Tuesday, October 22, 2019
Time  Items 

All day 

4pm 
10/22/2019  4:00pm Abstract: Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$. For $l$ a prime, a result of Deligne implies that upon replacing $E$ by a finite extension, we obtain a representation $\rho_l:\mathrm{Gal}(\overline{E}/E)\rightarrow G(\mathbb{Q}_l)$ where $G$ is the Mumford–Tate group of $A$. For $v\nmid l$ a prime of $E$ where $A$ has good reduction, we show that the conjugacy class of $\rho_l(\mathrm{Frob}_v)$ in $G(\mathbb{Q}_l)$ is defined over $\mathbb{Q}$ and is independent of $l$. This is joint work with Mark Kisin. Location:
LOM 205
10/22/2019  4:15pm I will talk about joint work of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface. Location:
DL 431

6pm 
10/22/2019  6:00pm Abstract: A natural integer is a congruent number if it is the area of a right triangle with three rational number sides. It is a problem recorded in an Arab manuscript of the 10th century to find all congruent numbers, which is still open today. In this talk, we will do some experiments on finding congruent numbers, prove that 1 is not a congruent number, and finally speculate a pattern of these numbers. We will also talk about the history and progress of this problem, and its connection with modern number theory. Location:
LOM 206
