Tree Metrics, the Four-Point Condition, and Cactus Metrics

The first note:

• explains how a local condition characterizes the tree realization of a metric;
• gives a short constructive proof, rebuilding the tree one point at a time through a geometric “projection” to the tree.

The second note gives a characterization of cactus from Khanfir (2026) and answers why four-point tests cannot recognize cactus metrics. I presented this topic at the combinatorics seminar.

Cubic Boolean Functions with Small Support on Ordinary and Rainbow Johnson Slices

This note:

• asks how few fixed-size sets can receive the value 1 under a nonzero Boolean function represented by a polynomial of degree at most three;
• proves sharp support bounds, attained only by boolean functions testing whether a set contains a fixed triple (star indicator).

These findings and discussions build on Yaokun Wu's work on low-degree Boolean functions on Johnson slices.

From Closed-Loop Integrals to Cyclic Monotonicity: Potential Energy and Revealed Preference Theory

This note:

• explains cyclic monotonicity as a discrete analogue of the closed-loop integral condition for conservative force fields;
• develops the parallel between potential energy in physics and utility recovery in revealed preference theory;
• briefly presents Afriat’s Theorem and explains how cyclic monotonicity connects GARP to utility rationalization.

I reorganized my handwritten notes into this document by ChatGPT.

Reducing Dimensions of Semidefinite Programs with Wedderburn-Artin Theorem

This note:

• provides a brief introduction to semidefinite programming;
• demonstrates how the Wedderburn-Artin Theorem can be used to reduce the dimensions of semidefinite programs;
• includes a simple example to illustrate the application: Lovász Number $\vartheta$ of a cycle graph.

I presented this topic at MATH3402: Representation Theory of Groups and Algebras.

Integer Formulation for the Houtman-Maks Index

This note:

• introduces the Houtman–Maks Index in revealed preference theory and its connection to rationalizability and Afriat’s inequalities;
• develops a geometric and combinatorial perspective based on sign vectors, hypersum spans, and convex hulls, and uses it to derive an integer programming formulation for the index;
• compares the resulting formulation with an existing formulation and illustrates its computational performance through numerical experiments.

I presented this topic at MATH6104: Algebraic Combinatorics.

Shapley-Folkman Lemma in Economics

This note:

• provides a brief introduction to Consumer Theory and General Equilibrium;
• demonstrates how the Shapley-Folkman Lemma ensures an approximate General Equilibrium in large non-convex economies.

I presented this topic at MATH6104: Algebraic Combinatorics.

Afriat's Theorem and Graph (Working)

This note provides:

• a brief introduction to Revealed Preference Theory and Afriat's Theorem;
• an elementary graphical proof of Afriat's Theorem;
• an $O(n^2)$ graph algorithm for testing GARP based on the proof;
• simple upper and lower bounds for the problem of covering choice data with the minimum number of preferences.

Talk at Micro Theory Reading Group (MTRG)

These are slides from my presentation at MTRG about "Preference for Flexibility and Randomization under Uncertainty" by Kota Saito.

Continuous Ramsey Model under Population Decline

Far from rigorous and probably containing a few mistakes, this ODE course project was my first attempt to independently explore a concrete problem in computational economics.