Title: On the Radon Transform

Abstract: In this talk we will introduce, motivate and discuss some of the elementary properties of the Radon transform operator. We will cover a little about how to invert this operator via its adjoint (i.e. back-projection operator) and approximate identities.

This week Mushfeq Khan is continuing his

seminar from 2 weeks ago on

“Turing degrees and Muchnik degrees of recursively bounded DNR functions”.

Summary:

This talk is based on a forthcoming paper by Steve Simpson. It contains

some results that shed light on a part of the Muchnik lattice that remains

poorly understood: the various degrees of recursively bounded DNR functions

obtained by varying the recursive bound.

Keller Hall 303

TITLE: The Algebraic Approach to Determining the Complexity of Constraint Satisfaction Problems

SPEAKER: William DeMeo

ABSTRACT: The “CSP-dichotomy conjecture” of Feder and Vardi asserts that every constraint satisfaction problem (CSP) is in P or is NP-complete. Sometime around the late 1990′s it was observed that a CSP is naturally associated with a general (universal) algebra via a certain Galois connection, and that this connection makes it possible to use algebraic methods to determine the complexity class of a CSP. This led to the “algebraic CSP-dichotomy conjecture” which, after more than a decade of substantial research, has been reduced to the following conjecture: the CSP associated with a finite idempotent algebra A is tractable if and only if the variety generated by A has a Taylor term.

In this talk we try to give most of the background required to understand the algebraic approach to CSP. We give some concrete examples that demonstrate how one uses properties of a universal algebra to determine the complexity class of a CSP. If time permits, we will highlight some of our most recent discoveries that have helped resolve the dichotomy conjecture for most idempotent varieties.

The talk should be fairly self-contained for anyone greater than or equal to a math or cs graduate student. Roughly speaking, if you have heard of the complexity classes P and NP and if you know what a universal algebra is, then you should understand most of this talk.

TITLE: Asymptotic Fixed Points, Part I.

ABSTRACT: By a standard exercise, a uniform contraction on a complete metric

space has a fixed point. Easy examples show that this fails if the

contraction is not uniform. I will discuss some recent results where the

uniform property is replaced by asymptotic conditions on the contraction.

These results are most easily framed and proved using methods from

nonstandard analysis, so I will also briefly brief review those methods. If

there is interest, I will actually prove some of the fixed-point results in

a second seminar (hence the “Part I.”)

Speaker: David Webb (Associate Professor of Mathematics Education Executive Director, Freudenthal Institute US University of Colorado Boulder School of Education)

Title: Infusing Active Learning Design Principles in the Undergraduate Calculus Sequence

Abstract: This interactive presentation will provide a brief overview of undergraduate mathematics at the University of Colorado Boulder, and related activities that we and other universities have designed and used in the freshman pre-calculus through Calculus 2 sequence. Using principles of Active Learning, students are encouraged to conjecture, explore and communicate their reasoning in the process of solving mathematics problems. Underlying this approach is research that has demonstrated how students who are involved in active learning techniques can learn more effectively in their classes, resulting in lower DFW rates, increased persistence in subsequent courses, and improved dispositions towards mathematics.