# Honors Mathematics

The Mathematics Honors Program provides opportunities and assistance to talented undergraduates hoping to graduate with Latin Honors in mathematics.

## Senior Latin Honors thesis

One requirement for graduating with Latin honors in mathematics is to:

- Take the required thesis course (HCOL 3101H or HCOL 3102H).
- Complete an honors thesis under the supervision of a faculty mentor.
- Summarize your thesis work in a printed scientific poster.
- Give a short public presentation of your work.
- Submit to an oral exam by your faculty committee (summa cum laude students only).

Normally students contact a faculty member who works in an area of interest. That faculty member, if willing, will suggest a topic of study, or in some cases, a topic of research that is suitable.

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Honors Math courses

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Lower division courses

#### MATH 1571H/1572H: Honors Calculus I/II

This is the introductory Honors Calculus sequence. It begins with a discussion of functions, limits, and continuity. The course then proceeds to the main topics of differentiation and integration of functions of a single real variable. One then studies the properties of these key operations, including the chain rule, the fundamental theorem of calculus, and methods of integration. Applications from the physical and biological sciences are emphasized. Applications include max-min problems, related rates, arc length, volumes, and surface area of solids of revolution. Sequences and series are also studied. The course strives to provide an introduction to the mathematical method of proof and to mathematical rigor.

#### MATH 2573H/2574H: Honors Calculus III/IV

This course is a two-semester course covering multivariable calculus, including differentiation of functions of several variables, multiple integration, discussion of chain rule, inverse function theorem, implicit function theorem, multiple integration, and Fubini’s theorem. Applications include max-min problems using Lagrange multipliers, volume, and surface area. The course also treats vector calculus, vector fields, div, curl, grad, Green’s theorem, Stokes’ theorem, and Gauss’ theorem. Also covered are linear algebra, including the study of eigenvalues and eigenvectors, and the application to diagonalization of suitable linear transformations. The course provides an introduction to ordinary differential equations, particularly first-order equations, constant-coefficient linear equations, variation of parameters, first-order systems, and Laplace transforms. Applications from the physical sciences will be emphasized.

#### MATH 3592H/3593H: Honors Mathematics I/II

This course is an Honors course designed for students who are interested in pursuing mathematics beyond basic calculus or are interested in why theorems hold and wish to understand the mathematical reasoning underlying mathematical results, or who enjoy doing mathematics for its own sake. As such, the course will help prepare students for advanced undergraduate courses and graduate courses in math. The course emphasizes mathematical proof and rigor. It is a 5-credit course and is more demanding than 2573H-4H, although both courses have a large overlap in content. While this course covers in depth the topics of multivariable calculus and linear algebra and vector calculus, it does not cover in any detail the introduction to ordinary differential equations. It is recommended that students needing or desiring this subject take an additional course (several excellent courses in ordinary differential equations are offered at the 5000 level). Some topics covered in 3592H-3H (which are not covered in 2573H-4H) include greater attention to n-variable cases (n greater than 3), manifolds, the spectral theorem for real symmetric matrices and applications to classification of real quadratic forms, applications to max-min-saddlepoint classification of critical points, differential forms on manifolds and exterior differentiation, integration on manifolds, and generalized Stokes’ theorem.

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Upper division courses

MATH 5285H/5286H - Honors: Fundamental Structures of Algebra I/II

This course introduces the study of basic algebraic structures, sub-objects, quotient objects, and maps between objects. In particular, groups, rings, modules, and fields are treated. Some topics covered are the Sylow theorems, factorizations in integral domains, principal ideal domains, unique factorization domains, chain conditions in commutative rings, structure theorem for finitely generated modules over a pid, applications to linear algebra (Jordan normal form), finite fields, and elementary Galois theory.

#### MATH 5345H - Honors: Topology

This course focuses on abstract topological spaces, both the concrete and the very formal, the non-intuitive and the geometric. Along with an emphasis on the ability to effectively communicate mathematical arguments, in this course, students will develop qualitative tools to characterize topological spaces (e.g., connectedness, compactness, second countable, Hausdorff...), develop tools to identify when two spaces are equivalent (homeomorphic), and explore examples and counter-examples that inform the development of the subject. Several important results will be proved, such as the Tychonoff theorem on products, but an equal focus will be placed on understanding examples coming from geometry, algebra, and number theory. Other topics include the fundamental group and, if time permits, covering spaces.

#### MATH 5615H/5616H - Honors: Introduction to Analysis I/II

This course gives a rigorous treatment of basic analysis. It covers metric spaces, convergence, connectedness, compactness, uniform convergence of sequences, and series of functions in one and several variables. Also covered are the Stone-Weierstrass theorem, rigorous development of differentiation and Riemann-Stieltjes integration, Taylor’s theorem, Implicit function theorem, and Stokes’ theorem.

## Latin Honors Adviser

Professor Dmitriy Bilyk

Latin Honors Adviser for Mathematics Majors