Welcome to the Applied Mathematics Reading Seminar, organized by Nathan Soedjak, Han Yong Wunrow, and Edith Zhang. The purpose of this seminar is to build community among the applied mathematics graduate students and to create a space for students to teach and learn topics in applied mathematics. There will be an emphasis on applications to supplement APAM coursework.
Graduate students will sign up for either 2-3 seminar dates to cover their proposed topic. Prior to each seminar, speakers will email the attendees an abstract of their talk and any relevant textbook chapters or papers for attendees to read prior to the seminar.
This summer we will cover linear programming, heat kernels, special function and group representations, and the adjoint state method. Our talks will be held in-person in Columbia University (Mudd 210) on Mondays from 3 p.m. to 4 p.m. EDT.
If you would like to come or to be added on the mailing list, please email [email protected].
| Date | Presenter | Title/Abstract | Materials |
|---|---|---|---|
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| Date | Presenter | Title/Abstract | Materials |
|---|---|---|---|
| 7/24 | Yan Cheng | Adjoint State Method |
Slides |
| 7/10 | Jonah Chaban | Special functions and group representations Solutions of certain linear PDEs can sometimes be expressed as ""superpositions"" of special functions which arise as the result of a separation of variables ansatz (e.g. Fourier series, Bessel series, Fourier transform). However, the separation of variables strategy only ""works"" in the case that the original PDE possesses suitable symmetry properties. This might cause one to wonder if there's a connection between special functions and the symmetries which characterize the situations in which they're most useful. We'll take a look at this connection using representation theory, which conveys information about symmetry (formulated in the language of groups) to the solution space of linear PDEs (structured as a vector space). |
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| 6/26 | Edith Zhang | Pattern formation and opinion dynamics: how small-scale interactions between individuals develop large-scale patterns in behavior. We'll discuss some simple opinion-dynamics models, which are used in, for example, voting patterns in social networks. We will learn about the PDE dynamics that arise from the individual interactions, stability analysis, and how opinion clusters form over time. This is the topic of a conference I'm attending the week prior to this presentation. |
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| 6/19 | Blake Sisson |
Linear Programming II We will continue our discussion on linear programming with how to handle integrality constraints and large scale problems. |
Slides |
| 6/12 | Blake Sisson |
Linear Programming I Linear programming is one of the premier areas of optimization for both its applicability and the robustness of its algorithms. While greatly developing in the past 70 years since its inception, linear programming still faces challenges tackling some of today's problems. We provide an overview of the Simplex Method for solving linear programs, explaining the algorithm's practical effectiveness despite a major theoretical shortcoming. Next we analyze the difficulty imposed by integrality constraints featured in many applications, and discuss some of the methods used for handling such cases. Finally we illustrate the shortfalls of state of the art algorithms when applied to large scale problems, and the challenges to overcome them. |
Slides |