Algebra Seminar

Ari Shnidman, Temple University

During the spring semester, we will run an "arXiv seminar",  with talks on papers from the last five years or so in any area of algebra/number theory. Papers will be exposited over 1-3 talks with the first talks devoted to the background material. This will be a short organizational meeting. We will go over some ground rules and then people can volunteer or request papers/topics. The organizers are happy to help choose a topic and prepare for the talk.

Violet Nguyen, Temple University

In this talk, we will walk through a new elementary proof of a theorem of William Chen concerning Markoff triples. After this short proof, we will discuss applications to Nielsen equivalences on $SL_{F_p}$ and to generalized Markoff-like surfaces.

Vasily Dolgushev, Temple University

The main motivation to explore the Grothendieck-Teichmueller group GT and its versions comes from the connection of GT to the absolute Galois group $G_{\mathbb{Q}}$ of rational numbers. In my talk, I will recall the definition of GT and say a few words about the Ihara embedding from $G_{\mathbb{Q}}$ to GT. I will talk about interesting versions of GT and formulate known
results about the versions of this group. Finally, I will say a few words about open questions related to GT.

Sean O'Donnell, Temple University

The goal of this talk is to give a description of the action of the absolute Galois group of the rational numbers on the set of child's drawings. We give several definitions of child's drawings, discuss why they are categorically equivalent, and give examples to illustrate the action of the Galois group on each.

Vasily Dolgushev, Temple University

Sage (or SageMath) is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography and numerical computation. Although the syntax of Sage is similar to that of Python, it runs much faster than Python with its libraries (e.g. SymPy). During this Zoom session, I will show examples of using Sage in algebra and number theory. (I plan to hit the record button in the beginning of the session.) Here is the Zoom link for our session: 
https://temple.zoom.us/j/95281041454

Holly Miller, Temple University

In this third talk on the GT group, we introduce GT-shadows. After a reminder on profinite completions of groups, we recall the definition of the gentle version of the GT group as a subgroup of automorphisms of the profinite completion of the free group on two generators. It is natural to ask if the gentle GT group is itself profinite, and GT-shadows are discussed through the perspective that comes with this question. Along the way, a few open questions in the area will be mentioned.

Holly Miller, Temple University

This talk covers the action of GT-shadows on child's drawings. We start by defining this action and that of the gentle GT-group on child's drawings, remarking on their compatibility. Following this, we describe the "hierarchy of orbits" to motivate the study of this action. The remainder of the time will be spent on the accessible details underpinning the theory.

Ari Shnidman, Temple University

The BSD conjecture is one of the Clay millennium problems and a central open question in number theory. It predicts that the number of independent rational points on an elliptic curve is determined by the asymptotic growth of the point counts mod p via the Hasse-Weil L-function. I'll go over the history and precise formulation of the conjecture, and what is currently known. This is background information for the next two talks by Ani and Coco.

Aniruddha Sudarshan, Temple University

Selmer groups are used to study ranks of ellptic curves. Selmer groups are defined to be a subgroup of a certain first cohomology group whose elements satisfy some local conditions. For any integer m, we define the m-Selmer group using the Kummer maps. We then stick to the case when m=2, and perform a 2-descent to prove the Weak Mordell-Weil theorem for certain elliptic curves. If time permits, we talk about Selmer groups of twists of elliptic curves.

Xiaoyu Huang, Temple University

Recent work of Burungale, Skinner, Tian, and Wan established the first infinite families of quadratic twists of non-CM elliptic curves over Q for which the strong Birch-Swinnerton-Dyer conjecture is known to hold. In this talk, I will describe how their criteria can be described as an explicit algorithm and apply it to the elliptic curve data in the L-functions and Modular Forms Database (LMFDB), thereby identifying curves of conductor at most 500,000 that admit infinitely many quadratic twists satisfying strong BSD. I will also present numerical evidence related to a conjecture of Radziwill and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, along with a systematic positive bias within the BSD-satisfying subfamily.

Aditya Sarma Phukon, Temple University

This is the first of two talks based on a paper on normal bundles on rational curves in Grassmannians by Coskun, Larson and Vogt. We shall introduce vector bundles on a manifold as locally free sheaves and talk about short exact sequences of sheaves. Thereafter we will talk about line bundles and then prove Birkhoff-Grothendieck's theorem which says that every vector bundle on $\mathbb{P}^1$ splits into line bundles. We then define a j-balanced vector bundle and present three very simple lemmas about these bundles. Finally, we shall introduce rational curves and thereafter Grassmannians and state a few lemmas about normal bundles. These surprisingly simple tools will be used to finally prove the main theorem in the next talk.

Aditya Sarma Phukon, Temple University

This is the second talk based on a paper on normal bundles on rational curves in Grassmannians by Coskun, Larson and Vogt. This talk will focus largely on specialized machinery required for the proof. We introduce the notions of generalized pointing bundles and modified vector bundles. We shall state some propositions and maybe sketch their proofs before discussing a technique called one-secant degeneration. This will be used to track how vector bundles deform as the base curve deforms. Finally, we shall recall the various lemmas we saw in the previous talk and piece together the proof of the main theorem which is about how the particular normal bundles in question are 2-balanced. We might end on some remarks on implications.