# 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.