## Greenslopes Seminar
[Department of Mathematics] |

For the seminar attendance sheet, click here.

Abstracts appear below.

Date | Speaker | Title | Advisor |

Aug 31 | Vance Blankers | Impossible Things and Where to Find them | Renzo Cavalieri |

Sep 7 | Sam Pine | 7 Things Engineers Want You to Know About Radar | Margaret Cheney |

Sep 14 | Andy Fry | Tropic(al) Thuder | Renzo Cavalieri |

Sep 21 | Wes Galbraith | Ions Bombardment and Patterns | Patrick Shipman |

Sep 28 | Sophia Potoczak and Zhouran Wang | Practice Talks for SIAM regional conference | Olivier Pinuad/ James Liu |

Oct 5 | Ben Sencindiver/Gabby Hernandez | What is Mathematics Education Research, What It Looks Like, and Why We Need It | Mary Pilgrim/Jess Ellis |

Oct 12 | Dean Bisogno | Abhyankar's Inertia Conjecture for the Group Mathieu 11 | Rachel Pries |

Oct 19 | Derek Handwerk | Conservation laws, the finite volume method, and the population balance equation. | Patrick Shipman |

Oct 26 | Catalina Comacho | Finding Rational Points on Curves | Rachel Pries |

Nov 2 | No Greenslopes - Math Day | ||

Nov 9 | Karleigh Cameron | Dan Bates | |

Nov 16 | Javier Alvarez | General Relativity: the Importance of Conceptual Understanding in Science | Chris Peterson |

Nov 23 | No Greenslopes - Fall Recess | ||

Nov 30 | |||

Dec 7 |

** **

** 8/31/2017 - Impossible Things and Where to Find them - Vance Blankers**

In this decidedly non-research talk, we'll discuss a handful of impossible straight-edge-and-compass constructions from antiquity, including trisecting the angle and squaring the circle. Establishing impossibility in these cases offers an introduction to Galois theory and the constructible numbers.

** 9/7/2017 - 7 Things Engineers Want You to Know About Radar - Sam Pine**

In accordance with the click-bait title, we'll cover just the basics of synthetic aperture radar (SAR). Starting from Maxwell's equations, we describe SAR as an inverse problem in which we attempt to reconstruct the reflectivity of a scene from measurements of its scattered electric field. Many of the open problems in SAR research are mathematical in nature, but the overwhelming majority of papers are written by engineers, for engineers. This purpose of this talk is to bridge that gap, following the book by Cheney and Borden.

**9/14/2017 - Tropic(al) Thunder -Andy Fry**

In this talk we will go over the basic definitions of tropical geometry and we will see what happens when algebraic geometry meets combinatorics! (Hint: it gets tropical.) We will see that by tropicalizing algebraic curves we obtain combinatorial tools that help with certain computations.

**9/21/2017 - Pattern formation during Ion bombardment experiments **

We extend a model of Bradley and Shipman for the normal incidence ion bombardment of binary compounds to account for the effects of competing instabilities caused by the Bradley-Harper effect and ion-induced phase separation. Through a weakly nonlinear analysis and numerical integrations, we find that the phase difference between surface height and composition may serve as a useful indicator of which physical mechanism primarily contributes to the formation of experimentally observed ripples and nanodot arrays. Our numerical simulations also suggest the existence of travelling wave solutions to our equations in the phase separation regime.

** 9/28/2017 - A Numerical Method for the Quantum Liouville-BGK Equation- Sophia Potoczak**

This talk will introduce a numerical scheme for solving the quantum Liouville-BGK equation, where the equilibrium in the relaxation term is obtained by minimizing a free energy functional under local constraints . The equation forms a system of coupled nonlinear PDEs and models the dynamics of a many-particle system. The numerical difficulties lie in the nonlinear coupling between the PDEs, and are addressed by using a splitting scheme in which the nonlinearity becomes a perfectly linear term. We will discuss the specifics of the scheme and present related numerical results.

** 9/28/2017 - The lowest-order weak Galerkin finite element method for Darcy
equation on general polygonal meshes Zhouran Wang **

In this talk, we present the lowest-order weak Galerkin (WG)
finite element method for solving the Darcy equation on general convex
polygonal meshes. In this approach, we approximate the pressure by
constants inside elements and on edges. The discrete weak gradients of
these constant basis functions are defined in an H(div)-subspace that
is constructed by normalized coordinates and Wachspress coordinates. This
new WG method is locally mass-conservative, and produces continuous normal
fluxes. Numerical results are presented to demonstrate its 1st order
accuracy in pressure, velocity, and flux.

**10/5/2017 - What is Mathematics Education Research, What It Looks Like, and Why We Need It **

This talk will give an overview of what mathematics education research is. We'll talk about theories on learning and how this affects research. The main two flavors (quantitative and qualitative research) will be discussed, and exemplary articles from both areas will be presented highlighting their analysis and results.... which is our research, obvi ..... and others, too.

**10/12/2017 - Abhyankar's Inertia Conjecture for the Group Mathieu 11 **

In topology we learn that the complex line is simply connected so it has a trivial fundamental group. Galois groups of unramified covers of the affine line correspond to quotients of the fundamental group. Consequently, there are no nontrivial unramified covers of the complex line. Zariski and Abhyankar investigated what happens when the complex numbers are replaced by a field with positive characteristic. In this case, there are nontrivial unramified covers; meaning that the fundamental group of the affine line in positive characteristic is far from simply connected. In this talk, we will study the inverse Galois problem for the group Mathieu 11 in characteristic 11. This talk is given in preparation for my MS defense. Even if the content is distant from your own work, I would appreciate feedback on any aspects of the presentation.

**10/19/2017 - Conservation laws, the finite volume method, and the population balance equation. **

I'll derive some common PDEs from conservation laws and see how this process leads easily into a numerical scheme, the finite volume method. This method is commonly used to solve poplulation balance equations which are in turn used to model things like nucleation and growth in colloidal substances.

** 10/26/2017 - Finding Rational Points on Curves **

By the Mordell-Weil conjecture, later proved by Faltings, the set of rational points over a curve of genus >1 is finite. However, Faltings proof is not constructive, so here we will review some methods on how tofind or at least bound the number of rational points, one of the main ones being the Chabauty-Coleman method.

** 11/9/2017 **

**11/16/2017 - General Relativity: the Importance of Conceptual Understanding in Science **

Formulated in terms of pseudo-Riemannian geometry, and regarded by many as the most beautiful of all physical theories, Relativity describes the most mysterious of the four fundamental forces of Nature by identifying gravitation with the curvature tensor of space-time. However the theory cannot be obtained a posteriori as a model fitting empirical data, only as a conceptual principled theory from physical and geometric arguments. We will review the importance of Philosophy in Science, formulate the theory mathematically, and compare its empirical success to the current crisis in fundamental theoretical physics, where experimentally unsuccessful “theories” dangerously disregard conceptual understanding in favor of "only" mathematical beauty.

** 11/30/2017 **

** 12/7/2017 **

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