My CV
ResearchGateGCMC conjecture
Information Theory with Algebraic Geometry
In the past decade, June Huh solved several long-standing open problems on log-concave sequences in combinatorics. The ground-breaking techniques developed in those work are from algebraic geometry: “We believe that behind any log-concave sequence that appears in nature there is such a Hodge structure responsible for the log-concavity”.
A function is called completely monotone if its derivatives alternate in signs, A fundamental conjecture in mathematical physics and Shannon information theory is on the complete monotonicity of Gaussian distribution (GCMC), which states that the Fisher information of random variables X+Z is completely monotone in t, where Z is a Gaussian noise with variance t, X and Z are independent.
Inspired by the algebraic geometry method introduced by June Huh, GCMC is reformulated in the form of a log-convex sequence. In general, a completely monotone function can admit a log-convex sequence and a log-convex sequence can further induce a log-concave sequence. The new formulation may guide GCMC to the marvelous temple of algebraic geometry.
- Algebraic structure on information theory
- Differential manifold and Hodge theory
- Hodge theory for matroids
Integrable Probability
Integrable probability is an area of research at the interface between probability theory, mathematical physics, combinatorics and representation theory. It refers to the study of probabilistic models that are exactly solvable.
The notion of exact solvability or integrability is somewhat vague. A model is called exactly solvable when observables of interest can be computed by a formula involving well-known functions (rational functions, exp, sin, Γ), so that the complexity of the formula does not increase as parameters go to ∞.
