We begin to chapter 4 (Dp-ranks) of Pierre Simon's notes.
Let G be a finitely generated group of polynomially bounded growth. We follow Kleiner's proof that G has an infinite homomorphic image with a faithful finite-dimensional representation. This is the main step of the proof.
Given a genus g curve, we can associate an abelian variety to it called the Jacobian of the curve. Using the Riemann relations established last week, we will prove these Jacobians are indeed abelian varieties. Time permitting I will go into more depth regarding the case where g = 1.
The word Jacobian also occurs in vector calculus. Is this a coincidence? Rob thinks so.
In recent work of Ford and Wooley, considerable progress has been achieved in the diagonal behaviour in VMVT. We will begin my two talks with the conjecture and best known results in this front. In the remainder, we will sketch their proof of the main theorem.
Throughout the spring 2013 term, we will (as a group) be reading through and lecturing on "The Geometry of Yang-Mills Fields" by Sir Michael Atiyah. All are welcome to attend.
We go over the material in section 2.2 of Pierre Simon’s notes again with an eye toward the o-minimal case.
We continue with Kleiner’s proof of Gromov’s thorem.
After Tys scintillating introduction to Jacobian varieties over the complex numbers, I will discuss the algebraic properties of Jacobians, with particular reference to number fields.
The study of both symplectic and complex geometry began hundreds of years ago, and have impacted many areas of mathematics and physics. Generalized complex geometry is a relatively new area of mathematics that encompasses both symplectic and complex geometry. In this talk we will examine some background of these areas and look at the structures on vector spaces and how they relate to each other.
In recent work of Ford and Wooley, considerable progress has been achieved in the diagonal behaviour in VMVT. In the first talk we discussed with the conjecture and best known results. In the second talk, we will sketch their proof of the main theorem.