There is a tradition of viewing mathematics as a subject rooted deeply in antiquity; on the other hand, the language of mathematics itself is characterized by a few prototypes, which were absent before modern era. One of such prototypes is the Euler's polyhedron formula V-E+F=2, which serves nowadays as a cliche to illustrate a stereotypical toolkit for working mathematicians: the Gauss-Bonnet theorem in differential geometry, the Riemann-Roch theorem in algebraic geometry, and the Lefschetz fixed point theorem in algebraic topology.
The Atiyah–Singer index theorem, proved in 1963, includes as special cases all the results just mentioned. Later on, it grew with rich applications and reunification with a broad range of subjects, from supersymmetry in particle physics to elementary number theory. Moreover, it was rewritten and interpreted in growing clarity, flexibility and accessibility. Nowadays, non-experts have already been able to grasp and appreciate this big picture.
In this project we are going to understand most of the components of a certain version, suitable for students according to their knowledge and interests, of the Atiyah–Singer index theorem. What we are going to carry out here, is to often omit most technical parts and proofs, so that we can concentrate on the conceptional components of the main results. At the end of the project, students will be able to draw a big picture in their theses, as well as to present it in a scientific way.
Gruppstorlek 2-6 studenter
Målgrupp GU- och Chalmersstudenter. För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Projektspecifika förkunskapskrav Basic knowledge of Fourier analysis.
Se respektive kursplan för allmänna förkunskapskrav. Utöver de allmänna förkunskapskraven i MVEX01 ska Chalmersstudenter ha avklarat kurser i en- och flervariabelanalys, linjär algebra och matematisk statistik.
Handledare Alexey Kuzmin: firstname.lastname@example.org, Jiacheng Xia: email@example.com
Examinator Maria Roginskaya, Ulla Dinger
Institution Matematiska vetenskaper