we study the quasilinear wave equation $\box_g u=0$, where the metric $g$ depends on $u$ and equals the schwarzschild metric when $u$ is identically 0. under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. this is joint work with hans lindblad.

we explore two problems with applications in detonation shock dynamics and
molecular dynamics. first, we discuss level set methods, which are a
popular approach to modeling evolving interfaces. we present a level set
advection solver in two and three dimensions using high-order finite
elements. our approach leads to stable front propagation and convergence
on high-order, curved, unstructured meshes. the solver's ability to
implicitly track moving fronts lends itself to a number of applications;
in particular, we highlight applications to high-explosive (he) burn and
detonation shock dynamics (dsd).

in the second half, we look at molecular dynamics (md) simulations, which
are widely used to study the motion and thermodynamic properties of
molecules. computational limitations and the complexity of problems,
however, result in the need for error quantification. we examine the
inherent two-scale nature of md to construct a large-scale dynamics
approximation as a means of error estimation.

conical singularities occur quite often and naturally. for example, according to cheeger-colding, under ricci curvature lower bounds, the limit spaces will generally carry singularity of conical type. this process of a family of smooth metrics limiting to a singular metric of conical type will be called conical degeneration. again by cheeger-colding, under rather general conditions, the basic analytic quantities such as the eigenvalues and eigenfunctions will converge. so will the heat kernels (ding). it is rather different story for global geometric invariants defined in terms of the eigenvalues. we will discuss some recent work in this direction.

a number of problems in statistical signal processing model the data as a wide-sense
stationary (wss) time series, whose power spectrum (or equivalently, the covariance
matrix) acts as a sufficient statistic for inferring parameters of interest. the covariance
matrix of such data exhibit toeplitz structure, which can be leveraged to design highly
efficient compressive samplers to reduce the dimension of the wss data, without
requiring it to have a sparse representation. unlike existing results in compressed
sensing, the goal here is to recover the high dimensional covariance matrix (or infer
parameters of interest from it), instead of reconstructing the data itself.

inspired by our past work on nested sensor arrays, i will describe a new sampling
techniques, known as the ``generalized nested sample'' (gns), to acquire compressive
measurements in such a way that it becomes possible to perfectly reconstruct the original
high dimensional covariance matrix from these compressed sketches. i will focus on lowrank
toeplitz covariance matrices and develop an efficient gns-based sampler which
allows the recovery of a rank-r toeplitz covariance matrix from a compressed sketch of
size $o(√r) x o(√r)$, where the size of the sketch has no dependence on the ambient large
dimension n. our reconstruction technique will use a regularizer-free framework,
combined with the ability to extrapolate additional “unobserved” entries of the nxn
covariance matrix. the algorithm has significantly lower computational complexity (that
does not scale with n) compared to recent nuclear-norm based compressive covariance
estimators, and is provably robust against bounded errors. finally, i will consider the
special case of rank-1 and sparse covariance matrices that arise in the important problem
of “phase retrieval” in optical imaging. the role of 2nd order difference sets in complex
phase retrieval will be demonstrated, inspiring the design of a new class of non-uniform
fourier sampler, that can provably recover a complex signal (upto a global phase
ambiguity) from its amplitude measurements with near-minimal number of samples. an
interesting connection with the so-called 4n-4 conjecture will also be established, which
hypothesizes 4n-4 to be the minimum number of measurements necessary to ensure
injectivity in n dimensions for complex phase retrieval.
joint work with heng qiao, graduate student, university of california, san diego.

we compute the picard group of the moduli stack of genus zero stable quotients to projective space, grassmannians, and any sl flag variety in the case of more than 2 markings. furthermore, in the case of exactly 2 markings, we calculate the picard group of the moduli stack of genus zero stable quotients to projective space, grassmannians, and to partial flag varieties where the ranks of the subspaces differ by more than 1. along the way we establish projectivity of the coarse moduli space.

i will summarize the work i have done both myself and with my coauthors during my ph.d. studies. this includes the estimates of moduli of continuity for viscosity solutions in domains in euclidean spaces and on manifolds, asymptotic behavior of nonparametric hypersurfaces moving by powers of gauss curvature, and the classification of four-dimensional shrinking gradient ricci solitons with positive isotropic curvature.

previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle. in this talk we will present the first results of the same type for a non-integrable discrete system: graph homomorphisms form $z^d$ to a regular tree. we will also explain how the technique used could be applied to other non-integrable models.

when utilizing the finite element method in two dimensions, one requires a suitable mesh of the domain they wish to solve on. in this talk we will go over the term suitable, an in depth approach to arrive at this goal, and strategies for programming implementations. this talk will additionally demonstrate the workings behind the culmination of this research: a program which allows users to create 2d triangle meshes for any domain they desire.

recently, mendel and naor, dumitriu and radcliffe, and radcliffe and williamson have begun the study of what could be termed the geometric fiedler vector (or spectral gap) for graphs. their principle observation is that the functional form associated with the graph can be expressed in terms of the distance function on $\mathbb{r}$. we give a partial structural characterization of when the geometric fiedler vector can be extended to a geometric spectrum. additionally, we provide applications of the geometric spectrum to community detection in graphs. this is joint work with tobias hagge, patrick mackey, kathleen nowak, carlos ortiz marrero, and jenny webster.

i will discuss recent joint work with pankaj vishe, in which we are able to say something
about the naive moduli space of rational curves on arbitrary smooth hypersurfaces
of sufficiently low degree, by invoking methods from analytic number theory.

the growth of bacterial colony exhibits striking patterns that are determined by the interactions among individual, growing and dividing bacterial cells, and that between cells and the surrounding nutrient and waste. understanding the principles that underlie such growth has far-reaching consequences in biological and health sciences. in this work, we construct a multi-scale model of the growth of e. coli cells on agar surface. our model consists of detailed, microscopic descriptions of the cell growth, cell division with fluctuations, and cell movement due to the cell-cell and cell-environment mechanical interactions, and macroscopic diffusion equations for the nutrient and waste. our large-scale simulations reproduce experimentally observed growth scaling laws, strip patterns, and many other features of an e. coli colony. this work is the first step toward detailed multi-scale computational modeling of three-dimensional bacterial growth with mechanical and chemical interactions.

one possible path forward for practical quantum computation is
the development of a topological quantum phase. in principle, such as
system will be more resistant to decoherence because of its inherently
topological nature. we highlight progress on two combinatorial questions
which arise naturally in the study of these systems.
joint work with paul bruillard and kathleen nowak.