Sep

14

Thu

Title: A Simple Proof of a Theorem of Woodin

Abstract: In a similar spirit as my talk last semester about computing

and non-standard models, I will relay Joel David Hamkins’ new proof of a

theorem of Woodin: that there is a function that enumerates any finite

set (if computed in the correct model M of arithmetic), and which can

enumerate any extension of that set (if run in the correct end-extension

of M).

Sep

15

Fri

Speaker: Monique Chyba

Title: Is control theory loosing control?

Abstract: We live in an era of exciting scientific advances such as discovering new planets and black holes far away in the universe or gaining a better understanding of our own biological system. Unsurprisingly, mathematics plays a dominant role in almost all of them. Control theory models, analyzes and synthesizes the behavior of dynamical systems. Those systems are described by sets of ordinary differential equations that include an additional parameter referred to as the ‘control’. It can be viewed as the ship’s wheel of the system in analogy to the navigation of a boat. A vast area of work takes place in optimal control theory. Indeed, since by using different controls we can achieve the same goal, optimization with respect to a given cost such as energy or time becomes a primary interest. I will present three specific examples to illustrate the field of control theory and its current limitations that call for an innovative way of thinking.

Sep

21

Thu

Title: Measure-Risking Arguments in Recursion Theory

Abstract: By way of introducing the idea of measure-risking, I will present a proof of Kurtz’s theorem that the Turing upward closure of the set of 1-generic reals is of full Lebesgue measure. Then I will show how a stronger form of the theorem (due originally to Kautz) can be obtained by framing the proof as a “fireworks argument”, following a recent paper of Bienvenu and Patey.

Sep

29

Fri

Speaker: Rufus Willett

Title: Positive curvature and index theory.

Abstract: Starting with two-dimensional surfaces, I’ll introduce positive (scalar) curvature. I’ll then discuss the relationship of this to index theory, a theory that counts the number of solutions to certain partial differential equations. Finally, I’ll mention the relevance of K-theory, a way of generalizing the notion of dimension of a vector space from fields to arbitrary rings.

## University of Hawaiʻi at Mānoa