• Course Code: 01:640:492
• Semester(s) Offered: Spring
• Credits: 1
• Counts toward math major/minor?: Honors track only
• Prerequisites: Special permission only. Topics and prerequisites vary.

Undergraduate Mathematics Seminar: Reading, presentation, and discussion of mathematical topics.

This is a one-credit honors-level seminar. The topics, and prerequisites, vary from semester to semester. Typically, the seminar focuses on a subject area in mathematics that is outside the usual undergraduate curriculum, and participants take turns lecturing.

Admission to the Junior-Senior Honors Seminar is by special permission. To apply, use the online special permission form for honors courses. Students in the mathematics honors track are automatically admitted; other students are admitted based on course record and recommendations by mathematics faculty. Students in the seminar are expected to participate actively by contributing to discussions, making presentations in the seminar, and collaborating with other students in preparing talks.

While almost all participants in the seminar are juniors or seniors, applications from exceptionally qualified freshman and sophomores are considered.

Questions may be sent to one of the Department of Mathematics honors advisors at .

Textbook and Syllabus

Textbook, syllabus, and content change each time the seminar is offered. See the individual course descriptions, or the current textbook list

Spring 2025 - Prof. Dima Sinapova

Textbooks: Jech, Set Theory, the Millenium Edition

We will cover topics from set theory, in particular combinatorial properties of infinite objects. Some of the concepts include introductory cardinal arithmetic, infinite trees, the Suslin hypothesis, regularity properties of the real numbers, the axiom of determinacy.

I will give the first couple of lectures, and then students will take turns presenting. We will be following various chapters in Jech, Set Theory. No prior knowledge of set theory is required.

Admission to the course is by application through the online special permission for honors courses form.

Spring 2021 - Prof Xiaojun Huang

In this one semester seminar course, we will read part of the materials from a very nice small book by S. Krantz: Complex Analysis, the Geometric Viewpoint, the Carus Mathematical Monographs (Number 23).  This viewpoint starts with a classical paper by Ahlfors (An extension of Schwarz's lemma, Trans of the AMS 43(1938), 359-364).  We will discuss  the connection between some classical subjects in one complex variable with those in differential geometry. The prerequisite or corequisite for this course is Math 403, or permission of the instructor.

Notes

This seminar satisfies an honors track requirement.

There is also a U-seminar, for first or second year students.

This course is offered each Spring Semester.
Information will be available during the registration period, through the Honors Track or the Undergraduate Office.

Schedule of Sections

 01:640:492 Schedule of Sections

Previous semesters:

• Spring 2017 Prof. Kontorovich, Number theory, group theory and Ramanujan graphs
• Spring 2016 Prof. Kiessling, Less is more -- the beauty of minimal design
• Spring 2015. Prof. Kahn Surprising mathematical applications of linear algebra.
• Spring 2014 Prof. Beheshti, Mathematical General Relativity.
• Spring 2013
• Spring 2012
• Spring 2011 Profs. Goodman and Wilson The Geometry of finite reflection groups Finite Reflection Groups
• Spring 2010, Prof. Borisov
  Representation Theory
• Spring 2009, Prof. Hoelscher
  Matrix Groups: where Geometry meets Algebra
• Spring 2008, Prof. Carlen
  Inequalities
  
• Spring 2007, Prof. Woodward
  Elementary Number Theory, Group Theory, and Ramanujan Graphs
• Spring 2006, Prof. Beck
  Discrepancy Theory: Uniformity versus Irregularity
• Spring 2005, Profs. Tunnell and Woodward
  Modern Number Theory
• Spring 2004, Profs. Goodman and Sahi
  Fourier Analysis on Finite Groups

• Course Code: 01:640:196
• Semester(s) Offered: Spring
• Credits: 1
• Counts toward math major/minor?: Honors track only
• Prerequisites: Special permission only. Students should have completed at least Calc II.

Undergraduate Mathematics Seminar: Reading, presentation, and discussion of mathematical topics.

This one-credit honors-level seminar is designed for first- and second-year students who are considering studies in advanced mathematics. There is also a Junior-Senior Honors Seminar. These seminars satisfy requirements of the Honors Track in Mathematics.

The First and Second Year Honors Seminar provides a glimpse into the world of mathematics that lies beyond elementary calculus. It aims to be informal, lively, and challenging.

There are no problem sets or exams in the course. Students are expected to participate actively by making presentations in the seminar, collaborating with other students in preparing talks, and contributing to discussions. Occasionally, we have guest speakers (professors and graduate students).

Admission to the First and Second Year Honors Seminar is by special permission. To apply, use the online special permission form for honors courses.

Questions may be addressed to Professors Janos Komlos and Michael Beals (Honors Committee chair).

Textbook and Syllabus

The Spring 2025 textbook is "Conjecture and Proof" by Miklós Laczkovich and is available on the AMS website: https://bookstore.ams.org/clrm-15

Notes

This seminar satisfies an honors track requirement.

This course is offered each Spring Semester.
Information will be available during the registration period, through the Honors Committee or the Undergraduate Office.

Schedule of Sections:

01:640:196 Schedule of Sections

• Course Code: 01:640:491
• Semester(s) Offered: Fall
• Credits: 1
• Counts toward math major/minor?: Honors track only
• Prerequisites: Special permission only. Students should have completed at least Calc II.

This is a one credit seminar in mathematical problem solving.  It is aimed at undergraduate students who enjoy solving mathematical problems in a variety of areas, and want to strengthen their creative mathematical skills, and their skills at doing mathematical proofs.

A secondary goal of this seminar is to help interested students prepare for the William Lowell Putnam Undergraduate Mathematics Competition , which is an annual national mathematics competition held every December. Any full-time undergraduate who does not yet have a college degree is eligible to participate in the exam. (However, you are free to participate in the seminar without taking the exam, and vice versa.)

The meetings of the seminar will be a mixture of presentations by the instructors, group discussions of problems, and student presentations of solutions/ideas.

The seminar qualifies as an honors seminar for the honors track. It does not count as one of the required 300-400 level courses for the major or minor.

Students who have taken the seminar previously may not register for it, but are very welcome to attend.

All students taking the seminar are expected to:

• Attend regularly.
• Participate actively in group problem solving.
• Present problem solutions (or partial solutions) to the class.
• Work on some of the assigned problems and turn in a carefully written solution for at least one problem per week.
• Read assigned material prior to class.

Students taking the seminar for honors track credit may have additional requirements consisting either of doing additional problems or doing more extensive class presentations.

Some Appetizers:

• When you multiply the numbers 1, 2, 3, ..., 400, how many trailing 0's does the answer have?
• Suppose you have a finite collection of points on the plane, such that whenever you draw a line through any 2 of them, that line passes through a 3rd point. Must all the points be collinear?
• Is the 50000th Fibonacci number odd or even?
• Can the product of 2 consecutive integers be a perfect square?
• Suppose you have n red points and n blue points in the plane. Can you pair up the red points with the blue points (each red point is paired with one blue point) so that all the line segments between the pairs are nonintersecting?

This course is offered during the Fall semester.

Schedule of Sections

Archive

• Fall 2008
• Fall 2006

• Semester(s) Offered: Occasional
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Topics and prerequisites vary.

General Information

Content varies widely.

In Fall 2026, there will be two sections of Math 495:

• Section 01: AI Tools for Mathematics
  Instructor: Prof. Carbone <>
  Course Description: see Syllabus
  Prerequisites: Background in proof-based mathematics, 640:300 Intro Math Reasoning or equivalent is required. Graduate students in CS and ECE may receive an exemption with instructor permission. Programming experience will be beneficial but is not essential. 
  If you have taken Math 300 but are missing some of the official prerequisites listed in Webreg (Math 250 and Calc IV = Math 244/252), please fill out the Prerequisite Override Form for assistance with registration.
  
  
• Section 02: Multilinear Algebra and Tensor Networks
  Instructor: Prof. Echeverria <>
  Course Description: see Syllabus
  Prerequisites: Math 251 and Math 250, but Webreg lists Calc IV = Math 244/252 and Math 250 as the prerequisites. If you are missing Calc IV but have taken the true prerequisites (Math 251 and Math 250), please fill out the Prerequisite Override Form for assistance with registration.

In Spring 2026, there were three sections of Math 495:

• Section 01: AI Tools for Mathematics
  Course Description: see Syllabus
  Prerequisites: Background in proof-based mathematics, 640:300 Intro Math Reasoning is strongly recommended. Programming experience will be beneficial but is not essential.
  
  
• Section 02: Combinatorial Game Theory
  

  Course Description: A combinatorial game is ordinarily a two-player game with no hidden information, ending when there are no possible moves remaining. These games have a tendency to break down into smaller pieces, which can be analyzed independently. Dots and Boxes is perhaps a familiar example; one core objective of this course is to seek awesomeness at Dots and Boxes. Combinatorial Game Theory has also been applied to Go, but that is way beyond the scope of this course.

  This course will cover both impartial games (in which both players have the same options) such as Nim, and partizan games, such as Hackenbush. A balance will be sought between deep dives on specific games (such as Sprouts or Hey! That's My Fish!) and general theory. The space of combinatorial games has a rich structure, with many surprising connections.

  (Please note that this topic is essentially disjoint from "game theory", as studied by von Neumann, Nash, etc. No one will earn a Nobel prize based on knowledge from this course.)

  Prerequisites: Math 300 or Math 428 or Math 454, but the website will list 244 and 250 as the prerequisites. In such case, please fill out the Prerequisite Override Form for assistance with registration.
  
  
• Section 03: Introduction to Topological Data Analysis
  Course Description: see Syllabus
  Prerequisites: Math 251 and Math 250, but the website will list 244 and 250 as the prerequisites. In such case, please fill out the Prerequisite Override Form for assistance with registration.
  
  

In Fall 2025, there were three sections of Math 495:

• Section 01: AI Tools in Mathematics
  Course Description: see Syllabus
  Prerequisites: Background in proof-based mathematics, 640:300 Intro Math Reasoning is strongly recommended. Programming experience will be beneficial but is not essential.
  
  
• Section 02: Tensor Networks as a bridge between Neural Networks and Quantum Physics
  Course Description: see Syllabus
  Prerequisites: Linear Algebra (Math 250) is the only prerequisite for this course, 
  
  
• Section 03: An Introduction to Machine Learning
  Course Description: see Syllabus
  Prerequisites: A course in Linear Algebra
  
  

In Spring 2025, there were two sections of Math 495:

• Section 01: A Mathematical Invitation To Machine Learning
  Course Description: This mathematics course covers topics related to machine learning. Some of these are multivariable calculus applications in neural networks, linear regression, principal component analysis and support vector machines. Emphasis will be on the mathematics aspects and connections.
  Textbooks: The pre-print versions of both textbooks are freely available for download for personal use. The primary textbook: "Mathematics For Machine Learning" by Deisenroth, Faisal, Ong. (Cambridge University Press). Secondary textbook: "Foundations of Data Science" by Blum, Hopcroft, Kannan. (Cambridge University Press).
  Pre-requisites: Math 152 or equivalent. Further courses such as linear algebra, multivariable calculus, probability or statistics, are a plus. Prior exposure to machine learning is not required. (If you have completed Math 152 but not the official prerequisite courses Math 244/252 and Math 250, fill out the Prerequisite Override Form for assistance with registration.)
  
  
• Section 02: From Gravitational Waves to Supersonic Flows: An Introduction to Hyperbolic PDEs
  Course Description: The sonic boom of jets, the rippling of ocean waves, and even the gravitational waves detected by astronomers are all described by Hyperbolic Partial Differential Equations (PDEs). This course offers an introduction to the mathematical theory of hyperbolic PDEs, focusing on simplified models in fluid dynamics and linear wave propagation. While the emphasis will be on mathematical rigor, no prior knowledge of PDEs will be assumed.
  Textbook: Hyperbolic Partial Differential Equations, Serge Alinhac.
  Pre-requisites: Multivariable calculus (Math 251), elementary ODE theory (Math 244/252), intro linear algebra (Math 250).
  Assignments: Final presentation on a topic of the students' choosing (a list of suggested topics will be provided). Optional weekly homework will be available for extra credit.
   

In Spring 2024, there were two sections of 495:

• Section 01: Proofs from THE BOOK
  Prerequisites - Math 300
  Syllabus
• Section 02: Mathematical Adventures in One-Dimensional Physics
  Prerequisites - (244 or 252 or 292) (ODEs) and (250 or 291) (Lin. Alg.)
  Syllabus

See the archives for details.

Archives

• Spring 2009: Connections Seminar, Prof. Cohen
• Spring 2008: Connections Seminar, Prof. Retakh
• Fall 2007 Financial Mathematics, Professor Rodriguez.
• Spring 2007
• Fall 2006 (Financial Mathematics)

Schedule of Sections:

 01:640:495 Schedule of Sections

• Course Code: 01:640:486
• Semester(s) Offered: Fall
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 285 and Probability (Math 477 or Stat 381)

General Information (Catalog listing)

Survival models, life tables. Valuation of benefits, premiums, policy values of standard, single life insurance products. Covers part of the syllabus for Exam MLC of the Society of Actuaries.

Prerequisites:

• Introduction To Interest Theory For Actuarial Science - 01:640:285 
• Mathematical Theory of Probability - 01:640:477 or Theory of Probability 01:960:381

Textbook: 

Textbook:  For current textbook please refer to our Master Textbook List page

Offered in the fall. Required for the actuarial specialization.

Schedule of Sections:

01:640:486 Schedule of Sections

• Course Code: 01:640:487
• Semester(s) Offered: Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: C or better in Math 486

Course description (Catalog):

01:640:487 Mathematics of Life Contingent Risk Models II

Continuation of Mathematics of Life Contingent Risk Models I. Policy values, multiple state models, pension mathematics, effect of interest rate risk, emerging costs of traditional life insurance.

Covers part of the syllabus for Exam MLC of the Society of Actuaries.

Prerequisite: 01:640:486. Grade of C or better required in prerequisite course.

Offered in the spring only.

Schedule of Sections:

01:640:487 Schedule of Sections

• Course Code: 01:640:350
• Semester(s) Offered: Fall, Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250 and Calc III and a C or better in Math 300

General Information

Math 350 is a proof-based continuation of Math 250, covering abstract vector spaces and linear transformations, diagonalization, Jordan canonical form, and inner product spaces. Math 350 is one of two courses most mathematics majors may take to satisfy the upper-level algebra requirement. The other is Math 351.

The focus of Math 350 is axiomatic linear algebra, starting from the abstract notions of vector space and of linear transformation. Students will be expected to write precise proofs, building on their proof-writing experience from Math 300. From this abstract viewpoint, linear algebra will be developed far beyond Math 250, with new insight and new applications.

The honors section of Math 350 covers the same topics as the non-honors sections, but in significantly greater depth. Enrollment in the honors section requires the approval of the Honors Advisor.

Prerequisites

• Math 250, Introductory Linear Algebra
• and a C or better in Math 300, Mathematical Reasoning
• and Math 251, or Math 291, Multivariable Calculus

Enrollment in the honors section requires the approval of the Honors Advisor.

Textbook

For current textbook please refer to our Master Textbook List page

General Syllabus

Topics, in approximate sequence:

• Review of the basic ideas and techniques of Math 250
• Abstract vector spaces
• Subspaces; span of subsets; linear independence
• Bases and dimension
• Linear transformations; matrix representation
• Composition of linear transformations; invertibility
• Change-of-coordinate matrices (change of basis)
• Theoretical aspects of systems of linear equations
• Determinants and their properties
• Eigenvalues and eigenvectors; the characteristic polynomial
• Diagonalizability
• Invariant subspaces; the Cayley-Hamilton Theorem
• Jordan canonical form
• Real and complex inner product spaces
• Normal and self-adjoint matrices; unitary and orthogonal matrices

Instructors have some flexibility in deciding which proofs to treat.

As time allows, with the instructor’s discretion, some subset of the following topics may also be covered: Dual spaces, Direct sums of subspaces, Minimal polynomial, Rational canonical form, Normal and self-adjoint operators, Bilinear and quadratic forms, Applications of the theory.

Schedule of Sections:

01:640:350 Schedule of Sections

• Course Code: 01:640:351
• Semester(s) Offered: Fall, Spring
• Credits: 4
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250 and Calc III and a C or better in Math 300

General Information

Math 351 is one of two courses most mathematics majors may take to satisfy the algebra requirement. The other is Math 350.
Math 351 was originally part of a two-course sequence, Math 351-352. The continuation, Math 352, is no longer offered.

Catalog Description

01:640:351-352 Introduction to Abstract Algebra I, II (4,3)
Abstract algrebraic systems, including groups, rings, fields, polynomials, and some Galois theory.
Prerequisites: CALC3; 01:640:250; and a C or better in 300 or permission of department.

Textbook

For current textbook please refer to our Master Textbook List page

Topics, in approximate order

Division
Primes and unique factorization
Congruence
Modular arithmetic
Rings
Properties of rings
Isomorphisms and homomorphisms
Division in F[x]
Irreducibles and unique factorization
Roots and reducibility
Congruence in F[x]
Congruence and Ideals
Ring isomorphism theorems, prime and maximal ideals
Groups
Properties of groups
Subgroups
Group isomorphisms and homomorphisms
Symmetric and alternating groups
Lagrange’s Theorem
Conjugacy classes
Normal subgroups
Quotient groups
Center and commutator subgroups
Group isomorphism theorems
Simplicity of alternating groups
Classification of finite abelian groups

Sample Course Page

Spring 2022: Prof. Borisov

• Course Code: 01:640:354
• Semester(s) Offered: Fall, Spring, Summer
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 250

General Information (Catalog Listing)

01:640:354 Linear Optimization(3)
Linear programming problems, the simplex method, duality theory, sensitivity analysis, introduction to integer programming, the transportation problem, network flows, and other applications.
Prerequisite: 01:640:250. Credit not given for both this course and 01:640:453 or 01:711:453.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus

Sample Syllabus from Spring 2006

Office Hours

The TA's for this course will host online office hours on the website Discourse Technologies.

Previous semesters:

• Spring 2010, Section 01
• Spring 2010, Section 05
• Summer 2009

Schedule of Sections:

01:640:354 Schedule of Sections

• Course Code: 01:640:356
• Semester(s) Offered: Spring
• Credits: 3
• Counts toward math major/minor?: Yes
• Prerequisites: Math 300 and Calc III

General Information (Catalog Description)

Priorities of the natural numbers, congruences, disophantine equations, and elementary arithmetical functions.

Prerequisite: CALC3 and Math 300 or permission of the department.

Math 300 (Mathematical Reasoning) or a very good background in mathematical proof is required. Students with a strong record in their mathematics courses who have not taken Math 300, but wish to take Number Theory, should request a prerequisite override from the Head Advisor ().

• Suggested Textbook
• Suggested Syllabus
• Sections Taught This Semester
• Previous Semesters

Suggested Textbook

Instructors in 356 sometimes use a different text and follow a different syllabus than the one shown.

Textbook:  For current textbook please refer to our Master Textbook List page

Suggested Syllabus

A suggested syllabus follows. However, anything posted on a page for the current semester supersedes this syllabus.

One midterm exam and a final exam seem appropriate for this course. A selection of the optional topics in the middle of the table may be inserted if the earlier material is completed before an exam can be given. Topics listed for lecture 14 should not be introduced before the midterm exam. This syllabus allows two weeks at the end of the course for additional topics. Such topics should be dictated by the interest of the class.

Schedule of Sections:

Previous Semesters

• Fall 2016 : Prof. Tunnell
• Fall 2013 : Doron Zeilberger
• Summer 2011 : Jerrold Tunnell
• Fall 2010 : Anders Buch
• Summer 2010 : Thom Tyrrell
• Fall 2009 : Prof. Weibel
• Summer 2009 : L. Medina
• Fall 2008 : Prof. Munshi
• Summer 2008 : Michael Weingart
• Fall 2007 : Prof. Sahi
• Fall 2006 : Prof. Sills (T Th 1:40-3 PM, SEC-217)
• Fall 2001 : Prof. Miller
• Summer 2000 : One section taught by David Nacin. A lecture schedule is available.
• Fall 1998 : Prof. Bumby (One section.)