Mathematics
Mathematics has become one of my favorite creative pursuits because it has the unmatched ability of illuminating difficult concepts or connecting disparate fields. In the future, I plan to upload notes to this site relating to the projects that I am working on.
Favorite Theorem - Matrix Representation Theorem
Let $V$ and $W$ be finite-dimensional vector spaces over a field $\mathbb{F}$, with ordered bases
\[\mathcal{B}=(v_1,\dots, v_n), \quad \mathcal{C} = (w_1\dots, w_m).\]
For any linear map $T:V\to W$, there exists a unique matrix $[T]_{\mathcal{C}\gets \mathcal{B}}\in \mathbb{F}^{m\times n}$ such that for every vector $x\in V$,
\[[T(x)]_{\mathcal{C}}=[T]_{\mathcal{C}\gets \mathcal{B}}[x]_{\mathcal{B}}\]
We owe almost all of modern science to this elegant theorem. Every linear transformation of finite fields can be described with a matrix! And that's how we get computer graphics!
Completed Courses
- Calculus 1-3 (Stewart)
- Differential Equations (Brannan)
- Linear Algebra (Axler)
- Real Analysis (Lay)
- Topology (Hatcher)
- Abstract Algebra 1 & 2 (Matsuura & Gallian)
- Number Theory (Burton)
Undergraduate Research
Decentralized multi-agent systems arise in distributed optimization and federated learning where agents collaboratively optimize a global objective using only local computation and peer-to-peer communication. Decentralized multi-agent systems can become vulnerable to faulty agents.
Consider a fully connected graph with $n$ agents, honest set $\mathcal{H}$, and Byzantine set $\mathcal{B}$ with $|\mathcal{B}|\leq f$. Each agent $i$ holds local loss $Q_i(\cdot)$ and parameter $x_i\in\mathcal{W}\subseteq\mathbb{R}^d$. The goal is to design a decentralized algorithm so each honest agent's local parameter approaches:
\[x^* \in \arg\min_{x\in\mathcal{W}} \sum_{i\in\mathcal{H}} Q_i(x).\]
Note that we consider heterogeneous loss functions, i.e. $Q_i(x)\neq Q_j(x)$, for $i\neq j$.
Our research studies how close do the minimizers of each local cost function need to be to each other to guarantee global, approximate convergence to the global minimizer.