Joel H. Shapiro

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Whatever happened to l^p for p < 1? Notes for a lecture in the Portland State Analysis Seminar on strange happenings inside the sequence space l^p for p<1.

Sperner's Lemma and Brouwer's Fixed-Point Theorem. Notes for a lecture in the Portland State Analysis Seminar on the proof of the Brouwer Fixed-Point Theorem via Sperner's Lemma.

Paradoxical Decompositions from Galileo to Banach & Tarski. Notes for lectures I gave in the Portland State Analysis Seminar on paradoxical decompositions, with special emphasis on the Banach-Tarski Theorem.

Burnside's Theorem on Matrix Algebras. Notes for a lecture I gave in the Portland State Analysis Seminar, presenting Lomonosov and Rosenthal's 2004 "simplest proof" of Burnside's famous 1905 theorem.

The Invariant Subspace Problem. Notes for several lectures I gave at the Portland State Analysis Seminar on the Invariant Subspace Problem, with special emphasis on the 1972 blockbuster paper of Victor Lomonosov.

Equilibrium for non-cooperative games. Notes for a talk presented at the Portland State University Analysis Seminar, October 18, 2013. They provide a brief introduction to the theory of non-cooperative games, with emphasis on the notion of Nash Equilibrium, and the proof of its existence by means of the Brouwer Fixed Point Theorem.

Fixated on Fixed points. Periodically updated notes based on lectures given by participants in the 2012-13 Analysis Seminar at Portland State University.

The Schroeder-Bernstein Theorem via fixed points. Notes based on a lecture given by John Erdman in the Analysis Seminar at Portland State University, showing how the fixed point theorem of Knaster and Tarski implies the Schroeder-Bernstein Theorem.

Notes on Oscillatory Integrals. Notes based on a lecture I gave in the Analysis Seminar at Portland State under the title "What your calculus book doesn't tell you about oscillatory integrals."

What your calculus book doesn't tell you about Taylor series. Notes based on a couple of lectures I gave in the Analysis Seminar at Portland State.

Algebraic Fredholm Theory. Notes which develop the purely algebraic aspects of the theory of Fredholm operators. Based on a couple of lectures I gave in the Portland State Univ. Analysis Seminar.

Which linear fractional transformations induce rotations of the sphere?
Notes to supplement the material on linear fractional maps presented in a first year graduate
course. You find here a proof that a LFT induces, through stereographic
projection, a rotation of the Riemann sphere if and only if it can be
represented by a *unitary* (two by two) coefficient matrix.

Notes on the numerical range. These lecture notes set out the basic properties of the numerical range of a Hilbert space operator, most notably the Toeplitz-Hausdorff Theorem which shows that the numerical range is always a convex set. Along the way, the numerical ranges of two by two matrices are characterized. A proof is given of Hildebrandt's theorem, which states that if you intersect the closures of the numerical ranges of all operators similar to an operator T, what results is the closed convex hull of the spectrum of T.

Notes on the dynamics of linear operators. Notes complementing lectures I gave in June 2001 at the Universities of Rome, Florence, and Padua in Italy. They discuss the notion of hypercyclicity for linear operators (some orbit is dense), paying particular attention to its origins in non-linear dynamics, and its connections with analytic function theory and functional analysis.

The Arzela-Ascoli Theorem.
Notes to supplement the discussion of
normal families and the Riemann Mapping Theorem in a first-year
graduate course in complex variables

Notes on connectivity. Supplementary notes for a first-year graduate course in complex variables. The notes develop basic facts about connectivity and components in metric spaces, with emphasis on Euclidean space.

Notes on Differentiation. Further supplementary notes for a first-year graduate course in complex variables. Quick review of real differentiation in several variables, Conformality of stereographic projection, application to Mercator map projection.

A Gentle Introduction to Composition Operators. Lecture notes for the first half of a course on composition operators that I gave at the University of Padua (Italy) in June 1998.

Nonmeasurable sets and paradoxical decompositions. Supplementary notes for a first-year graduate course in real analysis.

Cyclic decomposition of a nilpotent matrix. Supplementary notes for the discusssion of Jordan Canonical form in a senior-level course in linear algebra.

Quadratic extension fields and geometric impossibilities. Supplementary notes for a first course in abstract algebra at the junior-senior level.

Cyclic inner functions in Bergman spaces. An exposition of the ``Korenblum-Roberts Theorem'' characterizing inner functions that are cyclic for the forward shift on Bergman spaces.