This seminar features expository talks on topics in analysis that reflect the participants' interests, but often don’t get into courses. In previous years we’ve heard talks on functional, complex, and harmonic analysis, probability, and dynamical systems. The atmosphere is relaxed and supportive; everyone is welcome to participate: give a talk, or just be part of the audience.

Friday, November 16 in East Hall Room 236: 2 PM – 3 PM

*Joel Shapiro* will speak on:

A.E. Convergence and Maximal Functions (you can’t have one without the other)

Abstract. Suppose you have a sequence \((T_n)\) of linear transformations taking a Banach space \(B\) into the space of functions measurable for some sigma-algebra. A famous theorem of Banach tells us that a certain type of uniform estimate on themaximal function\[ T^*f=\sup_nT_nf \qquad (f\in B) \] characterizes those operator sequences \((T_n)\) for which \((T_nf)\) converges a.e. for each \(f\in B\).In this talk I’ll explain Banach’s Principle, and show how it lurks behind the proofs of both the Lebegue Differentiation Theroem and the Birkhoff Ergodic Theorem.

*Notes* for this talk are here.

Friday, November 9 in East Hall Room 236: 2 PM – 3 PM

*Peter Oberly* will speak on:

The Pointwise Ergodic Theorem and its Applications

Abstract. At the intersection of dynamical systems and measure theory, the pointwise ergodic theorem is a powerful tool with applications ranging from statistical mechanics to number theory. In this talk we will outline a proof of the theorem, and then illustrate its utility by proving Borel’s theorem on normal numbers. If time permits, we will use the ergodic theorem to derive the law of large numbers.

*Notes* to accompany this talk are here.

Friday, November 2 in East Hall Room 236: 2 PM – 3 PM

*Prof. Mau Nam Nguyen* will speak on:

Smoothing Techniques and Applications to Optimization

Abstract. I will present smoothing techniques based on the Fenchel conjugate of convex functions with applications to a number of optimization problems in facility location, machine learning and image reconstructions.

Friday, October 26 in East Hall Room 236: 2 PM – 3 PM

*Pieter VandenBerge* will speak on:

The Hartman-Grobman Theorem

Abstract. Numerous important phenomena in the real world—such as predator-prey relationships, atmospheric dynamics and electrical signals—can be effectively modeled with systems of non-linear ordinary differential equations. However unlike their linear counterparts, such systems may be impossible to solve analytically. They may nevertheless be “locally equivalent” to linear versions of themselves. The Hartman-Grobman Theorem tells us when such “linearization” is possible.In this talk, we’ll explore this theorem in depth, quantify what it means for two systems to be “locally equivalent,” and look at counter-examples that show linearization is not always possible.

Friday, October 12 & 19 in East Hall Room 236: 2 PM – 3 PM

*Logan Fox* will speak on:

An Introduction to the Hausdorff Measure

Abstract. This talk will cover the definition and basic properties of the Hausdorff measure; show the uniqueness of the Hausdorff dimension of a set; and prove that the sigma-algebra generated by the Hausdorff measure includes the Borel sets. This is followed by the result that, given the appropriate correction factor, the n-dimensional Hausdorff and Lebesgue measures agree on Borel sets. And, given time, the Hausdorff dimension and measure of the Cantor set will be explored as an example.

*Notes* to accompany these talks are here.

Friday, September 28 in East Hall Room 236: 2 PM – 3 PM

*Joel Shapiro* will speak on:

The Basel Problem Made

~~Easy~~ Complex

Abstract. Complex analysis offers powerful tools that can be applied to problems in real analysis. To illustrate, I’ll show how complex methods can be used to evaluate \[ \sum_{n=1}^\infty \frac{1}{n^{2k}} \] for each positive integer k. For k=2 this is theBasel Problempopularized by Jacob Bernoulli 1689, and solved in 1735 by Leonhard Euler, who went on to discover the summation formula for general even powers.

*Prerequisite*: The arithmetic of complex numbers. All else will be explained.

*Slides* for this talk are here.