## MATH 83N: 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 seminar develops 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 algebraic notions such as groups, rings, and fields. This seminar may be paired with
Math 51; though that course is not a pre- or co-requisite.

Terms: Aut
| Units: 3
| UG Reqs: WAY-FR

## MATH 104: Applied Matrix Theory

Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems.
MATH 113 offers a more theoretical treatment of linear algebra.
MATH 104 and
EE 103/
CME 103 cover complementary topics in applied linear algebra. The focus of
MATH 104 is on algorithms and concepts; the focus of
EE 103 is on a few linear algebra concepts, and many applications. Prerequisites:
MATH 51 and programming experience on par with
CS 106.

Terms: Aut, Win, Spr, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Kazeev, V. (PI)
;
Taylor, C. (PI)
;
Velcheva, K. (PI)
...
more instructors for MATH 104 »

Instructors:
Kazeev, V. (PI)
;
Taylor, C. (PI)
;
Velcheva, K. (PI)
;
Ying, L. (PI)
;
Guijarro Ordonez, J. (TA)
;
Liu, Y. (TA)
;
Luo, S. (TA)
;
Sloman, L. (TA)
;
Truong Vu, N. (TA)
;
Wang, G. (TA)
;
Wolf, A. (TA)

## MATH 109: Applied Group Theory

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.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Taylor, C. (PI)
;
Libkind, S. (TA)

## 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; eigenvectors and eigenvalues; diagonalization. (
Math 104 offers a more application-oriented treatment.)

Terms: Aut, Win, Spr, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Manners, F. (PI)
;
Schaeffer, G. (PI)
;
Varolgunes, U. (PI)
...
more instructors for MATH 113 »

Instructors:
Manners, F. (PI)
;
Schaeffer, G. (PI)
;
Varolgunes, U. (PI)
;
Vondrak, J. (PI)
;
Falcone, P. (TA)
;
Helfer, J. (TA)
;
Izzo, Z. (TA)
;
Lolas, P. (TA)

## MATH 115: Functions of a Real Variable

The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 21.

Terms: Aut, Spr, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Schaeffer, G. (PI)
;
Thorner, J. (PI)
;
Datta, I. (TA)
...
more instructors for MATH 115 »

Instructors:
Schaeffer, G. (PI)
;
Thorner, J. (PI)
;
Datta, I. (TA)
;
Nuti, P. (TA)
;
Yang, K. (TA)
;
Zhang, S. (TA)

## MATH 116: Complex Analysis

Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. (
Math 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Eliashberg, Y. (PI)
;
Cotner, S. (TA)

## MATH 120: Groups and Rings

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 109 but altered content and more theoretical orientation. 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.

Terms: Aut, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

## MATH 145: Algebraic Geometry

An introduction to the methods and concepts of algebraic geometry. The point of view and content will vary over time, but include: affine varieties, Hilbert basis theorem and Nullstellensatz, projective varieties, algebraic curves. Required: 120. Strongly recommended: additional mathematical maturity via further basic background with fields, point-set topology, or manifolds.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Solis, P. (PI)
;
Qian, L. (TA)

## MATH 146: Analysis on Manifolds

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Wieczorek, W. (PI)

## MATH 171: Fundamental Concepts of Analysis

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 61CM or 61DM or 115 or consent of the instructor. WIM

Terms: Aut, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

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