## Grants and Contracts Details

### Description

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

Status | Active |
---|---|

Effective start/end date | 9/1/21 → 8/31/24 |

### Funding

- National Science Foundation: $160,000.00

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