# Distribution of Rational and Holomorphic Curves in Algebraic Varieties (15w5169)

## Organizers

(University of Alberta)

(Université du Québec à Montréal)

(Queen's University)

(University of Houston)

## Description

The Banff International Research Station will host the "Distribution of Rational and Holomorphic Curves in Algebraic Varieties" workshop from March 15th to March 20th, 2015.

The search for integral or rational solutions to algebraic equations is a recurring question
inherited from the mathematicians of antiquity.
As a concrete example one can imagine trying to find all the rational solutions to $x^5+y^5=z^5+1$.
The beautiful insight that fuels arithmetic geometry is the realization that these Diophantine problems
are intimately connected with the intrinsic properties of the algebraic varieties (i.e., the geometric shapes)
that the equations define. To return to the example, suppose we could draw all the real or complex solutions
to the equation above. At any point on the resulting surface we can investigate its geometric properties: How
curved is the surface at that point? Which kinds of curves can be made to pass through it? How large
a holomorphic disk can be mapped into the surface injectively at this point? The surprise is that
these geometric quantities govern the arithmetic problem of finding rational solutions.

Guiding ideas of Lang and Vojta have built up a vast (and largely conjectural)
dictionary between the arithmetic properties of a variety and its purely differential-geometric ones.
Recently there has been spectacular progress in establishing parts of this deep and beautiful correspondence.

This workshop will bring together the world's top experts in Nevanlinna theory, hyperbolicity, and rational
curves on varieties, in an effort to disseminate and share the latest developments and techniques, and stimulate
breakthroughs on the substantial questions which remain.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).