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MAT/03 - Geometry

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Description

Geometry is a branch of mathematics concerned with the properties and classification of both continuous and discrete geometric structures, such as algebraic varieties, topological spaces, differential manifolds, and block designs. It has deep connections with all major areas of mathematics, ranging from partial differential equations to number theory, and provides powerful tools for both physics and engineering. The MAT/03 research group in the Engineering Department is dedicated to studying various fields of geometry, including general and set-theoretic topology, schemes, and combinatorial designs.

Research Topics

There are three lines of research that are currently in progress:

  • The first one focuses on schemes with unexpected behavior with respect to either a general fat point or a general projection. The latter includes the geproci sets, which draw from algebraic geometry, commutative algebra, combinatorics, representation theory, and quantum mechanics. We also study varieties supported by numerical, algebraic, and combinatorial structures.
  • Our second line of research involves studying questions in general topology using techniques from both topology and mathematical logic. Specifically, we investigate topological cardinal invariants, convergence properties, and Corson compacta, utilizing set-theoretic tools such as infinitary combinatorics, elementary submodels, and forcing. Some of our main ongoing projects include studying cardinal invariants on the G_delta topology of a space, exploring the impact of infinite games on the cardinal properties of topological spaces, and investigating the existence of dense metrizable subspaces in certain subclasses of Corson compacta.
  • The third line of research concerns additive block designs and includes the study of Steiner triple systems of order 15 containing at least one Fano plane. This is done in terms of the corresponding 1-factorization of the complete graph K_8 and the Pasch configurations. We also study the simultaneous automorphisms of two orthogonal Fano planes and the oriented Fano plane. One of our goals is to characterize the affine and projective Steiner triple systems in terms of missing configurations.

Keywords

Combinatorics; cardinal invariants; elementary submodels; schemes; block designs.