A great resource to this is the list of open questions composed by my colleague, Sergey Bolotin [list].
See also Section 5 in the short lecture notes by Craig Evans [Ev].
It is known now that the periodic homogenization theory of Hamilton-Jacobi equations and weak KAM (Aubry-Mather) theory are much related through the cell (ergodic) problems, studied in an unpublished paper of Lions, Papanicolaou and Varadhan [LPV].

In Paper 31, we study further the aforementioned connection and obtain the optimal rate of convergence in various situations in periodic homogenization of Hamilton-Jacobi equations in convex setting.
We find a natural connection between how fast the average of backward characteristic converging to its rotation vector and rates of convergence in the homogenization problem.
In Paper 40, we continue exploring this connection for the homogenization of first-order front propagation problems.
We show that if the effective front is a polytope, then we have optimal rate of convergence.

In Paper 52, we finally obtain the optimal rate of convergence $O(\epsilon)$ for the general convex case.
We also get the optimal rate for a class of nonconvex Hamiltonians.
We rely on an equal curve cutting lemma.

Stochastic homogenization: See Papers 10, 12, 15, 16, 18, 25, 28, 37.
Papers 12, 18, 28 study stochastic homogenization of nonconvex Hamilton-Jacobi equations (see Question 32 in [list] by Takis Souganidis, which also has been asked by Lions, Varadhan, and some others).
Besides, homogenization in dynamic random environment is quite tricky, and there are many questions to be studied.

Periodic homogenization: See Papers 6, 31, 40, 52, 53, 56.

I am extremely interested in nonconvex Hamilton-Jacobi equations (see Papers 1, 2, 12, 18, 24, 28, 39).
Paper 2 addresses weak KAM theory in the nonconvex setting (see nonconvex Hamiltonians in Section 5 of [Ev], and a related question, Question 11 in [list] by Albert Fathi).

Paper 28 is the first attempt to systematically study properties of the effective Hamiltonian arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations.
Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex effective Hamiltonians.
Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry.
Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations.
Some conjectures and problems are also proposed. We are just at the beginning of this direction. Papers 12, 18 belong to this direction.

In Paper 52, we get the optimal rate of convergence in periodic homogenization for a class of nonconvex Hamiltonians.
See a conjecture (Summer 2022) I posed in this direction pdf

Up to now, the properties of the cell problems and the effective Hamiltonians are still quite mysterious and not yet well-studied.
As it was shown explicitly in an example in [LPV], the cell problem does not have unique solutions (even up to additive constants) and thus how can we select one solution out of these many?
This leads naturally to a question on the selection problem of the vanishing discount approximation procedure.
Roughly speaking, does the vanishing discount procedure select one and only one solution out of these infinitely many solutions of the cell problem?
In Paper 17, we get the convergence result for degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians.
In Papers 23 and 26, we have been able to handle the general convex,
possibly degenerate fully nonlinear elliptic equations under various types of boundary conditions (periodic, state constraint, Dirichlet, Neumann boundary conditions).
This means that convex cases are more or less complete.
In Paper 24, we have studied some specific nonconvex cases (quasiconvex, double-well type Hamiltonians in one dimension).

In Paper 32, we provide a simple way to find uniqueness sets for ergodic problems of first and second order Hamilton-Jacobi equations by using a PDE approach.
Paper 36 is also along this line, in which we deal with a generalized ergodic problem of contact Hamilton-Jacobi equation type.

** **

We basically look at the linearized operator the nonlinear equation around the solution and study the adjoint equation to this linearized operator.
By using this new adjoint equation and integration by parts, we derived some new estimates and conservation laws.
These new estimates are quite helpful in obtaining some fine properties of solutions.
See Papers 1, 2, 3, 5, 8, 9, 17, 24, 32, 36, 44.

** **

Large time behavior of solutions to Hamilton-Jacobi equations has received a lot attention recently. See Papers 4, 7, 8, 9.
In Papers 8, 9, we were able to obtain the large time behavior result for viscous Hamilton-Jacobi equations with possibly degenerate diffusions and uniformly convex Hamiltonians.
In Paper 44, we obtained the large time profile for Hamilton--Jacobi--Bellman equations.

** **

Policy iteration for the deterministic control problems -- a viscosity approach: Paper 58.

Numerics for Hamilton-Jacobi equations via the convexification method: Paper 47. See also Paper 51.

State-constraint static Hamilton-Jacobi equations in nested domains: In Paper 39, we study quantitative behaviors of state-constraint solutions in the given nested domains.
Both convex and nonconvex cases are analyzed.

Viscosity solutions of general viscous Hamilton-Jacobi equations: In Paper 11, we present comparison principles, Lipschitz estimates and study state-constraint problems for degenerate, second-order Hamilton-Jacobi equations.

Weakly coupled systems and random switchings: See Papers 4, 6, 7, 14, 19.
In the convex setting, a weakly coupled system of Hamilton-Jacobi equations corresponds to an optimal control problem,
which appears in the dynamic programming, for the system whose states are governed by random changes.
The random changes of the states are controlled by a continuous-time Markov chain.

The basic question is that how can we read off microscopic information if we are given macroscopic information?
We propose some inverse problems to study properties of the effective Hamiltonians from a different angle (sort of an extrinsic way).
A simple way to address this kind of question is that if the two different Hamiltonians give the same effective Hamiltonian,
then what can we say about the relations between the two? This has rich connections to dynamical systems, KAM, weak KAM theories.
See Papers 20, 30.
Paper 20 formulates this kind of questions for the first time.
In Paper 30, we prove a rigidity result in two dimensions, which partially resolves a conjecture stated in Paper 20.

In Paper 22, we study the effective burning velocity from a combustion model.
The level sets of the effective burning velocity are convex, but not much else is known.
We study the non-roundedness, flatness of the level sets of the effective burning velocity, which is a type of inverse problems.
A similar question to that of Paper 20 is also studied.
Paper 28 also belongs to this category.

In Paper 40, we show that in three dimensions or higher, one can always obtain effective fronts of centrally symmetric polytopes with rational coordinates and nonempty interiors.

In Paper 53, we show that in two dimensions, one can always obtain effective fronts of centrally symmetric polygons with rational coordinates and nonempty interiors. The two dimensional case is much harder because of the topological restriction.

In Paper 56, we study the differentiability of effective fronts in the continuous setting in two dimensions.
Our main result says that that the boundary of the effective front is differentiable at every irrational point.
Combining with the sufficiency result in Paper 53, our result implies that for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.

Level-set mean curvature flows and crystal growth: See Papers 21, 34, 38, 65.
We are interested in the asymptotic growth speed of crystals.
The equation of interest is of birth and spread type nonlinear PDEs.
In Papers 21, 34, we show that the asymptotic growth speed exists under fairly general assumptions.
The asymptotic growth speed is the first term in the asymptotic expansion of the solution.
Then, in Paper 38, we study large time behavior of the solution in some specific cases.
This is roughly the next term in the asymptotic expansion of the solution, and the large time behavior is quite complicated since the equation is degenerate and not convex in the gradient variable.
The general case is still open.

For the spiral crystal growth model, we provide a systematic study in Paper 65.
We obtain the uniform Lipschitz estimate of the solution (under a strong condition on the forcing term $c$), the existence of the asymptotic growth rate $S_c$, some estimates of $S_c$ and its strong dependence on the boundary of $W$.
We also understand that this problem is very different from the birth-and-spread model because of the failure of large time behaviors through a clear example.

Related problems: See Paper 35.

In Paper 50, we study the level-set forced mean curvature flow with the Neumann boundary condition.
We first show that the solution is Lipschitz in time and locally Lipschitz in space.
Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz.
We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations.
Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

In Paper 61, we study the bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows.
We show that the effective burning velocity associated with shear flows in dimensions three or higher ceases to exist when the flow intensity surpasses a bifurcation point.
The characterization of the bifurcation point in three dimensions is closely related to the regularity theory of two-dimensional minimal surface type equations.
As a consequence, a bifurcation also exists for the validity of full homogenization of the curvature G-equation associated with shear flows.

We are interested in a critical Coagulation-Fragmentation (C-F) equation and study the wellposedness of mass-conserving solutions.
More precisely, we focus on the case of multiplicative coagulation kernel and constant fragmentation kernel.
For this particular case, there is a sharp conjecture saying that the result depends on the mass $m(0)$ of the initial data.
In particular, the conjecture says that if $m(0)>1$, then there is no mass-conserving solutions; and if $m(0) \leq 1$, then there is a unique mass-conserving solution.

In Paper 41, we use the Bernstein transform to transform the critical C-F equation to a singular Hamilton-Jacobi equation.
This critical Hamilton-Jacobi equation is new in the literature of viscosity solutions.
We prove that indeed, if $m(0)>1$, then there is no mass-conserving solutions.
We show that if $m(0) \leq 1$, then there is at most one mass-conserving solution.
Then, we obtain existence of such a mass-conserving solution for $0< m(0) < 1/2$.
The existence of such solutions in case where $1/2 \leq m(0) \leq 1$ is still open.

In Paper 43, we study large time behavior for this critical Hamilton-Jacobi equation in the critical mass case $m(0)=1$.
Thus, large time behavior is well understood for all $0< m(0) \leq 1$.

In Paper 55, we study further the case $m(0)>1$ and prove that there exists a unique mass-conserving solution locally in time. Clearly, in this case, there is no global mass-conserving solution and hence the understanding is rather complete in this case.

A conjecture (Summer 2020) related to Papers 42, 46, pdf.

This conjecture was proved in Paper 54. Further analysis of the diffusion matrices were studied here.

Stochastic integrability of heat kernel bounds: Paper 57.

Optimal convergence rates in stochastic homogenization in a balanced random environment: Paper 59.

Late-time cosmological solutions and bound of the $\epsilon$-parameter: Papers 60, 62.

Analytic bounds on the equation of state of a cosmological fluid on general effective theories of ekpyrosis: Paper 64.

In Paper 27, I introduce a nonconvex Mean Field Games system by studying a
model with a large number of identical pairs of players who are all rational, and
each pair plays an identical zero-sum differential game. Wellposedness is then studied
for a simple system of this kind.

Partial regularity for minimizers of singular energy functionals and application to liquid crystal: See Paper 13.

Degenerate linear parabolic equations in divergence form: Papers 48, 49.

Degenerate linear parabolic equations in nondivergence form: Paper 63.