PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: In the morning background talk (10am in Park 337), I will provide an introduction to symplectic embeddings. I will explain the background to the symplectic embedding problem and some basic results about embedding four dimensional ellipsoids.

In the afternoon research talk (2:30pm in Park 300), I will discuss calculations of the embedding capacity function for ellipsoids into balls and related target spaces.

Abstract: When you want to solve a Diophantine equation in practice, one of the most powerful modern techniques is the Chabauty—Kim method, which proceeds by constructing p-adic functions vanishing on the rational solutions of your equation. In this talk, I will discuss a heuristic which predicts when these functions cut out exactly the set of rational points. As we will see, the naive version of the heuristic is not quite correct, since there can occasionally be extra "unexpected" algebraic irrational points where the functions vanish (in accordance with a conjecture of Stoll). I will outline what is known about these unexpected points, including some ongoing work with Jennifer Balakrishnan.

Abstract: Heuristics of Cohen, Lenstra, Martinet, Malle and others predict the distribution of the p-part of the class group in families of number fields with prescribed degree and Galois group G, at least when p does not divide the order of G. Results in this direction are few and far between, most notably the results of Davenport--Heilbronn and Bhargava. We give a new data point by computing the average size of the 2-torsion in the class group of the cubic fields Q(n^{1/3}). Along the way, we prove a "reciprocity law" for 2-torsion ideal classes in general cubic fields, which indicates how the existing heuristics should be adjusted when p = 2. This is joint work with Artane Siad.

Abstract: When you want to solve a Diophantine equation in practice, one of the most powerful modern techniques is the Chabauty—Kim method, which proceeds by constructing p-adic functions vanishing on the rational solutions of your equation. This talk will focus on giving specific examples where we can understand these functions quite explicitly. First, we will explain how to produce abelian integrals vanishing on rational points: the so-called Chabauty—Coleman method. Then we will explain one example going beyond the abelian level: the dilogarithm for once-punctured elliptic curves. This latter is the historical starting point of the quadratic Chabauty method.

Abstract: Selmer groups are a cohomological tool for finding rational points on elliptic curves and abelian varieties in general. I'll explain how they work in down to earth terms and give some applications to concrete questions such as: What is the average rank of an elliptic curve over Q? How many integers can be written as a sum of two rational cubes? Is there an algorithm to determine solubility of diophantine equations over a given finitely generated ring? This is also setup for the afternoon talk, where we will apply the Selmer framework to study class groups of number fields.

Join us for a talk by Alyssa Kovalchick from the Department of Chemistry at Zdilla Laboratory. And, of course, there will be free pizza!

Abstract: The structural analysis technique X-ray Crystallography has revolutionized the solid-state scientific community in recent decades. Advances to said technique in the world of biology and chemistry have catapulted innovation, allowing for more effective medications, energy sources, etc. X-ray crystallography is based in higher-level physics and mathematical applications, allowing for ease of structure solutions that are accessible to a trained user. In this talk, discussions of these mathematical roots of X-ray crystallography will be discussed, including the coordinate system utilized and the physical explanation for the output of data. Illustrations of the use of the complex plane and its importance to structure solution will be outlined through examples and explanations, ultimately informing the use of applied Fourier theory in an effort to create a structural model of the desired compounds. 

We construct a model of the feedback mechanisms that regulate the abundance of ribosomes in E.coli, a prototypical prokaryotic organism. The translation process contains an important feedback loop: ribosomes are made up of proteins, which need to be translated by ribosomes. The model accounts for the main feedback loops that control abundance of ribosomes in response to external conditions. It includes the concentrations of free ribosomes, ribosomal RNA (rRNA), and ribosomal proteins. We include direct negative feedback loops where ribosomal proteins, when in excess, slow down their own translation. The effect of the signaling molecule ppGpp is also included as a negative feedback mechanism, along with the effect of the abundance of building blocks for mRNA and rRNA synthesis. The model consists of a system of six differential equations parameterized by 23 parameters. An equlibrium analysis shows that for all values of parameters, the model system has either one equilibrium S, or two equilibria S and P in the biologically feasible region of parameters.

Aniruddha Sudarshan, Temple University

We go over results of Section 1.4 from the book "Cohomology of Number Fields" by Neukirch, Schmidt and Wingberg. We start with the basic definitions of cup-product and their functorial properties. If we have time, we end with a result on cohomology of cyclic groups.

Join us for a special event where faculty will discuss the upper-level courses they are teaching in the spring. Advisors will be available to answer questions about the program and scheduling. And, of course, there will be free pizza!

Abstract: Given a Hilbert modular form $f$ of weight two over a totally real field $F$, we can associate to it a finite module $\Phi(f)$ known as the congruence module for $f$, which measures the congruences that $f$ satisfies with other forms. When $f$ is transferred to a quaternionic modular form $f_D$ over a quaternion algebra $D$ via the Jacquet-Langlands correspondence, we can similarly define a congruence module $\Phi(f_D)$ for $f_D$. Pollack and Weston proposed a quantitative relationship between the sizes of $\Phi(f)$ and $\Phi(f_D)$, expressed in terms of invariants associated to $f$ and $D$.

In this talk, I will outline the ideas underlying the proof of this relationship. The approach combines a method of Ribet and Takahashi with new techniques introduced by Böckle, Khare, and Manning.