# Research

My advisors are Alexei Novikov and Yuri Suhov. My interests cover several topics in partial differential equations and probability, including stochastic processes, random matrices, KPZ equations, etc.

From Gowers’ Blog

One of the holy grails of mathematics is to prove universality, which basically says that if you stand back from a self-avoiding walk so that the mesh of the lattice is too small for you to see it, then you will not be able to tell what the model is.

The large-scale behaviour of a statistical system should be largely insensitive to the precise small-scale geometry of that system, after normalising some key parameters.

Check out my publication list (currently empty).

### Notes

These include conference notes, research notes, or any significant notes that I want to write down to better understand the materials. Every note will have a respective blog post; I will sometimes $\LaTeX$ the notes for convenience. Mistakes are unavoidable, but I will write anyway.

Topic (Blog Post) PDF (if available)

### Random Notes

(these are shorter notes with less specific topic, sometimes a quick proof, sometimes outside of my main research line or even in other subjects)

≡ Mathematics Categories (borrowed from arXiv)
math.AG - Algebraic Geometry
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
math.AT - Algebraic Topology
Homotopy theory, homological algebra, algebraic treatments of manifolds
math.AP - Analysis of PDEs
Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
math.CT - Category Theory
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra
math.CA - Classical Analysis and ODEs
Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
math.CO - Combinatorics
Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
math.AC - Commutative Algebra
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
math.CV - Complex Variables
Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
math.DG - Differential Geometry
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
math.DS - Dynamical Systems
Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
math.FA - Functional Analysis
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory
math.GM - General Mathematics
Mathematical material of general interest, topics not covered elsewhere
math.GN - General Topology
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties
math.GT - Geometric Topology
Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
math.GR - Group Theory
Finite groups, topological groups, representation theory, cohomology, classification and structure
math.HO - History and Overview
Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics
math.IT - Information Theory
Covers theoretical and experimental aspects of information theory and coding.
math.KT - K-Theory and Homology
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
math.LO - Logic
Logic, set theory, point-set topology, formal mathematics
math.MP - Mathematical Physics
Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics
math.MG - Metric Geometry
Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces
math.NT - Number Theory
Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
math.NA - Numerical Analysis
Numerical algorithms for problems in analysis and algebra, scientific computation
math.OA - Operator Algebras
Algebras of operators on Hilbert space, C^*-algebras, von Neumann algebras, non-commutative geometry
math.OC - Optimization and Control
Operations research, linear programming, control theory, systems theory, optimal control, game theory
math.PR - Probability
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
math.QA - Quantum Algebra
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
math.RT - Representation Theory
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
math.RA - Rings and Algebras
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
math.SP - Spectral Theory
Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
math.ST - Statistics Theory
Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
math.SG - Symplectic Geometry
Hamiltonian systems, symplectic flows, classical integrable systems

Topic (Blog Post) Category PDF (if available)