## Austin Christian

MS 3969

### Interests

Broadly speaking, I am interested in topology and geometry. Specifically, I'd like to know more about the interactions between contact/symplectic geometry and topology.

Here's some relevant expository writing:

### Teaching

Listed below is my teaching and TA experience.

#### UCLA

All teaching at UCLA is as a teaching associate.
• Fall 2017: MATH 32A (Calculus of Several Variables)
• Summer 2017: MATH 95 (Transition to Upper Division Mathematics)
• Spring 2017: MATH 123 (Foundations of Geometry)
• Winter 2017: MATH 32B (Calculus of Several Variables)
• Fall 2016: MATH 32A (Calculus of Several Variables) and MATH 33A (Linear Algebra and Applications)
• Spring 2016: MATH 32B (Calculus of Several Variables)
• Winter 2016: MATH 31A (Differential and Integral Calculus)
• Fall 2015: MATH 31A (Differential and Integral Calculus)

#### Pepperdine University

Teaching at Pepperdine is as adjunct faculty.
• Summer 2017: MATH 141 (Probability, Linear Systems, and Multivariable Optimization)
• Summer 2016: MATH 141 (Probability, Linear Systems, and Multivariable Optimization)

#### UT Tyler

At UT Tyler I was the instructor for the following courses:
• Spring 2015: MATH 1324, Mathematics for Business and Economics I
• Fall 2014: MATH 1324, Mathematics for Business and Economics I
• Spring 2014: MATH 2113, Calculus I Lab
• Fall 2013: MATH 2113, Calculus I Lab

I'm a third-year graduate student in the mathematics department at UCLA. Before UCLA I studied at the University of Texas at Tyler.

I live in the San Fernando Valley region of Los Angeles with my wife Kristin. My hobbies include cycling, following spectator sports (in particular, the Texas Rangers and Dallas Cowboys), as well as exploring LA with Kristin.

Photo credits: A nice lady at the Grand Canyon (left); Kristin Christian at the Getty Center (right).
Instructor: Mike Hill

Text: Algebraic Topology, Allen Hatcher

Topics: (Co)homology: universal coefficients theorem, cup/cap products, Künneth theorem, Milnor sequence, Poincaré duality, (co)homology with twisted coefficients. Homotopy theory: (co)fibrations, Whitehead's theorem, spectral sequences, Hurewicz's theorem, Steenrod squares.

Instructor: Peter Petersen

Text: Riemannian Geometry, Peter Petersen

Topics: Riemannian metrics, local representations, tensor notions, Lie derivatives, connections and covariant differentiation, Riemannian, sectional, Ricci, and scalar curvatures, geodesics, the exponential map, completeness, the Hopf-Rinow theorem, first and second variations of energy, the Hadamard-Cartan theorem, Synge's theorem, Preissmann's theorem.

Instructor: Ko Honda

Text: Algebraic Topology, Allen Hatcher

Topics: Fundamental groups, Seifert-van Kampen theorem, covering spaces, simplicial, singular, and cellular homology with their various long exact sequences, axioms of homology theories.

Instructor: Terence Tao

Text: An Epsilon of Room, I, Terence Tao

Topics: Uniform boundedness principle, open mapping and closed graph theorems, Stone-Weierstrass theorem, Prokhorov's theorem, interpolation in $L^p$ spaces, Haar measure and the Fourier transform on LCA groups, distribution theory.

Instructor: Alexander Merkurjev

Text: Algebra, Serge Lang

Topics: Hilbert's basis theorem and Nullstellensatz, Dedekind rings, semisimple rings, representation theory of finite groups, simple algebras, Morita equivalences, and Brauer groups.

Instructor: Kefeng Liu

Text: Geometry of Differential Forms, Shigeyuki Morita

Topics: Differential forms and operations on them, the Frobenius theorem, integration of differential forms and Stokes' theorem, de Rham cohomology, singular and simplicial homology, the de Rham theorem, Riemannian metrics, Laplacian and harmonic forms, the Hodge $*$-operator, the Hodge theorem, the Hodge decomposition, Poincaré duality.

Instructor: Mario Bonk

Text: Real Analysis, Gerald Folland

Topics: Radon-Nikodym theorem, $L^p$-$L^q$ duality, Hardy-Littlewood maximal functions, Riesz representation theorem for BLFs on a Hilbert space, Lebesgue differentiation theorem, FTC for Lebesgue integrals, Riesz representation theorem for locally compact Hausdorff spaces, weak and weak-$*$ topologies.

Instructor: Alexander Merkurjev

Text: Algebra, Serge Lang

Topics: Unique factorization domains, localization, factorization of polynomials, free, projective, and injective modules, field theory, Galois theory.

Outline: Here

Instructor: Ko Honda

Text: Differential Topology, Victor Guillemin and Alan Pollack; Ko Honda's notes.

Topics: Topological and smooth manifolds, transversality, oriented intersection theory, Lefshetz fixed point theory, degree theory.

Notes: Here

Instructor: John Garnett

Text: Complex Analysis Notes, Donald E. Marshall

Topics: Analytic and holomorphic functions, contour integration, Cauchy's theorem, conformal maps, residue calculus, Riemann mapping theorem.

Instructor: Sheldon Davis

Text: Real Analysis, Royden and Fitzpatrick, Chapters 17-20

Topics: General measure spaces and integration, $L^p$ spaces, the Riesz Representation Theorem, product measures.

Instructor: Nathan Smith

Text: Introduction to Mathematical Statistics, Hogg, et al., Chapters 6-10

Topics: Maximum likelihood methods, sufficiency, linear models.

Instructor: Sheldon Davis

Text: Real Analysis, Royden and Fitzpatrick, Chapters 1-6.

Topics: Set theory, topology of real numbers, Lebesgue outer measure, the Cantor function, Littlewood's principles, the Lebesgue integral, convergence in measure, differentiation, differentiation of an integral, absolute continuity, convex functions.

Instructor: Nathan Smith

Text: Introduction to Mathematical Statistics, Hogg, et al., Chapters 1-5.

Topics: Probabilities, distributions, important inequalities, multivariate distributions, various special distributions, statistical inferences, convergence in distribution and convergence in probability.

Instructor: Kristen Stagg

Text: Lecture Notes in LIE ALGEBRA, Misra, Chapters 1-7.

Topics: Lie algebras and subalgebras, derivations, ideals, homomorphisms, solvability and Lie's theorem, nilpotency and Engel's theorem, the Jordan-Chevalley decomposition, Cartan's Criterion, the Killing form, semisimplicity.

Instructor: David Milan

Text: Abstract Algebra, Dummit and Foote, Chapters 8-9, 13-14.

Topics: Fields, Euclidean domains, principal ideal domains, unique factorization domains, integral domains, irreducibility criteria, splitting fields, Galois theory.

Instructor: Regan Beckham

Text: Modeling MEMs and NEMs, Pelesko and Bernstein, Chapters 1, 3-8.

Topics: Modeling micro- and nano-electromechanical systems. Thermally driven systems, elastic structures, thermal-elastic systems, electrostatic-elastic systems, magnetically actuated systems.

Supervisor: David Milan

Text: Differentiable Manifolds, Conlon, Chapters 1-3, 5-7, 10-11.

Topics: Topological and smooth manifolds, vector fields, Lie groups and their Lie algebras, covectors, multilinear algebra, differential forms, Riemannian geometry, principal bundles, structure equations.

Instructor: David Milan

Text: Abstract Algebra, Dummit and Foote, Chapters 1-4, 7.

Topics: Groups, subgroups, Lagrange's theorem, quotient groups, the Holder program, group actions, Sylow p-groups, rings.

Instructor: Regan Beckham

Text: Mathematics Applied to Deterministic Problems in the Natural Sciences, Lin and Segel, Chapters 1, 4-5, 6-14.

Topics: Dimensional analysis, regular and singular perturbation theory, field equations, continuum mechanics, heat flow, Fourier analysis.

Instructor: Sheldon Davis

Text: Modern Set Theory, Judith Roitman, Chapters 1-7.

Topics: Theories and models, the Zermelo-Fraenkel axioms, the axiom of choice, the well-ordering principle, Zorn's lemma, Hausdorff maximal principle, infinite numbers, cofinality, the constructible universe, measurable cardinals.