Convex Bodies and Representation Theory (14w5013)


Megumi Harada (McMaster University)

Kiumars Kaveh (University of Pittsburgh)

(University of Toronto)


The Banff International Research Station will host the "Okounkov Bodies and Applications " workshop from February 2nd to February 7th, 2014.

Many problems in both applied and pure Mathematics involve the solution of a system of equations. Algebraic
Geometry is precisely the study of the space of solutions to systems of algebraic equations, and is therefore a
core area of Mathematics. For instance, the proof of Fermat's Last Theorem gives properties of an algebraic-
geometric object (an elliptic curve) associated to a non-trivial solution of Fermat's equation. Also, the theory of
Mirror Symmetry in theoretical physics uses the language of algebraic geometry in a fundamental way.
Algebraic geometry also has applications in quantum computing, mathematical biology, cryptography, and
image and signal processing. Combinatorial and convex geometry, on the other hand, includes the study
of polytopes, which are generalizations of the figures in plane geometry such as triangles, trapezoids, and
parallelograms. The convex geometry of polytopes is important in many research areas, such as optimization
theory. These two core areas of mathematics are related in many ways. The proposed workshop is on a new and exciting research topic which
lies at the intersection of these two areas: Okounkov bodies.

The proposed workshop concerns a research topic which forms an effective training program for future scientists. The long-term benefits of this research
area are two-fold: first, the results of this research will bring to light many new combinatorial techniques
for analyzing the algebraic geometry of important spaces which arise in many real-world applications (e.g.
Mirror Symmetry, cryptography, fluid mechanics, optimization theory), and secondly, the training of young scientists in this area
would result in highly trained individuals at the undergraduate, graduate, and postgraduate level,
who possess competitive research and technical skills in important areas of geometry.

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).