Noncommutative Ring and Representation Theory

• The orbit method for the Virasoro algebra.

  ArXiv

  We constructed a version of the orbit method for the Witt and Virasoro. We constructed a well-defined Dixmier map from the Poisson primitive spectrum of the symmetric algebra of Vir to the primitive spectrum of the universal enveloping algebra of Vir.

  Our method involves new ring homomorphisms from U(W) to the tensor product of a localized Weyl algebra and the enveloping algebra of a finite-dimensional solvable subquotient of W. We further show that the kernels of these homomorphisms are intersections of the primitive ideals constructed from natural subsets of W*

• On the problem of computing the Ext-groups in a category of tensor modules over a Mackey Lie algebra

  - Bachelor thesis with Dr. Ivan Penkov

Computer Science projects

• What do Graph Neural Networks learn? Insights from Tropical Geometry

  NeurIPS 2024, with Vikas Garg.

  Graph neural networks (GNNs) have been analyzed from multiple perspectives, including the WL-hierarchy, which exposes limits on their expressivity to distinguish graphs. However, characterizing the class of functions that they learn has remained unresolved. We address this fundamental question for message passing GNNs under ReLU activations, i.e., the de-facto choice for most GNNs. We first show that such GNNs learn tropical rational signomial maps or continuous piecewise linear functions, establishing an equivalence with feedforward networks (FNNs). We then elucidate the role of the choice of aggregation and update functions, and derive the first general upper and lower bounds on the geometric complexity (i.e., the number of linear regions), establishing new results for popular architectures such as GraphSAGE and GIN. We also introduce and theoretically analyze several new architectures to illuminate the relative merits of the feedforward and the message passing layers, and the tradeoffs involving depth and number of trainable parameters. Finally, we also characterize the decision boundary for node and graph classification tasks.

• Semantic Middleware for Optimizing Green Electricity Usage in Manufacturing

  -with Tina Boroukhian, Kritkorn Supyen, Christopher William Mclaughlan, Atit Bashyal and Hendro Wicaksono (2024).