Much of the week will be spent working individually and in groups, and in informal discussions with mentors.
There will be roughly two additional events per week. Some will be introductions to research tools (from writing with latex to the use of various software packages). Others will be lectures from researchers in academia and industry on what research is actually about --- how it is done, how to do it, and what it is like. The exact details can be found in the calendar of events. The SURIM group will also have access to various classrooms during the summer, see the space/time schedule .
Can you hear the shape of a drum?
In this project we will investigate what happens to a shape in the plane when it is "drummed." This exploration leads to the natural question of whether one can recover the shape of the drum from the sounds it makes. We'll learn how this is can be viewed as a question about the eigenvalues of the Laplacian, one of the simplest partial differential operators. In particular we will attempt to construct isospectral non-isometric polygons (planar polygons which do not look the same but support the same noises when drummed). Students will learn about constructions of such examples and explore the wealth of existing and new examples to examine what properties are "spectrally determined," using combinatorial, analytic and numerical (computer) methods. Other possible research directions include studying isospectral flat tori, which leads to an exploration of lattices and their length spectrum. This program has limited prerequisites, a student who has background in multivariable calculus and linear algebra, at the level of the Math 50, series will find the material accessible. Students with more focused interests in analysis, PDEs, or geometry will find that the subject material lends itself easily to more advanced questions as well. Participants should expect to learn about infinite dimensional eigenvalue problems, combinatorial, analytic, and numerical methods in spectral geometry, lattices and much more! More information can be found here !
Optimal transportation (guided by Prof. Yanir Rubinstein)
Suppose you are given a detailed map of California's lakes and ponds, and another map with the state's water consumption per town. What would be the most efficient water canal system that would transport the available water to all inhabitants? The theory of Optimal Transportation grew out of questions of this sort, whose earliest treatment is probably that of the French scientist Gaspard Monge in 1781, employed by the French army at that time. Since then much progress has been made and the mathematical theory has greatly evolved. Nowadays, optimal transportation techniques have found many deep and unexpected applications ranging from differential geometry (including, e.g., Ricci flow), probability, partial differential equations, economics, and even meteorology. At the same time, there are fascinating open problems, some of which are rather geometric, and are not hard to formulate and understand with little background. Students working in this project will be expected to learn the basics of the theory in the first few weeks, through guided reading and a working seminar where students will lecture to each other. In the second part of the program the students will be working on research problems. Some of these problems will also involve an experimental component where computer programming/visualization could be useful. This project will be overseen by Dr. Yanir Rubinstein who will suggest the problems (based to some extent by the students interests and background) and give some presentations during the summer. The students will be mentored on a regular basis by a graduate student (Otis Chodosh). Prerequisites: Students should be familiar with the basics of measure theory and some functional analysis (although the latter is not absolutely necessary). Familiarity with the basics of partial differential equations would be advantageous, but is not necessary. Computer programming skills (Matlab, Maple or Mathematica) could prove useful as well.
Explorations in Number Theory
In this project students will focus on two areas of number theory. Depending on the students interests, this could include studying the class groups of quadratic number fields (which measure the failure of unique prime factorization and are related to binary quadratic forms) and examining the remarkable number-theoretic properties of integer linear recurrences. Part of the class group project will be computational, students will learn to compute both by hand and via the use of computer algebra system SAGE. Students will notice some patterns, and try to prove that they hold in general. For example, there is a relationship between 2-torsion in the class group and the factorization of the discriminant. The integer linear recurrence project will involve examining the rich literature on zeros of linear recurrence sequences, including the beautiful theorem of Mahler-Skolem-Lech. Students will analyze the periodicity and divisibility properties of linear recurrence sequences, and relations to linear algebra, Galois theory, computability theory, and combinatorics. For example, students can explore the question of what is a sharp bound on the periodicity of linear recurrences mod n, and when is it attained? Prerequisites are minimal; both projects have aspects ranging from the immediately accessible to the bleeding edge of mathematics. A background in linear algebra, algebraic number theory, Galois theory, or complex analysis would open up more avenues for exploration, but are not required.
Mentors: Jeremy Booher, Otis Chodosh, Chris Henderson, Seungki Kim, Sam Lichtenstein, Daniel Litt, Cary Malkiewich, Khoa Nguyen, Simon Rubinstein-Salzedo.