Collaborative Research: Toric Geometry, Tropical Geometry, and Combinatorial Buildings

The proposal addresses some foundational problems in the intersection of algebraic geometry, combinatorics and representation theory. It aims to explore new con- nections between three important areas of algebra and geometry, namely study of families of algebraic varieties, tropical geometry and theory of buildings. We hope that this will signicantly enhance our understanding of these areas and extend the scope of their appli- cations. Intellectual merit: Algebraic geometry is the study of solution sets of polynomial equa- tions called algebraic varieties. It has applications in many elds as diverse as high energy physics, coding, cryptography and mathematical biology. Understanding of how algebraic varieties behave in families is one of the oldest and central problems in the eld. They appear in all branches of algebraic geometry and its applications, e.g. the geometric Lang- lands program is concerned with understanding principal bundles on curves, a very special class of families. Bundles are also main players in gauge theory in high energy physics. Toric varieties are a large class of varieties whose geometry is intimately connected with combinatorics of convex lattice polytopes. They play a central role in contemporary al- gebraic geometry. Tropical geometry is a relatively recent area of research and concerns study of piecewise linear geometry and has roots in convex optimization. Tropical geome- try translates numerous problems in algebra and geometry into combinatorial and convex geometric problems which are many times more tractable. It is one of the most active re- search areas in contemporary algebraic geometry. The theory of buildings initiated in the pioneering works of Tits and Bruhat, is an area of combinatorial representation theory and has deep connections with topology and dierential geometry. It aims to unravel hidden combinatorial geometry structures in matrix groups and related spaces. The projects in the proposal revolve around the common theme of study of families of algebraic varieties over a toric variety base, or a toric family for short. Our insight is that the combinatorial gadgets for understanding toric families come from tropical geometry and building theory. This far extends the pioneering work of Klyachko on toric vector bundles (ICM 2002 invited talk). It gives new perspectives on the subject, incorporates the separate works of many others into one unied frame work and provides powerful tools to attack old problems as well as suggesting numerous new avenues for further research. Broader impacts: The project involves elementary ideas from linear algebra and convex geometry, which makes it accessible to a wide mathematical audience. Parts of this project have already been woven into undergraduate research experiences where they can serve as an introduction to algebraic and convex geometry. Along these lines, the second PI directs the UK Math Lab, an undergraduate mathematics research and visualization lab at the University of Kentucky. This project will help the UK Math Lab produce implements for introducing concepts from geometry and combinatorics to a younger audience, aid undergraduates in carrying out their rst research projects, and provide for their travel to conferences to present their ideas to their peers. 1