Join us for a relaxing game night to take a break from studying/midterms. And, of course, there will be free pizza!

The rational points on an elliptic curve ($y^2 = x^3 + Ax + B$) form a finitely generated abelian group. Heuristics imply that 50% of elliptic curves have rank 0 and 50% have rank 1. For example, this follows from Poonen-Rains' heuristics for the dimensions of the p-Selmer groups of elliptic curves, for all primes p. Bhargava-Shankar computed the first moment of these distributions for p = 2,3,5 and recently Bhargava-Shankar-Swaminathan gave an upper bound on the second moment when p = 2. I'll give a leisurely explanation of the definitions and the methods, and then describe work in progress with Bhargava-Ho-Swaminathan where we prove something close to the full Poonen-Rains distribution for p = 2 in certain large families of elliptic curves.  

Violet Nguyen, Temple University

In this talk, we continue with Section 2 of Armand Borel's "Linear Algebraic Groups." We start by defining the group closure of a subset of a $k$-group $G$ and listing many properties of this operation. Using a result of R. Baer, we then extend the notion of solvability and nilpotence to $k$-groups.

This week we'll be having a movie night, watching and discussing a math video (there are several options, topic tbd). And, of course, there will be free pizza!

Sean O'Donnell, Temple University

Continuing from last week's talk, we will start by covering some of the necessary pre-requisite algebraic geometry, primarily focusing on k-structures when the field k is not algebraically closed. We will define algebraic groups and their morphisms, sketch proofs for some basic results and review a couple of particularly important examples. We will then discuss actions of algebraic groups on varieties and go over a result analogous to Cayley's Theorem in group theory.

Abstract: A codimension-1 submanifold embedded in a symplectic manifold is called “contact type” if it satisfies a certain convexity condition with respect to the symplectic structure. Given a symplectic manifold $X$ it is natural to ask which manifolds $Y$ can arise as contact type hypersurfaces. We consider this question in dimension 4, which appears much more constrained than higher dimensions; in particular we review evidence that no homology 3-sphere can arise as a contact type hypersurface in $\mathbb{R}^4$ except the 3-sphere. We exhibit an obstruction for a contact 3-manifold to embed in certain closed symplectic 4-manifolds as the boundary of a Weinstein domain — a slightly stronger condition than contact type — and explore consequences for the symplectic topology of small rational surfaces and potential applications to smooth 4-dimensional topology.

The morning introductory talk (at 11:00) will review symplectic structures, symplectic convexity, and the related notion of pseudoconvexity, together with some aspects of “embedding questions” for 3-manifolds in $\mathbb{R}^4$ or other 4-manifolds.

Abstract: If $L$ is a link in a 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations? In this talk, I will discuss the ziggurat phenomenon: if one restricts to foliations transverse to a fixed flow on the link complement, the set of multislopes typically has a fractal staircase shape with rational corners. In work in progress with Thomas Massoni, we explain the ziggurat phenomenon in some contexts using tools from contact geometry.

In the morning background talk (at 9:30), I'll introduce my two favorite codimension 1 structures on 3-manifolds — contact structures and foliations —  and the Eliashberg—Thurston theorem which relates them.

Vasily Dolgushev, Temple University

Let K be an algebraically closed field and k be a subfield of K. I will give a brief review of varieties and their morphisms over K. I will talk about k-structures on varieties and introduce morphisms of varieties defined over the smaller field k. If time permits, I will also talk about the functor of points. I will follow Borel's presentation of this material from Chapter "AG" in his book "Linear Algebraic Groups".  

We consider the convergence rate of a random non-divergence form difference equation on $\mathbb{Z}^d$ to its "effective" differential equation on $\mathbb{R}^d$. We will discuss the optimal convergence rate when the coefficient field has a finite range of dependence. Moreover, when the coefficient field is i.i.d. and exhibits isotropic symmetry, we prove strictly faster convergence, improving the generic finite-range rate. Joint work with Hung V. Tran (Wisconsin) and Timo Sprekler (Texas A&M).