Algebra, Combinatorics, and Geometry

Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh. A number of the ongoing research projects are described below. The group also offers a PhD program.

PhD in mathematics with the the Algebra, Combinatorics, and Geometry Group >

Research

Applications of Algebraic Geometry to Group Theory
In a surprising recent development by Borisov and Mark Sapir, periodic orbits of self-maps of affine varieties have been related to the finite quotients of mapping tori of algebraic groups.
Applications of Analysis to Number Theory
The analogy between number fields and fields of meromorphic functions on complex algebraic curves has been known for more than a century.
Combinatorial and Statistical Designs, Set and Graph Partitions
Combinatorial and Statistical Designs, Set and Graph Partitions
Cryptography and Quantum Computation
Kaveh has a side interest in applications of algebraic geometry and representation theory in cryptography and quantum computation.
Equivariant Cohomology
The equivariant cohomology along with the celebrated localization formula provides a strong tool in computing usual cohomology of a geometric object equipped with action of a group.
Formal Theorem Proving
In a formal proof, all of the intermediate logical steps of a proof are supplied. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive and yet less susceptible to logical e
Geometry
One of the principal goals of algebraic geometry is to classify all algebraic varieties up to isomorphism or birational equivalence.
Lattice Polytopes and Toric Varieties
Toric varieties are a particular class of algebraic varieties that are determined by a combinatorial data based on a collection of convex rational cones in the standard n-dimensional real vector space.
Lie theory, Representation theory
Ion's main research area is Lie theory/representation theory. Most recently, he has been interested in Macdonald theory, which provides an uniform framework for the study of several questions regarding the spherical harmonic analysis of real/p-adic r
Motivic Integration and Representation Theory
Several years ago, M. Kontsevich created a new type of integration, called motivic integration, where the values of integrals are not numbers but geometric objects.
Newton-Okounkov Bodies
The theory of Newton-Okounkov bodies attempts to generalize the correspondence between toric varieties and convex polytopes, to arbitrary varieties (even without presence of a group action).
Non-Commutative Algebra and Geometry
Ion maintains an active interest in several topics in non-commutative algebra/geometry: deformation quantization, (finite dimensional) Hopf algebras, graded rings, and categories.
Polynomials with Integer Coefficients
Polynomials with integer and rational coefficients appear in many areas of modern mathematics.
Sphere Packings and Discrete Geometry
The Kepler conjecture asks what is the densest packing of congruent balls in three-dimensional Euclidean space.
Symmetries and Dualities in Physics
Sati's research is interdisciplinary and lies in the intersection of geometry and mathematical/theoretical physics.

Contact Us

The Dietrich School of
Arts and Sciences
301 Thackeray Hall
Pittsburgh, PA 15260
Phone: 412-624-8375
Fax: 412-624-8397
math@pitt.edu

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