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Mathematics Department

Basic Notions Seminar

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 "basicnotion in the talk is an example of, or hints at, deep mathematical content.

  • Mar
  • Binomial Proofs and the Area Principle
    Will Traves
    USNA Math
    Time: 12:00 PM
  • Feb
  • TBA
    Evelyn Lunasin
    USNA Math
    Time: 12:00 PM
  • Jan
  • 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.
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