Basic Notions Seminars

Spring 2019

All talks are from 1200--1300 in the Seminar room, unless otherwise specified.

The Basic Notions seminar features ideas which can be viewed as fundamental to some area of mathematics, or fundamental to a connection between areas of mathematics, or fundamental to an application of mathematics. The definition of "mathematics", "connection", and "application", is broad in this context. Perhaps "Fundamental Notions" is the correct synonymous title. Talks are aimed at a "graduate-level", though frequently the "basic" notion in the talk is an example of, or hints at, deep mathematical content.

• Apr

  30
• Positive Definite Kernels: An Introduction for Machine Learning Applications

  Nick Wood

  USNA Math

  Time: 12:00 PM

  View Abstract

  Positive definite kernels are advantageous in machine learning applications for at least two reasons. First, they allow us to use linear methods to generate non-linear decision boundaries. Second, they provide a general framework which can be used for data in any form, e.g. text, image, graph, etc. In this talk, we will show how positive definite kernels can be used in machine learning with two examples. With these examples as motivation, we will then define positive definite kernels, provide methods for proving a kernel is positive definite, and finally we will give examples and proofs showing that several kernels are positive definite.

• Apr

  23
• Part II: Representations of Operator Algebras and the Gelfand-Naimark-Segal Theorem

  Mitch Baker

  USNA Math

  Time: 12:00 PM

  View Abstract

  This shortened session (we will dismiss at 1240 for administrative reasons beside our pay grade) will be a continuation of Professor Baker's talk from 09 April 2019; also in the Basic Notions seminar.

• Apr

  16
• Mathematical Existence: A Philosophical Discussion

  Darren Creutz

  USNA Math

  Time: 12:00 PM

  View Abstract

  "Everyone knows" that ZFC is the foundation of modern mathematics, yet virtually no one actually knows what those axioms are (except choice of course which is the one most often questioned, albeit erroneously). The reality of the situation is more nuanced: we all know what a proof is in that we recognize it when we see it. But do we really all agree? The issue centers on the meaning of "there exists" and the various interpretations that can be given. I will present an overview of the various schools of mathematical philosophy, and their accompanying notions of proof, ranging from the classical logic ZFC approach to constructivism/intuitionism (it only exists if you can construct it) to the bizarre ultrafinitism (there is a largest number) over to game formalism (proofs are all a game of symbol pushing). Holding some not-so-standard views myself, my intent is for the "talk" to be much more a guided group discussion rather than a traditional talk. All are welcome, no background (other than having done some mathematics) will be assumed.

• Apr

  09
• Representations of Operator Algebras and the Gelfand-Naimark-Segal Theorem

  Mitch Baker

  USNA Math

  Time: 12:00 PM

  View Abstract

  We introduce the notion of a C*-algebra, and linear functionals thereon. We then explore the relationship between positive linear functionals on such algebras, and representations of these algebras on Hilbert spaces - via the famous and beautiful Gelfand-Naimark-Segal Theorem. A few surprising connections between representations of certain group-invariant C*-algebras and representations of infinite-dimensional Lie Groups will be mentioned at the end of the talk.

• Mar

  26
• Binomial Proofs and the Area Principle

  Will Traves

  USNA Math

  Time: 12:00 PM

  View Abstract

  I'll discuss some elementary geometric ideas which were new to me when I first heard about them several years ago. In particular, I'll explain the role that determinants play in plane geometry and how these ideas lead to an elegant proof technique. Along the way, we'll see why determinants deserve their name, encounter the Fundamental Theorem of Invariant Theory, and learn an important application of Cramer's rule.

• Mar

  19
• Data-Driven Methods for Prediction and Control of Complex Systems

  Evelyn Lunasin

  USNA Math

  Time: 12:00 PM

  View Abstract

  In the first part of the talk I will discuss a few examples of how one integrates data into PDE models in order to accurately predict and control complex systems. In the second part of the talk I will give an overview of an emerging data driven modeling strategy which utilizes classical tools from linear algebra – the Singular Value Decomposition (SVD). Considered as the most impactful theorem in data science, SVD is an essential starting point to characterize complex fluid flows purely from collected measured data and with no governing equations needed.

• Jan

  22
• How discontinuous can an integrable function be? -- An intro to measure theory as the natural extension of calculus.

  Darren Creutz

  USNA Math

  Time: 12:00 PM

  View Abstract

  One of the few problems Riemann could not solve was a question about his own integrals: if f(x) is a bounded function on [a,b], when does the integral from a to b of f(x) dx exist (when are the lower and upper sums equal)? If f is continuous then the integral exists, and indeed f can have jump discontinuities. But if f is discontinuous at every point on a subinterval then the integral won't exist, so where is the line? Can an integrable function have infinitely many discontinuities? Could it be discontinuous on something as large as a Cantor set? I will answer these questions by presenting Lebesgue's notion of measurable sets, developed, initially, precisely to answer Riemann's question. No background beyond calculus will be assumed. If you haven't seen measure theory before, or it's just been a very long time, and you'd be interested in knowing what us analysts are doing, please join.