Spring 2015

Prerequisites: 1A.

Syllabus: Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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Prerequisites: Continuation of 10A.

Syllabus: Elementary combinatorics and discrete probability theory. Introduction to difference and differential equations, matrix algebra, and linear system of equations. 

Office: 925 Evans Hall (Bernd Sturmfels)

Office Hours: Monday 8-9 and Thursday 10-11:30

Required Text: Notes are posted on BCourses 

Recommended Reading: K. Rosen: Discrete Mathematics and its Applications.

Grading: Homework (10%), Quizzes (10%), Midterm 1 (20%), Midterm 2 (20%), Final Exam (40%).

Homework: Weekly homework will be posted on the course webpage.

Course Webpage: on bCourses

Prerequisites: 16A.

Syllabus: Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization.

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Prerequisites: Some calculus and physics background will be useful but not required.

Syllabus:

We will meet on ten Tuesdays from 2:10 to 3:30 in Room 891 Evans Hall, from January 27th through April 7th, every week except Spring Break.

Our main focus will be a careful reading and discussion of the book "Geometry, Relativity, and the Fourth Dimension," by Rudolf von Bitter Rucker. The author may be better known to some students as Rudy Rucker, described in Wikipedia as "...an American philosopher mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement." Depending upon student interests, we may look at other work of the author, and/or other material on relativity and higher dimensions (or lower dimensions, as in Flatland, by the repetitively named Edwin Abbott Abbott).   

Students should acquire the required text in time to read and prepare to discuss Chapter 1 at the first meeting.The remaining class meetings will be devoted mostly to chapter by chapter presentations by groups of students.   In past versions of the seminar, students have found it helpful to include multimedia in their presentations.   

Office: 825 Evans Hall

Office Hours: TBA

Required Text: Rudolf v.B. Rucker, Geometry, Relativity, and the Fourth Dimension, Dover Publications, 1977.  (Used copies available online at low price.)

Grading: P/NP

Homework: Preparation of presentations, reading

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Prerequisites: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or UC/CSU diagnostic examination.

Syllabus: Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences.

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Prerequisites: Mathematics 1B.

Syllabus: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites: 1B.

Syllabus: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

For more information on the course content, see the course web page.

Office: 883 Evans

Office Hours: M 12:30–1:30, WF 11–12

Required Text: Lay, Nagle, Saff, & Snider, Linear Algebra and Differential Equations (UC Berkeley custom edition). See the bookstore web site for exact details.

Grading: Grading will be based on a first midterm (10%), a second midterm (20%), the final exam (45%), and results from the discussion section (25%).

Homework: Homework will be assigned weekly, and likely will involve WeBWorK, a web-based system for homework assignments.

Course Webpage: http://math.berkeley.edu/~vojta/54.html

Prerequisites: 1B.

Syllabus: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Syllabus: Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on

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Syllabus: Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on

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Prerequisites: 53 and 54.

Syllabus: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 53 and 54.

Syllabus: Completeness of real numbers. Metric spaces. Open and closed sets. Convergence. Compactness. Connectedness. Continuous functions. Uniform continuity. Uniform convergence. Numerical series and series of functions. Differentiation. Riemann Integral.

Office: 827 Evans

Office Hours: Wednesdays, 10-11am. Otherwise by appointment.

Required Text: Rudin, Principles of Mathematical Analysis.

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Grading: The grade will be based 40% on the Final Exam, 25% on a Midterm, 25% on Quizzes, and 10% on weekly homework.

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Prerequisites: 53 and 54.

Syllabus: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 104.

Syllabus: Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable.

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Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

Office: 927 Evans Hall

Office Hours: MWF 9:00-10:00 927 Evans Hall. (For quick questions ask me before or after class.)

Required Text: Abstract Algebra by Dummit and Foote. (third edition)

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Homework: Assigned each week on the course webpage

Course Webpage: https://bcourses.berkeley.edu/courses/1277697

Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Required Text: J.B. Fraleigh, "Abstract Algebra", 7th Edition

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Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

Office: 1071 Evans Hall

Office Hours: Monday, 9-12am

Required Text: A First Course in Abstract Algebra, John Fraleigh (7th Edition)

Recommended Equipment: A standard 3x3x3 Rubik's cube.

Course Webpage: https://math.berkeley.edu/~libland/teaching/math-113/

bCourses site: https://bcourses.berkeley.edu/courses/1301479

Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: Algebra and “algebraic thinking” permeates all of the modern Mathematics and now is also deeply affecting Mathematical Physics. The course is designed to be a broad and intensive introduction into “algebraic thinking”. The first weeks will be spent on introducing the student to the language of structures, explaining what constitutes a “structure”, and illustrating the corresponding concepts with a number of key examples. From the very beginning we will be building material to illustrate the concepts of a category and a functor. To make the student familiar with the language of categories is one of the aims of this course.

In the second part of the course I will consider a few case studies, demonstrating some fascinating applications of Algebra (for example the proof of a famous result in Logic to the effect that “every tautology has a proof’) and the techniques used in the classification of finite groups (we shall prove, for example, that a nonabelian simple group of order ≤ 60 is canonically isomorphic to the alternating group on a set of 5 elements naturally associated with the group).

Along the way I will be constantly “opening windows” and leading students into various “wings” of the edifice of modern Mathematics. After successfully completing the course, the student will have ample opportunities to develop in various directions.

This course is aimed at an enthusiastic, ambitious and a diligent student. It is expected that the students enrolled in the class will be actively following the lectures. There is no text that could replace them.

Office: 995 Evans Hall

Office Hours: TBA

Required Text: no required text; lectures are a primary source, various sets of notes distributed by the instructor — the secondary source for the material covered in class

Recommended Reading: Mac Lane, Birkhoff, “Algebra” (an excellent text for self study written by two great masters)

Grading: TBA

Homework: assigned and collected regularly throughout the semester; your course grad will directly reflect the quality of your homework.

Course Webpage: TBA

Prerequisites: 53 and 54.

Syllabus: Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images.

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Prerequisites: 53 and 54.

Syllabus: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Special functions, series solutions of ordinary differential equations, partial differential equations arising in mathematical physics, probability theory.

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Prerequisites: 53 and 54.

Syllabus: Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.

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Prerequisites: 53, 54, and 55.

Syllabus: Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.

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Prerequisites: 113.

Syllabus: Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.

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Prerequisites: 151, 152.

Syllabus: The real line and least upper bound, limit and decimal expansion of a number, differentiation and integration, Fundamental Theorem of Calculus, characterizations of sine, cosine, exp, and log.

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Prerequisites: 55.

Syllabus: Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional topics at the discretion of the instructor.

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Prerequisites: 104.

Syllabus: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

Office: 1053 Evans Hall

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Required Text: Complex variables and Applications, Brown and Churchill (9th Edition)

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Prerequisites: 104.

Syllabus: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

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Office: 935 Evans
Office Hours: TBA
Prerequisites:Ability to write proofs and willingness to work on open-ended problems a must. At least one 100 level math course strongly recommended. Math 55 or other exposure to combinatorics required.
Syllabus:Historically, the first topological invariants used by mathematicians were from combinatorial construction. More recently, combinatorial topology has been used to attack problems in knot topology, partially ordered sets, polytopes, and matroids. In class we will develop some of the classical combinatorial techniques used in topology. Possible topics covered in this class include planar graphs, colorability, Kuratowski's theorem, simplicial complexes, combinatorial fixed point theorems, simplicial homology, knot theory, braid groups, knot polynomials, fundamental groups, and additional topics based on student's input. Projects include applications in algebraic combinatorics, algebraic topology, computational topology, differential geometry, and knot theory.

Students will apply their knowledge from class to learn advanced methods independently or work on open ended projects. Each student will have considerable latitude in finding problems that fit his or her interest. Students will work in small groups on these problems, with the majority of this work completed outside of class.
Course Webpage: http://math.berkeley.edu/~jhicks/classes/spring15math191/courseindex.html
Grading: One short and one long project. Weekly progress reports, an in-class presentation, and a final write-up using LaTeX will be required for each project. Exercises will occasionally be assigned.

Prerequisites: Math 185.

Syllabus: This is a second course in complex analysis. Among the many possible choices for such a course, I will focus on conformal mapping, including the Riemann mapping theorem and on a few important classes of special functions.

Office: 783 Evans

Office Hours: TBA

Required Text: No required book. The basis for the course will be the notes you will take in class. Many of the topics covered can be found in the book by Ahlfors.

Recommended Reading: Lars V. Ahlfors "Complex Analysis" (3rd edition)

Grading: homework and participation in the discussions in class.

Homework: there will be homework

Course Webpage: 

Prerequisites: 202A.

Syllabus: Smooth manifolds; tangent space and differentials; immersions and embeddings; transversality and Sard's theorem; vector fields; Lie groups and Lie algebras; flows and the Lie derivative; vector bundles and tensors; Riemannian metrics; differential forms and Stokes' theorem; de Rham cohomology; distributions and foliations; introduction to Morse theory.

Office: 923 Evans

Office Hours: TBA

Required Text: John Lee, Introduction to Smooth Manifolds, 2nd edition

Recommended Reading: M Spivak, A comprehensive introduction to differential geometry, vol. I

Grading: The course grade will be based on peer-graded homework assignments.

Homework: Homework will be assigned weekly.

Course Webpage: Will be linked from http://math.berkeley.edu/~hutching.

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Syllabus: The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Also listed as Statistics C205B.

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Prerequisites: 105 or 202A, as well as 222A

Syllabus:  Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, Hamilton-Jacobi equations and conservation laws

Office: Evans 841

Office Hours: TBA

Required Text: L.C.Evans, Introduction to Partial Differential Equations

Recommended Reading: M.E. Taylor, Partial Differential Equations

Grading: Homework

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Course Webpage:  https://bcourses.berkeley.edu/courses/1303124

Prerequisites: Math 104 (analysis), Math 185 (complex analysis), Math 110 (linear algebra). 224A is not a prerequisite.

Syllabus: The course will survey basic theory and practical methods for solving the fundamental problems of mathematical physics. It is intended for graduate students in applied mathematics, physics, engineering or other mathematical sciences. The overall purpose of the course will be to develop non-numerical tools for understanding and approximating solutions of differential equations. A rough outline for 224B is:

• Linear and Nonlinear Waves (KdV, NLS, the inverse scattering method). (refs 1-2)
• Potential theory, integral equations, compact operators. (refs 3.4)
• Euler equations, potential flow, water waves, conformal mapping methods, stability of traveling waves. (refs 5-6)
• Stokes equations, Lam'e equations, weak solutions, Lax-Milgram theorem, Korn's inequality, saddle-point problems, inf-sup conditions. (refs 7-8)
• Semigroup theory, parabolic equations, porous medium equation. (refs 9-10)
• asymptotic behavior of solutions of ordinary differential equations, WKB theory. (refs 11-12)
• asymptotic behavior of integrals, method of steepest descent, method of stationary phase. (ref 12)
• asymptotics behavior of elliptic systems near corners, Mellin transform. (ref 13)
• singular perturbation theory and boundary layer theory. (refs 14-15)

Office:  1051 Evans

Office Hours: Mon 11:10-11:55, Wed 2:15-3:30

Required Text: I will post excerpts from the following books and papers on bCourses.

Recommended Reading:

1. G. B. Whitham, Linear and Nonlinear Waves
   
2. Novikov, Manakov, Pitaevskii, Zakharov,  Theory of Solitons, The Inverse Scattering Method
3. G. B. Folland,  Partial Differential Equations
4. R. Kress,  Linear Integral Equations 
5. R. S. Johnson,  A Modern Introduction to the Mathematical Theory of Water Waves
6. Dyachenko, Kuznetsov, Spector, Zakharov, Analytic description of the fres surface dynamics of an ideal fluid, Phys. Lett. A, 221:73-79, 1996.
7. L. C. Evans  Partial Differential Equations 
8. D. Braess,  Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics 
9. Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
10. F. Otto,  The Geometry of Dissipative Evolution Equations: the Porous Medium Equation, Comm. Partial Diff. Eq., 26, 101-174, (2001).
11. F. W. J. Olver,  Asymptotics and Special Functions
12. C. Bender and S. Orszag,  Advanced Mathematical Methods for Scientists and Engineers 
13. Kozlov, Maz'ya, Rossmann,  Elliptic Boundary Value Problems in Domains with Point Singularities 
14. Kevorkian and Cole,  Multiple Scale and Singular Perturbation Methods 
15. J. Neu,  Singular Perturbation Theory (unpublished notes)

Grading: 100% homework

Homework: 6-7 assignments, due roughly every two weeks. The lowest homework score will be dropped.

Course Webpage: http://math.berkeley.edu/~wilken/224B.S15, http://bcourses.berkeley.edu

Prerequisites: Math 128A/B or equivalent knowledge of basic numerical analysis, MATLAB or Python programming experience.

Syllabus: Theory and practical methods for numerical solution of partial differential equations. Finite difference methods for parabolic and hyperbolic problems, finite volume methods for hyperbolic conservation laws, finite element methods for elliptic equations, discontinuous Galerkin methods for first and second order systems of conservation laws.

Office: 1089 Evans

Office Hours: TBD

Required Text: 

Recommended Reading:

• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007. ISBN 978-0-898716-29-0.
• R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002. ISBN 978-0521009249
• C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009. ISBN 978-0486469003.

Grading: Problem sets.

Homework: 7 problem sets.

Course Webpage: 

Prerequisites: Graduate standing in Math, MCB, and Computational Biology; or consent of the instructor.

Syllabus: A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof. Also listed as Molecular and Cell Biology C243.

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Prerequisites: 250A.

Course Webpage: https://math.berkeley.edu/~atopaz/spring2015.html (includes a tentative list of topics)

Prerequisites: 256A.

Syllabus: Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

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"An introduction to Gauge Field Theory". Joint course, Lectures by Peter Teichner (first 7 weeks) and Kolya Reshethikin

Prerequisites: Basic geometry, topology and algebra.

Syllabus: Part 1: Introduction to classical field theories in the BV formalism, Part 2: Path integral quantization and Feynman diagrams

Office: 703 (Peter) and 917 (Kolya)

Office Hours: By appointment

First References, more will be provided in class:

Albert Schwarz, "Geometry of Batalin-Vilkovisky Quantization"

Daniel Freed, "Classical Chern-Simons theory, Part 1"

Kevin Costello and Owen Gwilliam, "Factorization Algebras in Quantum Field Theory"

Course Webpage

Prerequisites: Hilbert space operators. Some basic classical probability notions may be useful background, but  will not be assumed.

Syllabus: I will provide an introduction to free probability and to the recent realization that the theory has an extension which may be called bi-free probability or free probability for pairs of faces. [ Free probability is the study of freeness, a highly noncommutative notion of independence. Applications include random matrices and the operator algebras of free groups. There is also a combinatorial side involving non-crossing partitions.]

Office: 783 Evans

Office Hours: TBA

Required Text: there is no required text. You will use the notes you will take in class from the lectures.

Recommended Reading: 

Grading: a few homeworks and participation in the discussions in class will be the basis for the grades.

Homework:  there will be a few homeworks.

Course Webpage: 

Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center.

Syllabus: May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.

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