A world expert on the development and implementation of algorithms for mathematics, John's scientific contributions are recognised via many awards including the CSIRO Medal (1993) the ATSE Clunies Ross Award (2001) the Richard D. The group is led by Professor John Cannon, the founder of Magma, and of its predecessor Cayley. It supports cutting-edge computations in algebra, number theory, algebraic geometry, and algebraic combinatorics and is used on a daily basis by thousands of research mathematicians in over 70 countries. Magma is a world-leading computer algebra system developed by the Computational Algebra Group at the University of Sydney. Modular curves and Galois representations We strongly encourage participation of young researchers. In the discussion meeting, we wish to bring many experts across the world in the area of arithmetic geometry together to share their ideas and the current state of research that will facilitate future research in this direction. We also attempt to strike a balance between the more advanced topics and the down-to-earth examples. These courses would include an introduction to the geometry of modular curves, their ℚ-rational points, classical and non-abelian Chabauty methods, and related computational aspects. In the mini-courses, we give an advanced introduction to the theory of rational points on modular curves, under both theoretical and computational aspects. The objective of this program is to understand the theoretical and computational aspects of determining K-rational points on modular curves X_H(K) for various fields K and subgroups H of GL_2(ℤ/Nℤ) for any natural number N. Modular curves can be thought of as moduli spaces for elliptic curves with extra level structures. In this program, we are mainly interested in the study of rational points on modular curves.Įlliptic curves, modular forms and modular curves are central objects in arithmetic geometry. Over the past few decades, many techniques have been used to decide whether a variety over a number field has a rational point or not, and even to describe those points completely. The study of rational points on varieties is a field of special interest to arithmetic geometers. This special year intends to bring together a mix of people interested in various facets of the subject, with an eye towards sharing ideas and questions across fields. These advances have already led to breakthroughs in multiple different areas of mathematics (e.g., significant progress in the Langlands program and the resolution of multiple long-standing conjectures in commutative algebra), have uncovered new phenomena that merit further investigation (e.g., the discovery of new structures on algebraic K-theory, new period spaces in p-adic analytic geometry, and new bounds on torsion in singular cohomology), and have made hitherto inaccessible terrains more habitable (e.g., birational geometry in mixed characteristic). The last decade has witnessed some remarkable foundational advances in p-adic arithmetic geometry (e.g., the creation of perfectoid geometry and the ensuing reorganization of p-adic Hodge theory). adic arithmetic geometry, organized by Jacob Lurie and Bhargav Bhatt, who will be the Distinguished Visiting Professor. During the 2023-24 academic year the School will have a special program on the p
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