printer friendly pageMATH 56: Proofs and Modern Mathematics
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as
Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with
Math 51; though that course is not a pre- or co-requisite.
Terms: Aut, Win
| Units: 4
| UG Reqs: WAY-FR
MATH 61CM: Modern Mathematics: Continuous Methods
This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with
Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor. This series provides the necessary mathematical background for majors in all Computer Science, Data Science, Economics, Mathematics, Natural Sciences, and Engineering.
Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Luk, J. (PI)
;
Greilhuber, J. (TA)
MATH 61DM: Modern Mathematics: Discrete Methods
This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with
Math 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.This series provides the necessary mathematical background for majors in Computer Science, Data Science, Economics, Mathematics, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering, majors requiring Math 50's beyond
Math 51, are recommended to take Math 60CM.
Terms: Aut
| Units: 5
| UG Reqs: WAY-FR
Instructors:
Fox, J. (PI)
;
Xu, M. (TA)
MATH 104: Applied Matrix Theory
Linear algebra for applications in science and engineering. The course introduces the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and nearly all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject. Topics include orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares methods, and algorithms for solving systems of linear equations; applications include clustering, principal component analysis and dimensionality reduction, regression.
MATH 113 offers a more theoretical treatment of linear algebra.
MATH 104 and
ENGR 108 cover complementary topics in applied linear algebra. The focus of
MATH 104 is on algorithms and concepts; the focus of
ENGR 108 is on a few linear algebra concepts, and many applications. Prerequisites:
MATH 51 and programming experience on par with
CS 106A.
Terms: Aut, Win, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Candes, E. (PI)
;
Cortinovis, A. (PI)
;
Cheng, R. (TA)
...
more instructors for MATH 104 »
Instructors:
Candes, E. (PI)
;
Cortinovis, A. (PI)
;
Cheng, R. (TA)
;
Goyal, S. (TA)
;
Malhotra, A. (TA)
;
Nehme, G. (TA)
;
Park, J. (TA)
;
Tse, D. (TA)
MATH 109: Groups and Symmetry
Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM. Prerequisite:
Math 51.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Swaminathan, M. (PI)
;
Yang, Y. (TA)
MATH 110: Number Theory for Cryptography
Number theory and its applications to modern cryptography. Topics include: congruences, primality testing and factorization, public key cryptography, and elliptic curves, emphasizing algorithms. Includes an introduction to proof-writing. This course develops math background useful in
CS 255. WIM. Prerequisite:
Math 51
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 113: Linear Algebra and Matrix Theory
Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. (
Math 104 offers a more application-oriented treatment.) Prerequisites:
Math 51
Terms: Aut, Win, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Malinnikova, E. (PI)
;
Swaminathan, M. (PI)
;
Vondrak, J. (PI)
...
more instructors for MATH 113 »
Instructors:
Malinnikova, E. (PI)
;
Swaminathan, M. (PI)
;
Vondrak, J. (PI)
;
Amar, S. (TA)
;
Niu, J. (TA)
MATH 115: Functions of a Real Variable
The development of 1-dimensional real analysis (the logical framework for why calculus works): sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Includes introduction to proof-writing. Prerequisite:
Math 51 or
Math 56.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 116: Complex Analysis
Holomorphic and analytic functions, power series, Cauchy integral and Cauchy integral formula, meromorphic functions and differential forms, calculus of residues and applications, analytic continuation, conformal mappings, Riemann mapping theorem, Laurent series and conformal classification of annuli, harmonic functions and Dirichlet problem, introduction to Riemann surfaces, theory of elliptic functions and integrals. (
Math 106 offers a less theoretical treatment). Prerequisites: 51,52 and 171, or 61cm and 62cm.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Eliashberg, Y. (PI)
;
Chen, Y. (TA)
MATH 120: Groups and Rings
Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced treatment of group theory than in
Math 109, also including ring theory. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM course. Prerequisite:
Math 51 and some prior proof-writing experience.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
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