Below is the tentative schedule of speakers for this semester. Titles and abstracts will be updated each week.
August 28 – Organizational Meeting
September 11
Speaker: Billy Duckworth
Title: A Tour of Filters in Analysis
"We will cover the definition of a filter, the two main intuitions behind them, and some basic properties of filters. Then, we will go on a quick, high-level tour of some uses of them in analysis such as simplifying notation, compactifying/completing spaces, and generalizing limits. There will be a quiz later, so take notes!"
September 18
Speaker: Billy Duckworth
Title: The Analysis Extended Universe (part 1)
"A common motif throughout mathematics is passing to extended spaces. If we cannot solve a problem because there is no solution or we cannot find one, we simply expand our search area to make sure that we can. Obviously, these extensions need to be done in a controlled way to give us information about the original problem. That is where monads enter the picture! Ubiquotous in algebra, monads are largely overlooked in analysis even though they are lurking in several of our common constructions. This talk will bring these into the light and show how they help organize results and generate interesting questions."
September 25
Speaker: Micah Coats
Title: Symmetrization on Hyperbolic Spaces
"Schwarz symmetrization of a real-valued function on Euclidean space is a rearrangement that is radially symmetric and decreasing. It is a useful tool for optimization problems and was classically important for proving the isoperimetric inequality. Many results that hold for Euclidean space carry over to hyperbolic space, in particular, Talenti's theorem for rearrangements. We follow the elegant technique outlined by Baernstein and verify that it holds in hyperbolic space. Some possible future directions for the technique will also be discussed."
October 2
Speaker: Scott Blair
Title: Dynamical Methods in Combinatorics
"In additive combinatorics we are concerned with various so-called partition regular properties. These are properties preserved by one part of a finite partition. It turns out that there is an intimate relationship between partition regularity and dynamics in the wide reaching Furstenberg correspondence principle. In this talk we will provide dynamical proofs of van der Waerden's, Szemeredi's, and Hindman's theorems."
October 9
Speaker: Micky Santiago-Zayas
Title: Building on the AEPG Framework
"We propose an extension of adaptive energy-based preconditioned gradient methods to optimization problems with nonlinear equality constraints through a preconditioned gradient descent in a Bregman geometry. Building on the AEPG framework, we introduce a structured approach that integrates geometric constraints through a generalized preconditioning operator and energy-based step size adaptation. We establish stability properties and provide convergence guarantees under mild assumptions."
October 16
Speaker: Khiem Nguyen
Title: When are Continuous Functions Dedekind Complete?
"A poset is called Dedekind complete if every nonempty set with an upper bound has a least upper bound. We will investigate when the poset of continuous, real valued functions on a topological space is Dedekind complete."
October 30
Speaker: Billy Duckworth
Title: Myriad Fixed Points: The Subtle Art of Staying Put
"Fixed point theorems are ubiquitous in mathematics. One of the most famous and useful is Banach's fixed point theorem for contractions on a complete metric space. This talk will explore a constellation of corollaries, variations, and converses of this wonderful result."
November 13
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November 20
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November 27 – Thanksgiving Break
December 4
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December 11
Speaker: Reserved
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