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SUMaC Student

Academic Tracks

SUMaC offers two courses, called Program I and Program II, with unique topics for each course. Each student attends one of two courses, both of which build on topics central to mathematics through their historical significance and their relevance to current lines of mathematical research. Students can indicate their preference between the two course options on their applications, and then the Admissions Committee will admit students into their final course.

Program I – Abstract Algebra & Number Theory

Program I topics are introduced through five motivating problems such as limitations of straight-edge and compass constructions, classification of patterns in two dimensions, error-correcting codes, cryptography, and the analysis of symmetry in structures.

The mathematics central to solving these problems comes from the areas of abstract algebra and number theory. Abstract algebra originated in the early part of the 19th century through the study of polynomial equations. This branch of mathematics lies at the core of many areas of modern mathematical research. Number theory concerns properties of the integers, and has its origins in ancient mathematics. Number theory remains a very active field of study with interesting open problems and important applications in computer science.

Recommended Prerequisite Mathematics Experience for Program I:

Students applying for Program I should have experience writing and reading mathematical proofs, and strong high school geometry and algebra mastery. SUMaC Program I applicants should be comfortable with:

  • Proofs by induction, contradiction, contrapositive, and more.
  • Logic used in mathematics such as basic logical symbols and their meanings like if, then, or, and, etc.
  • Notation for subsets, supersets, and intersections.

Students accepted to Program I have typically studied number theory, and are comfortable with modular arithmetic and some basic theoretical results involving modular arithmetic. Prior participation in mathematics competitions or contests is not required.

Program II – Algebraic Topology

Program II centers on algebraic topology, a major area of current mathematics research.

Topology is the study of the properties of shapes that remain unaffected by deformations. For example, a sphere made out of rubber can be deformed into the shape of a cube. While it may appear that a sphere and a cube don't have that much in common, it turns out that they are topologically equivalent and in a way that can be made precise mathematically. This course will explore different ways of analyzing topological properties of shapes using algebraic concepts, such as the notion of group.

Recommended Prerequisite Mathematics Experience for Program II:

Students applying for Program II should have enough mathematics experience to learn new mathematics quickly. SUMaC Program II applicants should have:

  • More proof experience than a Program I student.
  • Deep, thoughtful interest in higher mathematics, while a Program I student might only have experience from mathematics contests or clubs.
  • Experience with group theory, but it is not required.

Program II students are typically previous participants of Program I or have experience with content from Program I. 

A student using a tablet to study graph theory

Stanford Pre-Collegiate Summer Institutes: Mathematics Courses

If you are not ready for Abstract Algebra & Number Theory or Algebraic Topology, we offer other math courses through Stanford Pre-Collegiate Summer Institutes. These courses are highlighted below. Please note: Stanford Pre-Collegiate Summer Institutes is another program that requires a different application from Stanford University Mathematics Camp. You can use the same application account to begin and submit applications for both programs. 

The deadline to apply for Stanford Pre-Collegiate Summer Institutes courses is March 15.

Number Theory

Number Theory is the study of the integers and their properties. For thousands of years, the greatest minds have been working to produce what many consider to be some of the most elegant and powerful ideas in all of mathematics. Number Theory continues to be an area of active research, and with the increasing power and availability of computers, there have been significant developments in applications of number theory that would not have been possible even 50 years ago. Our overarching goal in this course will be to observe, investigate, conjecture, and prove the patterns and relationships we see occurring in the integers. By the end of this course, students should gain a deeper understanding of number theory, and gain valuable experience writing proofs, presenting solutions to mathematical problems, and communicating clearly with peers about mathematical concepts.

Discrete Mathematics

Discrete mathematics encompasses a broad range of mathematical fields centered on discrete (non-continuous) mathematical structures with an eye toward applications in applied and theoretical computer science. Topics include number theory, set theory, logic, graph theory, and combinatorics. Problems encountered in this field range from easy to very difficult, so this course provides an opportunity to hone mathematical problem-solving skills. Additionally, the course will help students develop proof-writing skills, and it will enable them to build a strong mathematical background for future study in computer science. The course will include applications in the analysis of computer algorithms.

Linear Algebra with Proofs

This course is intended to serve as an introduction to pure mathematics. We will develop the foundations of linear algebra with a proof-based approach, which will include introducing the notion of a vector space over a field, along with theorems on linear independence and dependence, bases, linear maps, and more. An emphasis will be placed on the logical framework of linear algebra instead of the more computational aspects. Students with an eagerness to engage with a theoretical approach to mathematics will enjoy this course. Problem sets will be assigned, and classes will consist of a mixture of lecture and group work in which students collaboratively work through problems. Students considering a math major in college will benefit from taking this course.