# TriangleCats

It is a happy coincidence that in Chinese *‘TriangleCat’*, or ‘三脚 猫’ (san1jiao3mao1), refers to a proverb translating approximately as *‘jack of all trades, master of none’,* because in mathematics a Triangulated Category can be associated with objects arising from disparate mathematical areas. Interestingly, from a given triangulated category it is technically difficult to extract or ‘decode’ the information one desires.

Chunyi LiLink opens in a new window has been awarded a Whitehead PrizeLink opens in a new window by the London Mathematics SocietyLink opens in a new window for his deep contributions to a wide range of questions in algebraic geometry. One of the main philosophical questions in Chunyi Li’s research is how to decode the geometric information from a given triangulated category.

and now some of the maths

The fundamental objects to study in algebraic geometry are algebraic varieties, classically defined as zeros of multivariate polynomials, for example the zeros of the polynomial* x*^{2}+*y*^{2}-*z*^{2}. Modern algebraic geometry adopts highly abstract algebraic structures and constructions. Each algebraic variety* *X is associated with a triangulated category D(X), which encodes much information on X itself.

The primary tool that Chunyi uses is the notion of stability conditions on triangulated categories introduced by Tom BridgelandLink opens in a new window around 2002. Roughly speaking, a stability condition σ on a triangulated category assigns a class of objects in the category to be stable, subject to some compatibility conditions. Then the moduli space M_{σ}(*v*) parametrizing the σ-stable objects with the same numerical invariant *v* is expected to be a good algebraic space. A seminal result by Bridgeland is that there is a natural topological structure on the space Stab(X) of all stability conditions on D(X) which makes Stab(X) a complex manifold. Many of Chunyi’s results in recent years have been inspired by the conjectures and questions concerning Stab(X) and M_{σ}(*v*).

The first question about Stab(X) is whether it is always non-empty. This is unknown in general even when X is of complex dimension three. Technically speaking, the question is to find a Bogomolov type bound for the Chern characters of stable sheaves on X, which is classically unknown for the third Chern character and above. One of Chunyi’s results was the existence of stability conditions on all quintic threefolds, which is one of the most interesting types of Calabi–Yau varieties. This is a fast-developing area. The existence of stability conditions on several important types of threefolds and a few types of fourfolds have been found only in the last five years.

The next question is about the topology of the whole manifold Stab(X). When X is a Calabi–Yau variety, a conjecture inspired by homological mirror symmetry is that Stab(X) is connected and simply connected (or more ambitiously, contractible), and a subspace of the stability manifold is conjectured to parameterize the mirror family of its ambient space. The first part of the conjecture is only known for very few cases, and the second part is only accurately stated for some types of Calabi–Yau varieties.

The manifold Stab(X) admits a well-behaved wall and chamber structure: the moduli space M_{σ}(*v*) remains unchanged until the stability condition σ crosses a wall. When X is a surface, there is a Gieseker chamber where M_{σ}(*v*) is isomorphic to the classical moduli space M(*v*) of Gieseker-stable sheaves. In a series of joint works with Xiaolei ZhaoLink opens in a new window, Chunyi described the birational geometry of M(*v*) on the projective plane and its noncommutative deformations via wall-crossing on the stability manifold. One of the important motivations of this project was to prove the strange duality conjecture between different M(*v*)’s on the projective plane. Although Chunyi has not focused on this conjecture for several years, he sincerely wishes some mathematician will solve it in the near future.

WMI Magazine staff

Published 8 October 2022

Surface associated with the polynomial equation *x*^{2}+*y*^{2}-*z*^{2}=0

(image created with GeoGebraLink opens in a new window)