Hello! I am a postdoctoral fellow in the Department of Mathematics at University of Toronto, working with Joel Kamnitzer and Eckhard Meinrenken. Before coming to Toronto, I spent a year at the Max Planck Institute for Mathematics in the research group of Christian Blohmann. I completed my Ph.D. in 2021 in the Department of Mathematics at the University of Chicago, under the direction of Victor Ginzburg.
I received my B.Sc. degree in mathematics from the University of Hong Kong in 2014.
Here is my C.V..
My research interest lies in the junction of geometric representation theory and symplectic/Poisson geometry. I am particularly interested in the interaction of representation theory and Poisson geometry with cluster algebras and combinatorics. The problems that I think about usually have to do with compatible cluster structures, deformation quantization, integrable systems, mirror symmetry, toric varieties, matroid Schubert varieties, oriented matroids, etc.
In particular, I am interested in anything that is even remotely related to the standard Poisson structure on a semisimple Lie group. Examples of such objects include generalized minors, the variety of Lagrangian subalgebras, the wonderful compactification of a semisimple Lie group and of a Cartan subgroup, additive/tropical analogues of the wonderful compactifications, quantum cohomology, the Yang-Baxter equation, quantized universal enveloping algebras, crystal bases, quantum boson algebras, abelian ideals, Kazhdan-Lusztig theory for an affine Weyl group, etc.
The following two paragraphs illustrate the typical kinds of projects that I work on.
Together with Yanpeng Li and Jiang-Hua Lu, we developed a new method for constructing polynomial integrable systems on the dual space of a class of finite dimensional Lie algebras equipped with the Kirillov-Kostant-Souriau Poisson structure. Let \((P, \pi)\) be an \(n\) dimensional complex Poisson manifold, and \(p\) a point of \(P\) where \(\pi(p) = 0\). The cotangent space \(T^*_pP\) to \(P\) at \(p\) is a Lie algebra whose Lie bracket is determined by \[[d_p \varphi, d_p \psi] = d_p \bigl( \{\varphi, \psi\}_{\pi} \bigr),\] for holomorphic functions \(\varphi, \psi\) defined near \(p\). The tangent space \(T_pP\) then carries the corresponding Kirillov-Kostant-Souriau Poisson structure \(\pi_0\).
For a holomorphic function \(\varphi\) defined near \(p\), its lowest degree term \(\varphi^{{\rm low}}\) consists of those terms with the lowest possible degree in the Taylor expansion of \(\varphi\) with respect to a local holomorphic coordinate system on \(P\) centered at \(p\). The expression \(\varphi^{{\rm low}}\) is a well-defined function on \(T_pP\). The lowest degree term of a (meromorphic) tensor field is definded analogously.
A log-canonical system on \((P, \pi)\) is a set \(\{\varphi_1, \ldots, \varphi_n\}\) of holomorphic functions (defined near \(p\)) such that \[\{\varphi_i, \varphi_j\}_{\pi} \in \mathbb C \varphi_i \varphi_j ~ \forall i,j \in [1,n] \quad \text{and} \quad d\varphi_1 \wedge \cdots \wedge d\varphi_n \ne 0.\] It is evident that \[\{\varphi_i^{{\rm low}}, \varphi_j^{{\rm low}}\}_{\pi_0} = 0\] for all \(i,j \in [1,n]\). We proved that if \({\rm deg} \bigl( (d \log \varphi_1 \wedge \cdots \wedge d \log \varphi_n)^{{\rm low}} \bigr) \leq r_0\), where \(2r_0\) is the rank of \(\pi_0\) in \(T_pP\), then any maximal algebraically independent subset of \(\{\varphi_1^{{\rm low}}, \ldots, \varphi_n^{{\rm low}}\}\) is an integrable system on \((T_pP, \pi_0)\).
An abundant supply of log-canonical systems have recently emerged in the study of compatible cluster structures. These are cluster or generalized cluster structures on the coordinate ring of an irreducible rational quasi-affine Poisson variety such that the cluster variables that belong to the same cluster are pairwise log-canonical. A nice feature of our method is that, for a compatible cluster structure on the coordinate ring of \((P, \pi)\), the number \({\rm deg} \bigl( (d \log \varphi_1 \wedge \cdots \wedge d \log \varphi_n)^{{\rm low}} \bigr) \) is invariant under cluster mutations. Hence if the inequality above is satisfied by one cluster, then taking the lowest degree terms of every cluster gives rise to an integrable system on \((T_pP, \pi_0)\). In particular, our method makes it possible to speak of "mutation of integrable systems": if two clusters are related by a mutation of clusters, then their lowest degree terms are related by a mutation of integrable systems. We are working on packaging all these into a formal notion of cluster integrable systems.
The easiest example of a compatible cluster structure to which our method is applicable is the standard cluster structure on the coordinate ring of a Bruhat cell \(BwB/B\) in the flag variety \(G/B\). The Poisson structure on \(BwB/B\) is the restriction of the standard Poisson structure on \(G/B\), and the point where the Poisson structure vanishes is \(wB/B\). Our method then gives rise to an infinite family of integrable systems on \((\mathfrak n_{w,-}^*, \pi_0)\), where \(\mathfrak n_{w,-} := \mathfrak n_- \cap {\rm Ad}_w \mathfrak n\).
Together with Sam Evens, we defined and studied the wonderful compactification \(\bar {\mathfrak h}\) of a Cartan subalgebra \(\mathfrak h\) of a semisimple Lie algebra \(\mathfrak g\), in the course of an attempt to understand a degeneration of the variety of Lagrangian subalgebras for \(\mathfrak g \oplus \mathfrak g\) into that for \(\mathfrak g \ltimes \mathfrak g^*\). The variety \(\bar {\mathfrak h}\) is, by construction, an additve/tropical analogue of the De Concini-Procesi wonderful compactification of the Cartan subgroup \(H\) of \(G\) with \({\rm Lie}(H) = \mathfrak h\).
We proved that \(\bar {\mathfrak h}\) is a matroid Schubert variety, an object that has drawn intensive research interests in matroid theory and combinatorial algebraic geometry. The real locus \(\bar {\mathfrak h}(\mathbb R)\) of \(\bar {\mathfrak h}\) admits an explicit combinatorial description. The permutohedron is a convex polytope whose vertices are the elements of the Weyl group orbit of the Weyl vector, the half-sum of positive roots. In a joint work with Aleksei Ilin, Joel Kamnitzer, Piotr Przytycki and Leonid Rybnikov, we proved that \(\bar {\mathfrak h}(\mathbb R)\) is homeomorphic to the permutohedron with parallel faces identified. More generally, let \(\mathcal A\) be a central essential real hyperplane arrangement. Together with Leo Jiang, we proved that the real locus \(\bar {\mathfrak h}_{\mathcal A}(\mathbb R)\) of the matroid Schubert variety associated to \(\mathcal A\) is homeomorphic to the zonotope generated by an arbitrary choice of normal vectors to the hyperplanes that belong to \(\mathcal A\), with parallel faces identified.
A consequence of the aforementioned results is an intimate interplay between the combinatorics of \(\mathcal A\) and the topology of \(\bar {\mathfrak h}_{\mathcal A}(\mathbb R)\). One can read off many topological invariants of \(\bar {\mathfrak h}_{\mathcal A}(\mathbb R)\) from the combinatorial data encoded in \(\mathcal A\). For example, the elements of the lattice of flats of \(\mathcal A\) form a graded basis of the free \(\mathbb Z\)-module \(H^*(\bar {\mathfrak h}_{\mathcal A}(\mathbb R); \mathbb Z)\).
Conversely, and perhaps more importantly, one expects to be able to extract combinatorial information about \(\mathcal A\) from the topology of \(\bar {\mathfrak h}_{\mathcal A}(\mathbb R)\). The central real hyperplane arrangements are precisely those oriented matroids which are realizable. For a not-necessarily-realizable oriented matroid \(\mathcal M\), the crinkled zonotope \(Z_{\mathcal M}\) of \(\mathcal M\) is a piecewise linear subspace of a cube whose dimension is equal to the cardinality of the ground set of \(\mathcal M\). An important observation in my work with Jiang is that, when \(\mathcal M\) is not realizable, although the notion of matroid Schubert variety associated to \(\mathcal M\) is undefined, the notion of parallel faces of \(Z_{\mathcal M}\) still makes sense. Moreover, the topology of \(Z_{\mathcal M}/(\text{parallel faces})\) is controlled by the combinatorics of \(\mathcal M\) in much the same way as the topology of \(\bar {\mathfrak h}_{\mathcal A}(\mathbb R)\) is controlled by the combinatorics of \(\mathcal A\). We are woking on understanding the structure theory of \(\mathcal M\) using the topology, in particular the intersection cohomology, of \(Z_{\mathcal M}/(\text{parallel faces})\).
My research statement is available upon request.
At University of Toronto, I taught the following courses.
At the University of Chicago, I taught the following courses.
At the University of Hong Kong, I T.A.'d the following course.
Email liyu [at] math [dot] toronto [dot] edu
Office PG 111