# Applied/GPS

# Graduate Applied Math Seminar

The Graduate Applied Math Seminar is one of the main tools for bringing together applied grad students in the department and building the community. You are encouraged to get involved! It is weekly seminar run by grad students for grad students. If you have any questions, please contact Jim Brunner (jdbrunner (at) math.wisc.edu).

The seminar schedule can be found here. We meet in Van Vleck 901 from 1:00 to 2:00 on Fridays.

## Spring 2016

date | speaker | title |
---|---|---|

February 12 | Jim Brunner | "Chemical reaction networks" |

February 19 | Xiaoqian Xu | "Mixing: A brief introduction from the PDE aspect" |

March 2 | Fan Yang | "Berry phase in quantum mechanics" |

April 1 | Will Mitchell | "An exercise in asymptotics for finding stagnation points in a Stokes flow" |

## Abstracts

Please add your abstracts here.

### Friday, Feb 12: Jim Brunner

"Chemical reaction networks"

Abstract: I will present an introduction to Mass Action Kinetic models of biochemical systems. Chemical reaction network theory draws on both the theory of dynamical systems as well as techniques from algebra, algebraic geometry and graph theory, so hopefully I will be able to convince even the more algebraically minded among us that dynamic models are interesting.

### Friday, Feb 19: Xiaoqian Xu

"Mixing: A brief introduction from the PDE aspect"

Abstract: I will discuss several definitions and examples of mixing, and introduce an interesting conjecture about mixing, and also talk about the way how we can use them for other complicated situations like social life of bacteria and reproducing of corals.

### Friday, April 1: Will Mitchell

"An exercise in asymptotics for finding stagnation points in a Stokes flow"

Abstract: In this two-part talk, I will describe the Stokes flow about a sphere which is held fixed in a background shear flow. The flow and associated tractions are known exactly. We wish to find where the tangential component of the traction vanishes. This leads to some fourth-order polynomial equations which are hard to solve and provides a seque into the second part of the talk, wherein we attempt to solve them approximately using a small parameter argument.

## Spring 2014

date | speaker | title |
---|---|---|

February 3 | Jim Brunner | "Chemical reaction networks" |

February 17 | Peter Mueller | "Optimal swimming and evolution" |

March 3 | Zhennan Zhou | |

April 7 | Will Mitchell | "Pade Approximants: How does your machine compute exp(A), with A a matrix?" |

## Abstracts

Please add your abstracts here.

### Monday, Feb 3: Jim Brunner

"Chemical reaction networks"

Abstract: Jim will be using Jeremy Gunawardena's notes to introduce the topic: http://www.jeremy-gunawardena.com/papers/crnt.pdf and then transition into talking about what Prof. Craciun is looking at.

### Monday, Feb 17: Peter Mueller

"Optimal swimming and evolution"

Abstract: We will be going over Christophe Eloy's paper: "On the best results for undulatory swimming" (https://www.irphe.fr/~eloy/PDF/JFM2013a.pdf).

### Monday, Mar 3: Zhennan Zhou

"Efficient computation of the semi-classical limit of the Schrödinger equation"

Abstract: After looking at previous techniques, we will try using the Gaussian Wave Packet Transform on the semi-classical Schrödinger equation.

## Fall 2013

date | speaker | title |
---|---|---|

September 20 | Peter Mueller | "Fluid dynamics crash course" |

September 27 | Peter Mueller | "Solutions to Stokes flow" |

October 25 | Zhennan Zhou | "Numerical approximation of the Schrodinger equation with the electromagnetic field by the
Hagedorn wave packets" |

November 1 | Zhennan Zhou | Part 2: "Numerical approximation of the Schrodinger equation with the electromagnetic field by the
Hagedorn wave packets" |

November 8 | Will Mitchell | "How do we make a mesh? Two fundamental schemes" |

November 22 | David Dynerman | "Semi-algebraic geometry of common lines" |

## Abstracts

Please add your abstracts here.

### Friday, Sept 20: Peter Mueller

"Fluid dynamics crash course"

Abstract: Deriving fundamental solutions to Stokes flow and using complex variable tricks to solve two-dimensional problems.

### Friday, Sept 27: Peter Mueller

"Solutions to Stokes flow"

Abstract: We will slowly traverse the steps to exactly solve flow past a cylinder (2D) or sphere (3D).

### Friday, Oct 25 and Nov 1: Zhennan Zhou

"Numerical approximation of the Schrodinger equation with the electromagnetic field by the Hagedorn wave packets"

Abstract: In this paper, we approximate the semi-classical Schrodinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics gives the asymptotic solution with error O(eps^{1/2}), where eps is the the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to O( eps^{k/2}), where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when eps is not very small.

### Friday, Nov 8: Will Mitchell

"How do we make a mesh? Two fundamental schemes"

Abstract: Meshing a bounded 2D or 3D region using triangles or tetrahedra is a fundamental problem in numerical mathematics and an area of active research. In this talk I'll discuss two now-classical (although only 10-year-old) algorithms which can succeed in addressing the challenges of irregular boundaries and variable densities. For those wishing to read ahead, see:

1) Persson and Strang, "A simple mesh generator in Matlab," SIAM Review, 2004

2) Du et al, "Constrained centroidal Voronoi tesselations for surfaces," SIAM Journal on Scientific Computing, 2003.

### Friday, Nov 22: David Dynerman

"Semi-algebraic geometry of common lines"

Abstract: Cryo-electron microscopy (cryo-EM) is a technique for discovering the 3D structures of small molecules. To perform this 3D reconstruction a large number of 2D images taken from unknown microscope positions must be correctly positioned back in 3D space. Although these microscope positions are unknown, the common lines of intersection of the image planes can be detected and used in 3D reconstruction. A major difficulty in this process is large amounts of noise in the common line data.

The set of all noiseless common lines forms a semi-algebraic set (a set defined by polynomial equalities and inequalities). We define and describe the geometry of this set, and briefly discuss applications.

## Spring 2013

date | speaker | title |
---|---|---|

February 1 | Bryan Crompton | "The surprising math of cities and corporations" |

February 8 | Peter Mueller | Mandelbrot's TED talk |

February 15 | Jim Brunner | "Logical Models, Polynomial Dynamical Systems, and Iron Metabolism" |

February 22 | Leland Jefferis | Video lecture on intro quantum mechanics + The postulates of quantum mechanics + Spin 1/2 systems |

February 29 | Leland Jefferis | Topics in quantum mechanics: Spin 1/2 systems + Uncertainty relations + Quantum harmonic oscillators + ... |

March 15 | Will Mitchell | FEniCS, my favorite finite element software package |

March 22 | ||

April 5 | Bryan Crompton | TBD |

April 26 | Peter Mueller | Stokeslets, flagella, and stresslet swimmers |

## Abstracts

Please add your abstracts here.

### Friday, Feb 1: Bryan Cromtpon

"The surprising math of cities and corporations"

Abstract: We'll watch Geoffrey West's TED talk and discuss some of the math in his papers.

### Friday, Feb 15: Jim Brunner

"Logical Models, Polynomial Dynamical Systems, and Iron Metabolism"

Abstract: I will introduce logical models and polynomial dynamical systems in the context of a model of iron metabolism in an epithelial cell.

### Friday, Feb 22 & Feb 29: Leland Jefferis

"Topics in Quantum Mechanics"

Abstract: I will introduce the key ideas of quantum mechanics and expose the fascinating mathematical framework behind the theory.

### Friday, Mar 15: Will Mitchell

"FEniCS, my favorite finite element software"

Abstract: The finite element method is mathematically elegant but can be thorny to code from scratch. The free, open-source FEniCS software takes care of the worst implementation details without constraining the freedom of the user to specify methods. I'll review the finite element method and then give some examples of FEniCS code.

### Friday, Apr 6: Bryan Crompton

"Fractional Calculus and the Fractional Diffusion Wave Equation"

Abstract: I'll talk about the equivalent formulations, the Grundwald-Letnikov and Riemann-Liouville, of fractional calculus. I will give some examples of fractional derivatives (and integrals) and then discuss the fundamental solutions to the fractional diffusion wave equation. Derivations will be done non-rigorously.

### Friday, Apr 26: Peter Mueller

"Stokeslets, flagella, and stresslet swimmers"

Abstract: I will be discussing time-dependent swimmers involving stokeslets as an approximation to flagella. We will then approximate the far-field by an oscillating stresslet and discuss some questionable results.