the elliptic PDE $\Delta u = c - he^u$.

Aug. 29, 2016

Organizational Meeting

Trevor Leslie : noon in SEO 1227

Sept. 7, 2016

Introduction to dispersive and Strichartz estimates

Paolo Antonelli : 4 p.m. in SEO 1227
Abstract We will give a short introduction to dispersive estimates for Schrödinger type equations and derive the corresponding Strichartz-space-time estimates from it. These will be used to obtain local well-posedness for the initial value problem in function spaces which are not feasible for more classical techniques.

Sept. 12, 2016

Measuring Small Sets: Some Basic Properties of Capacity and Hausdorff Dimension

Trevor Leslie : noon in SEO 1227

Sept. 19, 2016

Banach Limits

Keaton Quinn : noon in SEO 1227
Abstract We give a cute application of the Hahn Banach Theorem by extending the limit operator from the space of convergent sequences to all bounded sequences.

Sept. 26, 2016

Oscillatory Integrals of the First Kind

Jack Arbunich : noon in SEO 1227
Abstract Oscillatory integrals have been an integral part of Harmonic analysis from its conception. The Fourier transform is a ripe example of such an oscillatory integral. One main use of oscillatory integrals is to obtain decay estimates for Fourier transforms of measures on surfaces. In a sense, oscillatory integrals link geometric properties of manifolds and the harmonic analysis related to them. We will prove a few basic yet noteworthy results on the asymptotics of oscillatory integrals in 1D.

Oct. 3, 2016

Asymptotic Values of Functions with Polynomial Schwarzian Derivative

Charles Alley : noon in SEO 1227
Abstract It is a classical result of Rolf Nevanlinna that an entire function whose Schwarzian derivative is a polynomial of degree $d$ has exactly $d+2$ asymptotic values. In this talk I will sketch a proof of this result and remark on the role of Stokes' phenomenon in understanding the set of asymptotic values.

Oct. 17, 2016

Interior $L^2$ Regularity for Elliptic Equations

Han Liu : noon in SEO 1227
Abstract We will first prove the Caccioppoli inequality, and use it along with the difference quotient method to show the interior $L^2$ regularity for elliptic equations.

Oct. 24, 2016

The Haar measure on compact groups

Keaton Quinn : noon in SEO 1227
Abstract We prove the existence and uniqueness of the Haar measure in the case of compact groups.

Oct. 31, 2016

Energy Balance Criteria for the Navier-Stokes Equations

Trevor Leslie : noon in SEO 1227
Abstract Let $u$ be a solution to the classical 3D Navier-Stokes equations. If $u$ is smooth, then we know that $u$ satisfies a certain energy balance relation. But what if $u$ is not smooth? It has been known since the 1960s (by a result of Lions) that $u$ satisfies the energy balance relation if $u$ is $L^4$ integrable in both time and space. We give new regularity conditions on $u$ that guarantee energy balance; our conditions depend on both the integrability of $u$ and the Hausdorff dimension $d$ of the singularity set. We recover Lions' result when $d=1$ and improve upon it when $d<1$ or when the singularity set is confined to a single time-slice. This is joint work with Roman Shvydkoy.

Nov. 28, 2016

Stokes' Phenomenon

Charles Alley : noon in SEO 1227
Abstract Stokes' phenomenon deals with subtleties of asymptotics in the complex plane. I will give a brief overview following chapter 15 of Wasow's book "Asymptotic Expansions for Ordinary Differential Equations".

Jan. 25, 2017

Spring Organizational Meeting

Trevor Leslie : 4 p.m. in SEO 512

Feb. 8, 2017

Isometries of Real Normed Vector Spaces

Keaton Quinn : 4 p.m. in SEO 512
Abstract We prove the Mazur Ulam Theorem which classifies all surjective (possibly non-linear) isometries between two real normed vector spaces.

Feb. 15, 2017

TBA

Ethan Fricker, David Reynolds : 4 p.m. in SEO 512

March 8, 2017

Modeling Nonlinear Wave Dynamics in Honeycomb Structures

Jack Arbunich : 4 p.m. in SEO 512
Abstract We will consider the nonlinear Schrödinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multiscale expansion together with rigorous error estimates, we will derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point. The aim of the talk will be to introduce the theory of periodic Schrödinger operators and to familiarize the audience with the formal technique of multiscale expansion.

March 15, 2017

Weak and Strong Compactness Theorems for Radon Measures

Trevor Leslie : 4 p.m. in SEO 512
Abstract After reviewing some basic properties of Radon measures (basic definitions, Riesz Representation Theorem), we discuss compactness properties of bounded sequences of Radon measures.

April 5, 2017

Bilinear Forms

Keaton Quinn : 4 p.m. in SEO 512

April 12, 2017

Convexity and Weak Solutions to Nonlinear PDE

Jack Arbunich : 4 p.m. in SEO 512

Sept. 6, 2017

Organizational Meeting

David Reynolds : 4 p.m. in SEO 512

Sept. 20, 2017

An Introduction to the Monodromy Representation

Charles Alley : 4 p.m. in SEO 512
Abstract The Monodromy Representation is a representation of the fundamental group of a Riemann surface into GL(n,C). In this talk I will survey the classical perspective on Monodromy from the theory of ordinary differential equations in the complex domain. This will be the first of two talks; the second talk will be focused on generalized monodromy which incorporates data arising from Stokes phenomenon.

Sept. 27, 2017

An Introduction to the Monodromy Representation

Charles Alley : 4 p.m. in SEO 512

Oct. 4, 2017

Solution Methods for $C^0$ Interior Penalty and Mixed Finite Element Discretizations of the Elliptic Monge-Ampère Equation

Eric Malitz : 4 p.m. in SEO 512
Abstract The Monge-Ampère equation plays a role in applications in geometry, optics, and other areas. The most natural finite element discretization of the Dirichlet problem for the Monge-Ampère equation is with $C^1$ elements, which lead to unruly discrete systems. We discretize the problem with a $C^0$ interior penalty method, allowing the use of simpler $C^0$ Lagrange elements. We then prove convergence of two methods for resolving the nonlinear discrete problem: an iterative time marching method that may capture more accurate results for nonsmooth solutions than does Newton's method, and a two-grid method which is computationally more efficient than Newton's method.

Oct. 11, 2017

Long time behavior of solutions to the Hall-MHD equations.

Han Liu : 4 p.m. in SEO 512
Abstract We investigate the long time behavior of solutions to the Hall-MHD equations using the Fourier splitting method, which had been widely used to show decay of solutions to dissipative equations.

Oct. 18, 2017

The Energy Measure for the Navier-Stokes Equations

Trevor Leslie : 4 p.m. in SEO 512
Abstract The potential failure of energy equality for a solution $u$ of the Euler or Navier-Stokes Equations (NSE) can be quantified using a so-called `energy measure': the weak-$*$ limit of the measures $|u(t)|^2\,dx$ as $t$ approaches the first possible blowup time. Focusing on the case of the 3-dimensional NSE, we discuss reasons why the energy measure is an object worthy of study and prove some of its basic measure-theoretic properties. We also state some new results relating to energy equality of the NSE. With the exception of a few remarks, the talk should be accessible to anyone who knows some measure theory and a bit of functional analysis. In particular, the necessary background for the NSE will be provided at the beginning of the talk.

Nov. 29, 2017

The Aubin-Lions-Simon Compactness Theorem

Trevor Leslie : 4 p.m. in SEO 512
Abstract The Aubin-Lions Theorem, proved in the 1960s, is a useful tool for extracting subsequences (of Banach space-valued functions) which converge in some sense. For example, it is a key step in one of the standard proofs of the existence of weak solutions to the classical Navier-Stokes Equations, namely showing that a subsequence of the relevant Galerkin approximations converges to a solution. Later, Simon removed the hypothesis that the underlying Banach spaces must be reflexive. The resulting theorem is consequently sometimes called the Aubin-Lions-Simon Theorem. In this talk, we state and prove the Aubin-Lions-Simon Theorem. If time allows, we will give an application that requires Simon's version of the Theorem.

Jan. 24, 2018

Organizational Meeting

David Reynolds : 4 p.m. in SEO 512

Feb. 14, 2018

Introduction to Fourier Multipliers and Associated Spaces (Part 1)

David N Reynolds : 4 p.m. in SEO 512
Abstract Over the course of two talks we will introduce the concept of Fourier Multipliers and prove some basic facts about the spaces of such multipliers. From there we will talk about Besov and Triebel-Lizorkin Spaces and some embedding theorems.

Feb. 21, 2018

Littlewood Paley Theory and Introduction to Besov and Triebel-Lizorkin Spaces

David N Reynolds : 4 p.m. in SEO 512
Abstract We will start by introducing the Littlewood-Paley decomposition operator and use this to define Besov and Triebel-Lizorkin spaces. From there we will prove some basic facts about the spaces and motivate the definitions of said spaces.

March 7, 2018

Introduction to High–Order Perturbation of Surfaces Algorithms (HOPS)

Xin Tong : 4 p.m. in SEO 512

March 21, 2018

Instant Hölder Regularization of Advection Fractional-Diffusion Equations, Part I

Trevor Leslie : 4 p.m. in SEO 512
Abstract We examine an equation of the form $u_t + b\cdot \nabla u + (-\Delta)^s u = f$. A 2011 result by Luis Silvestre says that any bounded solution of the equation satisfies a certain Hölder estimate. In this talk, we prove an intermediate estimate that is used in the proof of Silvestre's main result. We also give an application of the result to a model in collective dynamics. The rest of the proof of the Hölder estimate will be given in a future talk (April 18) at the Graduate Analysis Seminar.

April 2, 2018

An Introduction to Stability Theory of Solitons

Shijun Zheng : 2 p.m. in SEO 1227

April 4, 2018

The Stokes phenomenon and Monodromy of the Airy Equation: A Case Study

Charles Alley : 4 p.m. in SEO 512
Abstract Several powerful theorems allow us to describe the asymptotics of solutions to ordinary differential equations in the complex domain. The Stokes phenomenon refers to the multivalued behavior of formal solutions of equations with irregular singularities or, put another way, that there are "jumps" in the asymptotics of the solutions. The associated Stokes data, which I will define, can be thought of as an enriching of the monodromy data of differential equations. Understanding this has led to a rich geometric program. In this talk, and a talk to follow, I will describe the analytic foundations of this program while focusing on the example of the Airy equation $u'' + zu=0$, which has an irregular singularity at infinity.

April 25, 2018

Stokes Phenomenon with the Airy Equation as an example (part 2)

Charles Alley : 4 p.m. in SEO 512

May 2, 2018

A Locally Adaptive Algorithm for Global Minimization of Univariate Functions

Xin Tong : 4 p.m. in SEO 512
Abstract In this talk, I will introduce a locally adaptive algorithm for global minimization of univariate functions on a bounded interval. These functions are smooth and not too highly oscillatory, but are not necessarily convex, in the Sobolev space $W_2^{\infty}$. This algorithm samples more function values where the function is small in value and more spiky. It provides an estimated minimum that satisfies a user-specified absolute error tolerance. This algorithm, called funmin_g, is implemented in the Guaranteed Automatic Integrate Library (GAIL). There are numerical examples that illustrate its superior performance in comparison to other more established solvers including MATLAB's fminbnd and Chebfun's min.

Sept. 5, 2018

Organizational Meeting

David Reynolds : 4 p.m. in 1227 SEO

Sept. 19, 2018

An Introduction to Modeling Collective Behavior with Differential Equations

David Reynolds : 4 p.m. in 1227 SEO

Sept. 26, 2018

Gross-Pitaevskii-type Model for Exciton-Polariton Condensates

Ryan Obermeyer : 4 p.m. in 1227 SEO
Abstract Bose-Einstein condensation is an active, exciting area of modern physics. Though the phenomenon was predicted by Einstein in 1925, BECs were only first produced in experiments in 1995. A new type of condensate consisting of exciton-polariton quasi-particles has recently been studied along with the more classical BECs of ultracold atomic gases. In this talk I will discuss a non-equilibrium model of exciton-polariton condensates based on the Gross-Pitaevskii equation with an additional dissipative term.

Nov. 7, 2018

The Riemann Zeta Function and Pade Approximations

Matthew Kehoe : 4 p.m. in 1227 SEO

Jan. 16, 2019

Organizational Meeting

David Reynolds : 4 p.m. in 1227 SEO

Feb. 1, 2019

Introduction to the water wave problem of finite depth and modulated waves

Yannis Giannoulis : 3 p.m. in 636 SEO
Abstract We will discuss the derivation of the water wave equations for finite depth, their different features depending on the relation between their characteristic lengths, and the notion of modulated waves, and aim to give an outline of the proof of Lannes’s well-posedness result.

Feb. 6, 2019

Kinetic Theory Chapter 1

Scott Rogers : 4 p.m. in 1227 SEO

Feb. 13, 2019

Kinetic Theory Chapter 2

Justin Richman : 4 p.m. in 1227 SEO

April 3, 2019

Floquet theory - a quantum mechanical point of view

Radu Purice : 3 p.m. in 1227 SEO
Abstract We discuss a subject at the confluence of the theory of induced group representations and spectral theory in Hilbert spaces, studying the problem of some differential operators on an affine space being invariant only for a discrete subgroup and concentrating on the case of interest for mathematical physics, that of periodic quantum Hamiltonians.

Sept. 14, 2022

Littlewood-Paley and Onsager's Conjecture

Qirui Peng : 4 p.m. in 1227 SEO

Sept. 28, 2022

Intro to Fully Nonlinear PDEs & Viscosity Solutions

Sam Wallace : 4 p.m. in 1227 SEO
Abstract Sam is going to present important ideas and facts about viscosity solutions to degenerate elliptic PDEs.

Oct. 5, 2022

Intro to Convex Integration & Weak solutions for Fluid Mechanics

Qirui Peng : 4 p.m. in 1227 SEO
Abstract Qirui Peng will be talking about applications of convex integration to constructing weak solutions to Euler Eqns and NSE.