Dr Chunyi Li, Royal Society University Research Fellowship
Stability Condition and Application in Algebraic Geometry
Algebraic geometry is one of the most active branches of modern mathematics. It has both great intrinsic beauty as well as many surprising applications to other areas of pure mathematics and real-world problems.
Through his Fellowship, Dr Chunyi Li (Warwick Mathematics Institute) intends to develop the theory of stability conditions by studying some intrinsic questions that motivated the development of this field.
The fundamental objects to study in algebraic geometry are algebraic varieties, classically defined as zeros of multivariate polynomials, for example, the Pythagorean equation x^2+y^2=z^2.
A basic idea of elementary algebraic geometry views the variety (locus of zeros) of the Pythagorean equation as the projective line.
One can write down the full list of Pythagorean triples by knowing two basic solutions, just as one draws the line through two points.
Dr Chunyi Li.
Cross sections of the quintic Calabi-Yau threefold.
Modern algebraic geometry uses highly abstract algebraic structures and constructions. Each algebraic variety has an associated derived category D(X), which encodes most of the information on the variety itself.
The stability condition introduced by Bridgeland is an elegant structure that one expects D(X) to carry. The theory itself guides one to study the principles of mirror symmetry. One of Dr Li’s main recent achievements proves the existence of stability conditions on the quintic 3-fold, the most classic object in mirror symmetry.
The tools developed in the theory also have useful applications in several classical topics in algebraic geometry.