# Analysis

This page contains a list of ideas for DRP projects, but is by no means exhaustive.

Topic: Hilbert Spaces
Suggested Text: Hilbert Space: Compact Operators and the Trace Theorem, J. R. Retherford
Suggested Background: MATH 3100 (Introduction to Analysis), MATH 2600 (Linear Algebra)
Description: In a few words, Hilbert spaces are complete inner product spaces. In the same way that the theory of vector spaces is extremely nice (the dimension of a vector space determines that vector space up to linear isomorphism), the theory of Hilbert spaces is similarly nice. Likewise, Hilbert spaces have diverse applications in both mathematics and physics.

Topic: Irrationality and Transcendence
Suggested Text: An Introduction to Number Theory, Ivan Niven, Herbert Zuckerman, & Hugh Montgomery, and Transcendental Numbers, M. Ram Murty & Purusottam Rath
Suggested Background: MATH 3300 (Abstract Algebra) is a prerequisite for the more advanced transcendence theory material
Description: Irrational numbers are real numbers which cannot be expressed as the quotient $$n/m$$ of two integers (where $$m$$ is non-zero). Transcendental numbers are real numbers which are not the roots of any polynomial of a single-variable whose coefficients are rational. Famously, both $$e$$ and $$\pi$$ are transcendental, but proving this is extremely non-trivial. In general, transcendentality is difficult to prove (it is known that at least one of $$e+\pi$$ and $$e\cdot \pi$$ are transcendental, but neither has been proven so). However, it ends up that almost all real numbers are transcendental!

Topic: Space-Filling Curves
Suggested Text: Space-Filling Curves, Hans Sagan
Suggested Background: MATH 3200 (Introduction to Topology)
Description: In the 1800s, Georg Cantor showed that the unit interval and unit square are in a one-to-one correspondence, astounding the mathematical community. Such a bijection, however, cannot be continuous. Weakening the hypothesis that such a function from the unit interval to the unit square is only surjective, there are in fact continuous such functions. These are ‘space-filling curves’. More generally, the Hahn-Mazurkiewicz Theorem states that a non-empty Hausdorff topological space is a continuous image of the unit interval if and only if it is compact, connected, locally-connected, and second-countable

Topic: Infinitesimals and Hyperreals
Suggested Text: Foundations of Infinitesimal Calculus, H. Jerome Keisler
Suggested Background: MATH 3100 (Introduction to Analysis)
Description: In its infancy, calculus was described in terms of ‘infinitesimals’, which were non-zero quantities which were smaller in magnitude than any real number. Using this intuition, limits, continuity, differentiation, and integration were developed and studied. As time wore on, a distrust of such notions grew and eventually was replaced with the modern approach ushered in by Weierstrass and Cauchy. In the mid 1900s, however, Abraham Robinson showed that infinitesimals could be placed on a rigorous foundation, bringing back the old motivations in the form of non-standard analysis.

Topic: $$p$$-adic Analysis
Suggested Text: $$p$$-adic Numbers – An Introduction, Fernando Q. Gouvea
Suggested Background: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
Description: $$p$$-adic analysis represents a different approach to correcting the failings of the field of rational numbers, with a resulting theory that looks wildly different from the classical analysis of the real line. The $$p$$-adic reals (for each prime $$p$$) are nevertheless rich objects of study, both from an analytic point of view, as well as an algebraic one.