Monday, February 3, 2014 at 4:00-4:50pm

Location: Weber 237

Title: Ramanujan graphs and error correcting codes

Abstract: While many of the classical error correcting codes are cyclic, a long standing conjecture asserts that there are no 'good' cyclic codes. In recent years the interest in symmetric codes has been promoted by Kaufman, Sudan, Wigderson and others (where symmetric means that the acting group can be any group). Answering their main question (and in contrary to the common expectation), we show that there DO exist symmetric good codes. In fact, our codes satisfy all the "golden standards" of coding theory. Our construction is based on the Ramanujan graphs constructed by Lubotzky-Samuels-Vishne as a special case of Ramanujan complexes. The crucial point is that these graphs are edge transitive and not just vertex transitive as in previous constructions of Ramanujan graphs. All notions will be explained. This is joint work with Tali Kaufman.

*(Reception prior to talk in 117 Weber from 3:30-4:00pm)*

Location: TILT 221

**Title:** Real applications of non real numbers

**Abstract**: Number theoretic considerations led mathematicians over a century ago to introduce the "field of p-adic numbers", which is just like the "field of real numbers" a completion of the familiar "field of rational numbers". This abstract system of numbers has found in the last 3 decades some unexpected applications in computer science and engineering. We will explain the basic ideas and some of these applications.

*(Reception following talk in 117 Weber from 5:00-6:30pm*)

Wednesday, February 5, 2014 at 4:00-4:50pm

Location: Weber 237

**Title:** From Ramanujan graphs to Ramanujan complexes

Abstract: Ramanujan graphs are optimal expanders (from a spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms. The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings. This gives finite simplical complexes, which on one hand are "random like", and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will describe these developments and give some details on recent applications. The work of a number of authors will be surveyed. Our works in these directions are in collaboration with various subsets of {S. Evra, K. Golubev, T. Kaufman, D. Kazhdan, R. Meshulam, S. Mozes, B. Samuels, U. Vishne}.

*(Reception prior to talk in 117 Weber from 3:30-4:00pm)*

The Arne Magnus Lectures are given annually in the Department of Mathematics at Colorado State University in honor of Dr. Arne Magnus, our friend and colleague for 25 years.

The lectures are supported by the Arne Magnus Lecture Fund and the Albert C. Yates Endowment in Mathematics.

Contributions to the Magnus Fund are greatly appreciated and may be made through the Department of Mathematics. Please contact Sheri Hofeling (hofeling@math.colostate.edu) at at (970)-491-7047 for specific information.

All lectures are free and open to the public.