Felipe A. Ramírez
present
Graduate Analysis: Measure Theory and
Functional Analysis
| problem set (solutions/guidance) | due |
|---|---|
| Stein–Shakarchi, Exercises 2.16, 2.18, 2.19 | February 9 |
| Stein–Shakarchi, Exercises 6.1, 6.2, 6.3 | February 16 |
| Stein–Shakarchi, Exercises 6.13, 6.14, 6.16 | Feburary 23 |
| Stein–Shakarchi, Exercises 3.4, 3.7, 3.8, 3.9 | March 6 |
| Stein–Shakarchi, Exercises 6.10, 6.11 | March 30 |
| Stein–Shakarchi, Exercises 4.1, 4.4, 4.5, 4.6, 4.10, 4.11 | April 20 |
| Stein–Shakarchi, Exercises 4.15, 4.18, 4.19, 4.20, 4.22, 4.29, 4.30 | May 4 |
| day(s) | topic(s) |
|---|---|
| 1 | Snow day! |
| 2 | Recollections; Fubini's theorem and applications |
| 3 | More applications and discussion of proof |
| 4 | Proof of Fubini; abstract measure spaces |
| 5 | Outer measures; premeasures |
| 6 | Algebras and premeasures |
| 7 | Integration, general Fubini |
| 8 | Lebesgue–Stieltjes measures, Borel measures |
| 9 | Snow day! |
| 10 | Lebesgue differentiation theorem, Lebesgue density theorem |
| 11 | Proof of Lebesgue differentiation theorem |
| 12 | Absolute continuity, signed measures |
| 13 | Hahn decomposition; Radon–Nikodym theorem |
| 14 | L2 is a Hilbert space |
| 15 | Hilbert spaces, examples, orthonormal sets |
| 16 | Gram–Schmidt; unitary mappings; projections |
| 17 | Orthogonal complement, orthogonal projection; operators |
| 18 | Bounded operators; functionals |
| 19 | Adjoint operators |
| 20 | Compact operators |
| 21 | Spectral theorem |
| 22 | Banach spaces |
Advanced Topics in Analysis: Ergodic
Theory
| day(s) | topic(s) (notes and questions) |
|---|---|
| 1 | Basic examples and motivation |
| 2 | More examples and motivation |
| 3 | Measure spaces |
| 4 | More about measures; null sets |
| 5 | Tent map; Definition of measure |
| 6 | Nonmeasurable sets; mixing |
| 7 | Leading digits of \(2^n\); more mixing |
| 8 | Recurrence |
| 9 | Szemeredi's theorem |
| 10 | Ergodicity |
| 11 | Equivalent formulations |
| 12 | Ergodicity of circle expanding maps |
| 13 | Measurable functions, invariant functions |
| 14 | Ergodicity and invariant functions |
| 15 | More functional analysis |
| 16 | Back to measurable functions |
| 17 | Integration |
| 18 | Interchanging limits and integrals |
| 19 | Some limit theorems |
| 20 | Continued fractions, Gauss map |
past
| year | fall | spring |
|---|---|---|
| 2025/26 | Probability Theory; Differential Equations |
Measure Theory and Functional Analysis (graduate); Advanced Topics in Real Analysis |
| 2024/25 | Complex and Real Analysis (graduate); Fundamentals of Analysis |
Sabbatical |
| 2023/24 | Fundamentals of Analysis; Elementary Statistics |
Advanced Topics in Real Analysis; Fractal Geometry (graduate) |
| 2022/23 | Sabbatical | Measure Theory and Functional Analysis (graduate); Differential Equations |
| 2021/22 | Probability Theory; Discrete Mathematics |
Metric Number Theory (graduate); Calculus II |
| 2020/21 | Complex Analysis; Fractal Geometry (graduate) |
Multivariable Calculus; Calculus II |
| 2019/20 | Advanced Topics in Real Analysis; Complex and Real Analysis (graduate) |
Fundamentals of Analysis; Calculus II |
| 2018/19 | Sabbatical | Functional Analysis (graduate); Differential Equations |
| 2017/18 | Probability Theory | Measure Theory (graduate); Fundamentals of Analysis |
| 2016/17 | Multivariable Calculus; Complex Analysis (graduate) |
Differential Equations; Vectors and Matrices |
| 2015/16 | Probability Theory; Dynamics and Diophantine Approximation (graduate topics) |
Fundamentals of Analysis; Vectors and Matrices |