QUANTUM K-THEORY AND OTHER TOPICS IN ENUMERATIVE GEOMETRY
National Science Foundation
BUCH, ANDERS S
RUTGERS THE ST UNIV OF NJ NEW BRUNSWICK
The investigator will attack problems in the broad area of enumerative geometry, especially problems concerning quantum rings and quiver cycles. Enumerative geometry aims to provide the tools required to count the number of geometric objects of given type that satisfy specified conditions. A typical example is the fact that exactly two lines (possibly with complex coordinates) meet 4 randomly chosen fixed lines in 3-space. Enumerative problems are often approached by constructing a moduli space, which has one point for each geometric object of the given type, and then translating the specified conditions into polynomial equations on the moduli space. This transforms the problem into one of counting the solutions to such equations. Quiver cycles can be understood as a way to organize equations that arise in many important situations, and knowledge about geometric invariants of quiver cycles adds to their utility for counting solutions. Ideas from string theory in physics led to the definition of the quantum cohomology ring of a homogeneous space, which provides an efficient tool for counting the number of curves of given degree that meet fixed subvarieties in the space, at least when this number is finite. When the number of solution curves is infinite, the set of these curves form a space called a Gromov-Witten variety, and geometric invariants of this space can tell us much about the problem. For example, if the Gromov-Witten variety has non-zero Euler characteristic, then solution curves do exist. Quantum K-theory is a generalization of quantum cohomology that encodes the Euler characteristic of Gromov-Witten varieties. The investigator plans to study these topics with a combination of geometric and combinatorial methods.
City: NEW BRUNSWICK
Country: UNITED STATES
Award Notice Date: 11-Aug-2009
Project Start Date: 15-Aug-2009
Budget Start Date:
Project End Date: 31-Jul-2012
Budget End Date:
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National Science Foundation
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