Coursework
Below are some notes I wrote in form of coursework.
General Algebra (Math 8201/8202)
Topology and Manifolds (Math 8301/8302)
Complex Analysis (Math 8701/8702)
Lie Groups and Lie Algebras (Math 8271/8272)
Calculus of Variations and Minimal Surfaces (Math 8385/8386)
Theory of PDE (Math 8583)
Topics in PDE (Math 8590)
Differential Equations and Dynamical Systems (Math 8501/8502)
Theory of Probability and Measure Theory (Math 8651/8652)
Functional Analysis (Math 8801/8802)
Mathematical Fluid Mechanics (Math 8431/8432)
Begining French (French 4001/4002)
The textbook is Lang "Algebra" Revised 3rd Edition.
| Lang, Problems 7,9 (page 75) and additional problems |
Solution |
| Lang, Problems 24,25,26,28,30 (page 77) and additional problems |
Solution |
| Lang, Problem 8 (page 115) and additional problems |
Solution |
| Lang, Problems 1,2 (page 114-115) and additional problems |
Solution |
| Lang, Problems 5,8,9,10,18 (page 213) and additional problems |
Solution |
| Lang, Problems 1,2,3,4(b),14,15 (page 165-169) |
Solution |
| Lang, Problems 9,10,12 (page 546) and 13,14,16,22 (page 568-570) |
Solution |
| Lang, Problems 3,7,8,10,20,26 (page 253-256) and an additional problem |
Solution |
| Lang, Problems 1,7,8,9,10,15,18 (page 320-323) |
Solution |
| Lang, Problems 3,7 (page 353); 1,3 (page 374) and additional problems |
Solution |
The textbooks are
John Lee "Introduction to Topological Manifolds" 2nd Edition,
John Lee "Introduction to Smooth Manifolds".
The textbooks are
Ahlfors "Complex Analysis" 3rd Edition,
Miranda "Algebraic Curves and Riemann Surfaces".
Ahlfors, Problems 3,4 (page 9); 1,4 (page 11); 1,5 (page 17); 1,2 (page 20); 2,4,7 (page 28)
| Solution |
Ahlfors, Problems 2,3,6 (page 32); 3,5 (page 37); 2,4,8,9 (page 41); 1 (page 44); 5,6 (page 47)
| Solution |
Ahlfors, Problems 1,7 (page 53); 3,4 (page 58); 3,4 (page 63); 1,3 (page 66); 1 (page 72)
| Solution |
Ahlfors, Problems 1 (page 78); 1,4 (page 80); 2,4,7 (page 82); 1,3 (page 84); 2,4 (page 88);
2,6,7 (page 96)
| Solution |
Ahlfors, Problems 1,2,3,4,7 (page 108); 1,3 (page 120); 1,2,3 (page 123); 2,3 (page 130)
| Solution |
Ahlfors, Problems 1,2,3 (page 133); 1,2,3,4,5 (page 136)
| Solution |
Two problems
| Solution |
Ahlfors, Problems 1 b,e,f (page 161); 3 c,d,e,g,h (page 161) and additional problems
| Solution |
Ahlfors, Problem 4 (page 186); 1,2 (page 190); 1,3 (page 193)
| Solution |
Ahlfors, Problems 1,3 (page 198); 1,2,3 (page 200); 2,3 (page 206)
| Solution |
Math 8702: Homework 1 |
Solution |
Ahlfors, Chapter 6, Exercises 6.1.1, 6.1.2 (page 232)
and additional problems |
Solution |
Ahlfors, Chapter 6, Exercises 6.2.3, 6.2.5, 6.2.6 (page 238)
and additional problems |
Solution |
Ahlfors, Chapter 6, Exercises 6.4.1, 6.4.2, 6.4.3, 6.4.5 (pages 247-248) |
Solution |
Ahlfors, Chapter 7, Exercises 7.3.1, 7.3.2(pages 274-275) and Exercises 7.3.3 1-6
(pages 276-277) and additional problems |
Solution |
R. Miranda, Chapter I, Exercises I.1.G, I.2.C, I.2.J, I.3.A, I.3.C, I.3.E |
Solution |
R. Miranda, Chapter II, Exercises II.1.C, II.3.I, II.3.J, II.4.D, II.4.G, II.4.K |
Solution |
| Take home final |
Solution |
The textbook is Goodman-Wallach "Symmetry, Representations, and Invariants".
Goodman-Wallach, Problems 2,3,4 (page 11)
| Solution |
Goodman-Wallach, Problems 1 (page 34); 2 (page 106); 1 (page 145)
| Solution |
Goodman-Wallach, Problems 1 (page 234); 1 (page 254); 2 (page 255); 1 (page 277); 1 (page 339); 1 (page 396); 1 (page 490); 1 (page 550)
| Solution |
The textbook is Jurgen Jost & Xianqing Li-Jost "Calculus of Variations", 1998.
There is no textbook used exclusively in the course, but rather a series of chapters by Professor Mikhail Safonov.
There is no textbook used in the course, but rather a series of lectures by Professor Vladimir Sverak.
Roughly speaking, the topics being discussed fall into 3 following papers.
Leray "On the Motion of a Viscous Liquid Filling Space", 1934.
Kato "Strong L^p Solutions of the Navier-Stokes Equation in R^m, with Application to Weak Solutions", 1984.
Jia-Sverak "Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions", 2012.
There is no textbook used exclusively in the course. The lectures by Professor Arnd Scheel range over various topics. Some of the topics and corresponding reference books are:
Theory of ODE: Hartman "Ordinary Diferrential Equations", 2nd eddition.
Bifurcation theory: Chow-Hale, "Methods of Bifurcation Theory".
Lyapunov-Schmidt reduction: Zeidler "Nonlinear Functional Analysis and Its Applications I".
Center manifolds, normal forms: Shub, "Global Stability of Dynamical Systems".
Circle homeomorphisms, horseshoe, kneading theory: Katok-Hesselblatt, "Introduction to Modern Theory of Dynamical Systems".
The class uses the textbook by Fristedt - Gray "A Modern Method to Probability Theory" in addition to a series of lectures by Professor Nicolai Krylov.
There is no textbook used in the course. The class follows a series of lectures by Professor Vladimir Sverak. The subject is presented with an emphasis on compact operators, Fredholm operators, and their applications in PDE. Some background in group representation on a Hilbert space is also introduced.
There is no textbook used in the course. The class follows a series of lectures by Professor Vladimir Sverak.
The equations of motion of a fluid are derived. The class also goes in depth into the regularity theory of the Navier-Stokes equations and turbulence theory.
I was not enrolled in this class, but took notes of the lectures carefully.
Here are some translation efforts made after I took the classes.
French translation of a few sections of the paper Jia-Sverak "Minimal L^3-initial data for potential Navier-Stokes singularities" 2013.
English translation of the paper Gallagher-Iftimie-Planchon "Non-explosion en temps grand et stabilité de solutions globales des équations de Navier-Stokes" 2002.
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