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 >


Bergman kernel and deformation quantization
The asymptotic expansions for the heat kernel and Bergman kernel have many applications.
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
Intersection theory of moduli space of curves
The intersection theory on moduli spaces of curves is connected to KdV hierarchy through the celebrated Witten-Kontsevich theorem.
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.
Principal bundles and the Langlands Program
The Langlands Program is a series of far-reaching conjectures, which first emerged in number theory but then extended to many areas such as algebraic geometry, representation theory, and mathematical physics.
Spectral graph theory and random walk
Spectral graph theory is a subfield of graph theory that mainly concerns properties of a graph pertinent to eigenvalues and eigenvectors of its adjacency or Laplacian matrix.
Sphere Packings and Discrete Geometry
The Kepler conjecture asks what is the densest packing of congruent balls in three-dimensional Euclidean space.


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