Logic Seminar: David Ross @ Keller Hall 303
Oct 6 @ 1:30 pm – Oct 6 @ 2:30 pm

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.”)

Colloquium: David C. Webb @ Keller 401
Oct 7 @ 2:30 pm – 3:30 pm

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.

Colloquium: Pamela Harris (Williams)
Oct 7 @ 3:30 pm – 4:30 pm
Colloquium: Victor Donnay (Bryn Mawr College) @ Keller 401
Oct 14 @ 3:30 pm – 4:30 pm

Speaker: Victor Donnay, William Kenan, Jr Professor of Mathematics and Director, Environmental Studies program, Bryn Mawr College

Title: Connecting Math and Sustainability

Abstract: How can we better inspire our students to study and succeed in mathematics? Victor Donnay will discuss his experiences in using issues of civic engagement, particularly environmental sustainability, as a motivator. He will present a variety of ways to incorporate issues of sustainability into math and science classes ranging from easy to adapt extensions of standard homework problems to more elaborate service learning projects. He will share some of the educational resources that he helped collect as chair of the planning committee for Mathematics Awareness Month 2013- the Mathematics of Sustainability as well as his TED-Ed video on Tipping Points and Climate Change. He has used these approaches in a variety of courses including Calculus, Differential Equations, Mathematical Modeling and Senior Seminar.

André Nies: A gentle introduction to randomness and $K$-triviality
Oct 18 @ 1:30 pm – Oct 18 @ 2:20 pm

Keller 303
Introduction for grad students and other beginners, as a prequel to Thursday’s talk.

Analysis Seminar: Thomas Hangelbroek @ Keller 313
Oct 19 @ 3:30 pm – 4:30 pm

Title: A curiosity of the trace operator II

Abstract: I’ll discuss the regularity of the trace operator on various smoothness spaces. In short, this is an operator which reduces (roughly) smoothness as measured in L_p by an order of 1/p. Its behavior on Sobolev spaces, especially on the Hilbert spaces W_2^s (i.e., where smoothness is measured in L_2), plays a critical role in approximation theory when boundaries are present, and in the stability, regularity and existence results for weak solutions for PDEs.

Previously I presented a somewhat negative results: that the trace operator from W_2^(1/2) to L_2 is not bounded (although it is bounded from W_2^{s+1/2} to W_2^s when s>0. In this talk, I’ll explain how a minor correction (a so-called “curiosity” according to Hans Triebel) works – namely, by reducing the domain to a certain Besov space and using results from the atomic decomposition of these spaces.

Logic Seminar: André Nies
Oct 20 @ 1:30 pm – Oct 20 @ 2:30 pm

Speaker: André Nies (University of Auckland), Fellow of the Royal Society of New Zealand

Title: Structure within the class of $K$-trivial sets

Abstract: The $K$-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity $K$ grows as slowly as possible. Since 2002, many alternative characterisations of this class have been found: properties such as low for $K$, low for Martin-Löf (ML) randomness, and basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially the class looked quite amorphous. More recently, internal structure has been found. Bienvenu et al. (JEMS 2016) showed that there is a “smart” $K$-trivial set, in the sense that any ML-random computing it must compute all $K$-trivials. Greenberg, Miller and Nies (submitted) showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches using cost functions.

The talk gives an overview and ends with open questions (of which there remain many).

Location: Keller Hall 303

Grad student seminar (all welcome) – Michelle Manes @ Keller 301
Oct 21 @ 12:30 pm – 1:30 pm

Hilbert’s third problem, scissors congruence, and the Dehn invariant