May

3

Tue

**Jason L Greuling Tuesday, May 3rd, 3pm** Keller 302

Title: A Discrete Regge Complex

This paper is interested in a differential complex that arises in the study of finite element methods for certain partial differential equations from Riemannian geometry involving curvature. Specifically, we derive a two dimensional Regge complex that includes the linearized curvature operator, curlTcurl, and relate it to a complex that includes the linearized discrete notion of curvature on a triangulation. We establish a correspondence between the two complexes, giving a relationship between the two linearized maps.

May

12

Thu

**Hien Ha Thursday ****May 12, 3pm Keller ****302**

Tile: p-adic numbers

Abstract: The field of real numbers is completed from the field of rational numbers with respect to the distance metric. The heart of the completion process is the limit of Cauchy sequences. Recall that a metric space (X, d) is complete if every Cauchy sequence in X converges to a point in X. We know that Q is not complete with respect to the distance metric. For example, the Cauchy sequence of rational numbers 3 , 31/10 , 314 /100 , 3141/ 1000 …. is converging to π which is not in Q . Filling all convergent points of Cauchy sequences we get R. The field of p-adic numbers Qp is also completed from Q with respect to a different metric called the p-adic metric which is induced from the p-adic norm. We also make use of the convergence of the Cauchy sequences for this process. In this project we will see how Qp is completed from Q with respect to the p-adic norm. We also describe two trees associated with Qp: the trees of balls in the field Qp and the trees of lattices in the vector space Qp × Qp.

## University of Hawaiʻi at Mānoa