Spring 2020

Prerequisites 1A or N1A

Description 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

Description The sequence Math 10A, Math 10B is intended for majors in the life sciences. Elementary combinatorics and discrete and continuous probability theory. Representation of data, statistical models and testing. Sequences and applications of linear algebra.

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Prerequisites Three years of high school math, including trigonometry. Consult the mathematics department for details

Description This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions.

Exams: Please note this section of Math 16A has an evening midterm on Wednesday, April 1, between 7 and 9pm.  Do not schedule conflicting activities at this time.

Office 785 Evans

Office Hours TBD

Required Text: 

Calculus with Applications
Lial, Greenwell, and Ritchey 
11th edition, ISBN: 9780321979421

Use other editions at your own risk. Please note the department has negotiated a greatly reduced price for this textbook, so the campus bookstore will most likely be the cheapest place to buy it and possibly to rent it. The campus bookstore price is about $95, so you should not pay more than that. You are NOT required to have an online access code.

Homework: Lots

Instructor's Webpage: https://math.berkeley.edu/~talaska/index.php

Prerequisites 16A

Description 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 Three years of high school mathematics

Description 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 or N1B

Description 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, N1B, 10B, or N10B

Description Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to partial differential equations.

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Prerequisites Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended

Description Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

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Prerequisites 

Description Directed Group Study, topics vary with instructor.

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Description 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 field trips to campus resources. Students are not required to be declared majors in order to participate.

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Prerequisites 53 and 54

Description 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

Description 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

Description 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

Description 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

Description 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

Description 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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Prerequisites 54 or a course with equivalent linear algebra content

Description 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 1055 Evans

Office Hours Tuesday-Thursday 12:40-2:10 PM

Required Text John B. Fraleigh, A First Course in Abstract Algebra, 7th edition

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Course Webpage https://math.berkeley.edu/~art/S20-Math-113.html

Prerequisites 54 or a course with equivalent linear algebra content

Description 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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Prerequisites 54 or a course with equivalent linear algebra content

Description 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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Prerequisites 54 or a course with equivalent linear algebra content

Description 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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Prerequisites 110 and 113, or consent of instructor

Description Further topics on groups, rings, and fields not covered in Math 113. Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields.

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Prerequisites 53 and 54

Description 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

Description Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform.

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Prerequisites 110 and 128A

Description Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer.

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Prerequisites 53, 54, and 55

Description 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

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

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Prerequisites 53, 54, and 113

Description History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.

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Prerequisites 104

Description 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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Prerequisites 104

Description 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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Prerequisites 104

Description 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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Prerequisites 104

Description 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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Prerequisites 104

Description Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

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Prerequisites 

Description 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 field trips to campus resources. Students are not required to be declared majors in order to participate.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 

Description 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 field trips to campus resources. Students are not required to be declared majors in order to participate.

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Prerequisites: Math 202A or equivalent. I have no restrictions on enrollment by undergraduates. See Math department staff advisors for any needed enrollment codes. Students who did not take Math 202A last Fall and want to enroll in this Math 202B should have a solid understanding of the following parts of the Lang text listed below: Chapter II, Section 3 of Chapter III, and Sections 1-8 of Chapter VI.

Office: 811 Evans 

Office Hours: TBA 

Description: This course, and Math 202A, are "tool courses", in that they cover some basic mathematical concepts that are of importance in virtually all areas of mathematics and its applications. Our Math 202B will follow on from where we left off at the end of Math 202A. The topics we will discuss include: The Hahn-Banach Theorem, duals of Banach spaces and weak topologies, Krein-Milman Theorem, Hilbert spaces, the Radon-Nikodym Theorem, Stone-Weierstrass Theorem, signed measures, Radon measures, operators on Banach and Hilbert spaces, additional topics as time allows. 
In my lectures I will try to give careful presentations of the material, well-motivated with examples.

Recommended Texts:  (available free on line for UCB students): See course web page, including for the Lang text.  

Grading: I plan to assign roughly-weekly problem sets. Collectively they will count for 50% of the course grade. Students are strongly encouraged to discuss the problem sets and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help. 
There will be a final examination, on Wednesday May 13, 11:30-2:30 PM, which will count for 35% of the course grade. There will be a midterm exam, which will count for 15% of the course grade. There will be no early or make-up final examination. Nor will a make-up midterm exam be given; instead, if you tell me ahead of time that you must miss the midterm exam, then the final exam will count for 50% of your course grade. If you miss the midterm exam but do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent in order to try to avoid a score of 0.

Comment: Students who need special accomodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each exam what accomodation they need for that exam, so that I will have enough time to arrange it.

Using TEX: I encourage students to write up their problem-set solutions in TEX, more specifically LATEX. This is a powerful mathematical typesetting program which is widely used in the sciences, engineering, etc., for documents that use a lot of mathematical symbolism. Thus learning to use TEX is a valuable skill if you work in such fields. For more information about this see the course web page.

Comment: The above procedures are subject to change.

Course Webpage: math.berkeley.edu/~rieffel 

Prerequisites 202A

Description Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. Morse functions, differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.

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Grading Letter grade.

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Prerequisites 

Description 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.

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Grading Letter grade.

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Prerequisites Consent of instructor

Description Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions.

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Grading Letter grade.

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Description The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.

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Grading Letter grade.

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Prerequisites 128A

Description Theory and practical methods for numerical solution of partial differential equations. Finite difference methods for elliptic, parabolic and hyperbolic equations, stability, accuracy and convergence, von Neumann analysis and CFL conditions. Finite volume methods for hyperbolic conservation laws, finite element methods for elliptic and parabolic equations, discontinuous Galerkin methods for conservation laws. Other topics include applications of the techniques to a range of problems.

Office uncica Canic, canics [at] berkeley [dot] edu, 911 Evans

Office Hours TBD

Required Text R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations,
Steady State and Time Dependent Problems, SIAM 2007.

Recommended Reading

- R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002.
– C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009.

Grading Letter grade.

Homework Project assignments

Course Webpage https://bcourses.berkeley.edu/courses/1490883

Prerequisites 250A or consent of instructor

Description This semester I will mostly concentrate on (I) Enumeration, generating functions and the theory of combinatorial species, (II) Symmetric functions, Young tableaux, and connections with representation theory, and (III) q- and q,t- analogs of combinatorial objects associated with the preceding.

Office 855 Evans Hall

Office Hours By appointment

Required Text None

Recommended Reading See course web page

Grading Based on homework

Homework Problems assigned from time to time

Course Webpage https://math.berkeley.edu/~mhaiman/math249-spring20/

Prerequisites 254A

Description Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.

What We'll Actually Be Covering Class field theory

Office Evans 883

Office Hours MWF 11:30–12:30

Required Text Neukirch, Algebraic number theory, Springer

Recommended Reading None

Grading Letter grade.

Homework Assigned weekly

Course Webpage https://math.berkeley.edu/~vojta/254b.html

Prerequisites 214

Description Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.

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We will explore and build connections between the MSRI program Higher Categories and Categorification and categorical structures in low dimensional topology and symplectic geometry - in particular those induced by geometric PDEs. The ideal participant would have research experience in one of these areas and an eager interest in building bridges. 

Depending on participant interests and expertise, we may follow ideas laid out in the survey Floer Field Philosophy or Bottman’s proposal of a symplectic (A_\infty,2)-category https://arxiv.org/abs/1811.05442, and from there find other literature or directions to explore collaboratively. 

Prerequisites:  fluency in manifolds and homology

Undergraduate prerequisites (if you want a letter grade): 202A, 214, 215A, 242, or equivalent reading courses