Michael Ching

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I have taught the following courses at Amherst College:

• Math 111: Introduction to Calculus (Fall 2022, Fall 2019, Fall 2017, Fall 2016, Fall 2015, Fall 2013, Fall 2012, Fall 2011)
• Math 121: Intermediate Calculus (Spring 2012)
• Math 205: Inequality (Spring 2018, Spring 2017)
• Math 211: Multivariable Calculus (Spring 2024, Spring 2023, Fall 2020, Fall 2017, Spring 2015, Spring 2013, Fall 2012, Fall 2011)
• Math 220: Mathematical Reasoning and Proof (Fall 2021, Spring 2021, Fall 2020, Spring 2016)
• Math 256: Combinatorial Geometry (Fall 2023)
• Math 271: Linear Algebra (Spring 2017, Fall 2016, Fall 2015)
• Math 355: Introduction to Analysis (Spring 2019, Spring 2012)
• Math 385: Mathematical Logic (Spring 2022, Fall 2019, Spring 2018)
• Math 410: Galois Theory (Spring 2016, Fall 2013)
• Math 455: Topology (Spring 2019, Spring 2015, Spring 2013)

I have also taught special topics (independent study) courses on:

• Math Education (for pre-service middle school math teachers)
• Game Theory
• Homotopy Type Theory
• Algebraic Topology
• Motivic Homotopy Theory
• Chromatic Homotopy Theory
• Category Theory

Math 111: Introduction to Calculus

This is a one semester introduction to differential and integral calculus. The main topics are:

• Limits: limit laws, the ε/δ definition, basic limit calculations
• The derivative: limit definition, geometric meaning, product/quotient/chain rules, implicit differentiation, related rates
• Maximum and minimum problems, optimization
• The Riemann integral, Fundamental Theorem of Calculus

Math 121: Intermediate Calculus

This is a second-semester calculus course covering:

• Integration techniques: integration by parts, trig and hyperbolic substitution, partial fractions
• Improper integrals, L'Hôpital's Rule, calculating volumes
• Sequences and series: convergence, integral test, comparison test, absolute convergence, root and ratio tests, alternating series
• Power series, radius of convergence, Taylor and MacLaurin series
• Parametric curves and arc length, polar coordinates

Math 205: Inequality

This is course focused on math education at the K-12 level. The course is cross-listed as History 209 and Black Studies 209. We considered some of the following questions:

• What is the purpose of math education?
• What inequality exists between different groups?
• Is the U.S. math education system in crisis?
• What can be done to reduce inequality?

Math 211: Multivariable Calculus

This is a standard course of multivariable calculus covering:

• Coordinate systems and vectors in two and three dimensions, dot and cross products, equations of lines and planes
• Vector-valued functions, their derivatives and integrals, velocity, acceleration, speed, arc length, curvature
• Functions of several variables, their limits and continuity, partial derivatives, tangent planes, differentiability, directional derivatives, gradient
• Extrema of functions of several variables, second-order partial derivatives, constrained extrema problems, Lagrange multipliers
• Iterated integrals in cartesian, polar, cylindrical and spherical coordinates, changing variables in multiple integrals
• Vector fields, path integrals, Green's Theorem, curl and divergence, Stokes' Theorem

Math 220: Mathematical Reasoning and Proof

A standard introduction to logic and proof for prospective math majors. This course was organized around the fundamental mathematical objects of 'number' and 'set'. Topics included

• Quantifiers and proof strategies
• Integers, rationals, real and complex numbers
• Sets, functions, equivalence relations

Math 271: Linear Algebra

This is a proof-based introduction to linear algebra covering:

• Matrices and linear systems, inverse matrices and determinants
• Vector spaces and subspaces
• Linear independence, bases, dimension
• Linear transformations, null space and range, injections, surjections and isomorphisms
• Matrix representations of linear maps and change of basis
• Eigenvalues and eigenvectors
• Diagonalization

Math 355: Introduction to Analysis

This is a proof-based course on mathematical analysis and one of the core requirements of the math major at Amherst. It covers:

• Completeness of the real numbers, rationals and irrationals, mathematical induction
• Sequences and series: convergence, bounded monotone sequences, Cauchy sequences, Bolzano-Weierstrass Theorem
• Sets of real numbers: countability, open and closed sets, compactness, Heine-Borel Theorem
• Functions and continuity, Intermediate Value Theorem, continuous functions on a compact set, uniform continuity
• Derivatives, Rolle's Theorem and Mean Value Theorem
• Riemann integration
• Uniform convergence of sequences and series of functions, continuity of the limit, derivatives and integrals of series, Weierstrass M-Test, power series

Math 385: Mathematical Logic

This is an introduction to first-order logic, including Gödel's Completeness and Incompleteness Theorems. It covers:

• First-order languages: terms, formulas and truth in a structure
• Formal deductive systems: axioms and rules of inference for first-order logic
• The Completeness Theorem for first-order logic
• Gödel Numbering and Incompleteness
• Peano Arithmetic and the Second Incompleteness Theorem

Math 410: Galois Theory

This is an upper-level elective with Math 350 (Groups, Rings and Fields) as a prerequisite. Galois Theory is the study of the beautiful symmetries that appear in the solutions to polynomial equations. The main topics we cover are:

• Polynomials and their roots: the quadratic and cubic formulas, minimal polynomials, symmetric polynomials
• Fields and their extensions: finite and algebraic extensions, splitting fields, algebraic closures, the Fundamental Theorem of Algebra
• Galois extensions: normality and separability, Galois groups, the Galois Correspondence
• Solvable extensions: primitive roots of unity, abelian extensions, insolvability of the general quintic
• Finite fields and their Galois groups
• Constructibility of regular polygons

Math 455: Topology

This is an upper-level elective with Math 355 (Analysis) as a prerequisite that fulfils the upper-level requirement for honors students in the math major. Topology is the study of 'shape'. More precisely, it is concerned with a notion of shape that is invariant under stretching, squashing and bending, but not under cutting or gluing. From this perspective a circle and a square are essentially the same shape, but a line is different. Many objects, both mathematical and otherwise, have an underlying shape that can be captured through the ideas of topology. The main topics are:

• Topological spaces and continuous functions: subspace, quotient and product topologies
• Connectedness and compactness
• Separation axioms and metrizability
• Homotopy and the fundamental group
• Classification of surfaces

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I taught the following classes at the University of Georgia:

• Math 2001: Geometry for Elementary School Teachers (formerly Math 5002)
• Math 2002: Algebra for Elementary School Teachers (formerly Math 5003)
• Math 2003: Arithmetic for Elementary School Teachers (also Math 5001)
• Math 4000: Modern Algsebra and Geometry I (a first course in abstract algebra)
• Math 5020: Arithmetic for Middle School Teachers
• Math 5030: Geometry for Middle School Teachers
• Math 5035: Algebra for Middle School Teachers
• Math 8200: Algebraic Topology (qualifying exam preparation class for graduate students)

I taught the following classes at Johns Hopkins University:

• Math 106: Calculus I for Biological and Social Sciences
• Math 107: Calculus II for Biological and Social Sciences
• Math 202: Calculus III
• Math 211: Honors Multivariable Calculus
• Math 401: Advanced Algebra I
• Math 402: Advanced Algebra II
• Math 413: Introduction to Topology

I was a TA for the following classes at MIT:

• 18.022: Calculus (Multivariable)
• 18.03: Differential Equations
• 18.821: Project Lab in Mathematics