# Combinatorics and Discrete Geometry

Combinatorics is the study of finite structures, many of which arise in other branches of mathematics or from problems arising in science or engineering. The study of combinatorics involves general questions of enumeration and structure, matroid theory and aspects of graph theory, partially ordered sets, set partitions and permutations and combinatorial structures such as finite geometries and designs. Techniques tend to be algebraic and topological, involving methods from commutative ring theory, algebraic topology, representation theory and Hopf algebras.

Discrete geometry is concerned with properties of finitely generated geometric objects such as polytopes and polyhedra, triangulations and polyhedral complexes, configurations of lines and, more generally, hyperplanes in Euclidean and other spaces, the theory of rigid and flexible frameworks, tilings and packings. Many problems in discrete geometry arise from questions in computational geometry related to algorithms for analyzing discrete geometric structures.

## Faculty Members

 Marcelo Aguiar Algebra, combinatorics, category theory Robert Connelly Discrete geometry, computational geometry and the rigidity of discrete structures Tara Holm Symplectic geometry Jon Kleinberg Networks and information Robert Kleinberg Algorithms and theoretical computer science Allen Knutson Algebraic geometry and algebraic combinatorics Lionel Levine Probability and combinatorics Karola Meszaros Algebraic and geometric combinatorics Edward Swartz Combinatorics, topology, geometry, and commutative algebra Éva Tardos Algorithm design and algorithmic game theory

## Emeritus and Other Faculty

 Louis Billera Geometric and algebraic combinatorics Ahmed Bou-Rabee Probability, elliptic PDE Christian Gaetz Algebraic combinatorics Nima Hoda Geometric group theory Marie MacDonald Number Theory, commutative algebra, combinatorial geometry, university mathematics education

## Activities and Resources:

Historically, there have been connections between combinatorics, in particular enumeration theory, and questions in probability. In recent decades, there have been close connections between certain areas of combinatorics and questions arising in theoretical computer science and discrete optimization. Even more recently, there have arisen links to biology, in particular, the study of phylogenetics.

The group at Cornell is particularly interested in algebraic and topological combinatorics, questions of enumeration in polytopes and, more generally, matroids, combinatorial Hopf algebras and rigidity in discrete geometric structures.

## Related people  Portia Anderson

Ph.D. Candidate Louis Billera

Professor Emeritus Ahmed Bou-Rabee

NSF Postdoctoral Fellow Andrew Chen

Ph.D. Student  Moriah Elkin

Ph.D. Student Joseph Fluegemann

Ph.D. Candidate Christian Gaetz

Klarman Fellow Raj Gandhi

Ph.D. Student Elena Hafner

Ph.D. Candidate Tara Holm

Professor  Yibo Ji

Ph.D. Student Robert Kleinberg

Associate Professor Jon M. Kleinberg

Tisch University Professor of Computer Science and Information Science and Interim Dean of Computing and Information Science Allen Knutson

Professor Lionel Levine

Professor Yichen Ma

Ph.D. Candidate  Karola Mészáros

Associate Professor Luis Perez

Ph.D. Student Edward Swartz

Professor Éva Tardos

Jacob Gould Schurman Professor Gabriel Udell

Ph.D. Student Alexander Vidinas

Ph.D. Student Prairie Wentworth-Nice

Ph.D. Candidate John Whelan

Ph.D. Candidate Fiona Young

Ph.D. Candidate Joy Zhang

Ph.D. Student

## All research areas

Algebra    Analysis    Applied Mathematics    Combinatorics and Discrete Geometry    Geometry    Logic    Probability and Statistics    Topology
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