I am interested in algebraic geometry and symplectic geometry inspired by Homological Mirror Symmetry (HMS).

My past research studies HMS for toric varieties under GIT wall-crossings from the perspective of the coherent-constructible correspondence (CCC), window categories, and noncommutative algebra. In short, the HMS/CCC for all birational models of toric varieties over the GKZ fan can be understood uniformly in terms of a combined Lagrangian skeleton and a combined "Cox category". The equivalence between the two is given by an exit (homotopy) path algebra description of the skeleton and its natural monomial labels matching the morphisms of the line bundles in the Cox category forming a full strong exceptional collection. 

Building on these, my current research has two guiding aims:

• To strengthen the connection between Cox-level HMS/CCC and its underlying noncommutative algebraic geometry concerning exceptional collections, mutations, Calabi-Yau completions, and noncommutative (crepant) resolutions.

• To extend this birationally enhanced perspective of toric HMS to a more robust framework connecting the moduli space of Lagrangian skeleta/Weinstein pairs to its mirror birational geometry, applicable to non-toric cases.

Achieving these goals involves using natural symplectic/microlocal constructions to identify meaningful algebraic properties of the coherent derived categories, which is the main challenge in this research philosophy. 

In the context of toric GIT wall-crossing, window theory provides an important tool for relating the derived categories of different birational models inside a common derived category of multigraded modules. One of my goals is to develop a more intrinsic version of window theory directly from the A-side. I hope that such an approach could offer new insights into the underlying geometry of mirror symmetry and lead to some desired applications in algebraic geometry.

Some of my perspectives are also strongly influenced by the rich interaction between quiver theory over toric Calabi–Yau threefold singularities and tilting, torus dimers, and tropicalized fibers of mirror Landau–Ginzburg models, as developed in the physics literature in the 2000s. I also hope to generalize the mathematics of this story in several natural directions connected to the above research aims. 

I am a mathematician interested in the algebra and geometry inspired by a categorical form of type IIA/IIB string duality known to mathematicians by the name of Homological Mirror Symmetry (HMS).

HMS asserts that for a pair of mirror Calabi–Yau manifolds (X,Y), the category of topological B-branes on X is equivalent to the category of topological A-branes on Y. This correspondence exchanges complex moduli with Kähler moduli, coherent sheaves with Lagrangian cycles, and extension groups with Floer cohomologies, providing a precise categorified form of superstring T-duality in the type II theories. HMS captures how two apparently distinct quantum field theories - the topological A-model and B-model - encode equivalent information about the same underlying superstring background. Morphisms in these categories correspond to open-string states, and the compositions record worldsheet instanton corrections.

My past research develops mathematical applications of this duality in settings where both sides can be computed explicitly, primarily for toric varieties.  Toric varieties admit combinatorial descriptions that make their mirrors accessible via Laurent polynomials, whose Fukaya categories can be modelled microlocally by constructible sheaves on real tori.  This makes it possible to realize D-brane gauge theories, wall-crossing phenomena, and noncommutative resolutions entirely within a unifying combinatorial symplectic geometry framework. My current research has two guiding aims:

• To solve concrete problems in commutative and noncommutative algebraic geometry using HMS as a mechanism;

• To develop the combinatorial symplectic geometry needed to describe categorical wall-crossing and phase transitions of (supersymmetric) gauge theories.

Achieving these goals involves using the correct mirror constructions to identify meaningful algebraic properties of the coherent derived categories. I have studied how this works concretely in the context of toric GIT wall-crossings using window categories. These categories correspond to different gauge theory sectors within a common "ultraviolet" category. Each infrared phase selects a subset of UV branes which survive under a particular stability condition and RG flow to that phase. My work uses symplectic and microlocal techniques to describe the geometry of the mirror A-brane transports and develop applications of this geometric description to birational geometry and noncommutative algebra. 

Some of my perspectives are heavily inspired by quiver gauge theory, brane tilings, brane webs and related geometry appearing in physics literature.