Topics in Number Theory, Algebra and Combinatorics

Project ID: 2531ad1558

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Research Theme: Mathematical Sciences

UCL Lead department: Mathematics

Department Website

Lead Supervisor: Yiannis Petridis

Project Summary:

The proposed research will investigate aspects of algebraic and analytic number theory, graphs theory, and algebraic combinatorics. Some computational methods may be involved. Potential directions include:

• The arithmetic of elliptic curves and of Jacobians of higher genus curves, and the Birch-Swinnerton-Dyer conjecture.

• The analytic theory of L-functions and automorphic forms.

• The arithmetic and cohomology of locally symmetric spaces.

• The p-adic Langlands programme, especially the cohomology of arithmetic groups and their p-adic Banach space representations.

• Shimura varieties and the Andre-Oort conjecture.

• The interface of combinatorics with analysis and in particular the positivity of Taylor series of inverse powers of various combinatorially defined polynomials.

• Discrete geometry including Tverberg-type theorems and problems concerning lattice points.

• Extremal and probabilistic problems in graphs and hypergraphs.