Abstract: G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered:
$$
G_t + \left(1-d\, \Div{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0.
$$
 In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows.  Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures.  In three dimensions,  the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value. The existence result is based on joint work with  Hongwei Gao, Ziang Long and Jack Xin, while the non-existence result is  from collaboration with Hiro Mitake,  Connor Mooney, Hung Tran, and Jack Xin.

In this talk, we present a new proof of the delocalization conjecture for random band matrices. The conjecture asserts that for an $N\times N$ random matrix with bandwidth $W$, the eigenvectors in the bulk of the spectrum are delocalized provided that $W \gg N^{1/2}$. Moreover, in this regime, the eigenvalue distribution aligns with that of Gaussian random ensembles (i.e., GOE or GUE). Our proof employs a novel loop hierarchy method and leverages the sum-zero property, a concept that was instrumental in the previous work on high-dimensional random matrices. 

This work is a joint collaboration with H.T. Yau.

Given a family of algebraic varieties over an irreducible scheme, a natural question to ask is what type of properties of the generic fiber, and how do those properties extend to other fibers. For example, the Hilbert irreducibility theorem states that a dominant map from an irreducible variety X defined over a number field to some projective space which is generically of degree d provides a Zariski dense set of degree d points on X. One can also get quantitative estimates for size of the complement which does not carry the generic property.

We will explore this topic from an arithmetic point of view by looking at several scenarios. For instance, suppose we have a 1-dimensional family of pairs of elliptic curves over a number field,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. One may ask how does the property of "being (non-)isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most B, such that the two elliptic curves at the specializations are isogenous?