The Illinois Geometry Lab (IGL) provides a framework for faculty and graduate students to engage local undergraduates in research.

I have been a graduate student mentor for the following projects as part of IGL.

Click on a project to see details.

The goal of this project is to map out the 'quantum channel zoo' by creating a website hosting a database of the known quantum channels and their mathematical and information-theoretic properties.

This project was an exploration of many interesting facts and conjectures related to uniform distribution and rigidity. There were numerous sub-projects, which included: looking for computational evidence for uniform distribution of fractional parts of powers of rationals, testing the efficacy of low discrepancy sequences for numerical integration, and numerically estimating the ratio of the smallest gap to the largest gap in the three-gap theorem.

There was also a theoretical component, which included understanding the connection between Farey sequence and diophantine approximation, and surveying statements involving the Farey series that are equivalent to the Riemann hypothesis.

Link: poster [PDF].Quantum teleportation is a fundamental task in quantum information theory wherein two parties Alice and Bob use a shared entangled quantum state and classical communication to teleport an unknown quantum state.

This task is mathematically equivalent to a certain quantum state discrimination problem, where the goal is to perform a measurement that optimally distinguishes among a given set of quantum states.

This problem in turn has a useful description in terms of a class of optimization problems called semidefinite programs (SDP).

These SDPs are a generalization of linear programs with a nice duality theory and efficient solvers available in python and MATLAB. In particular, we numerically determine the optimal measurement, as measured by fidelity, for various quantum states with a given form as described in the reference below. We show that adding noise to the state can in fact increase fidelity, confirming and extending previous results. Furthermore, we characterize the set of states that attain the highest fidelity and plot related values.

Link: poster [PDF].What is our department good at? The question is not as simple as it seems: the research group composition changes fast, and their impact on the global scale might be smaller or larger than it seems.

The goal of this project is two-fold: on one hand, to detect the intrinsic research clusters (i.e., groups of people working in close areas, talking to the same communities, publishing in the same journals) within the department. On the other hand, to see how significant these clusters are in the context of their respective fields: which are strong, which are growing...

Link: code.I've been a teaching assistant for the following courses at UIUC:

Fall 2020, Math 220, Calculus

Spring 2021, Math 234, Business Calculus

Fall 2021, Math 221, Calculus I

Spring 2022, Math 257, Linear Algebra with Computational Applications

Summer 2022, NetMath 415, Linear Algebra

and I've been a grader for:

Spring 2021, Math 444, Elementary Real Analysis

Fall 2022, Math 564, Applied Stochastic Processes

Fall 2022, Math 285, Differential Equations

Fall 2022, Math 442, Partial Differential Equations

Spring 2023, Math 415, Applied Linear Algebra

Spring 2023, Math 540, Real Analysis

Spring 2023, Math 466/564, Applied Stochastic Processes

In Spring 2023, I was a participant mentor for Twoples, an online directed reading program organized by Stephen McKean and Julius Frank. I was a mentor for a project on numerical methods for ordinary and partial differential equations.

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