a ridge in a data cloud is a low-dimensional geometric feature that generalizes the concept of local modes, in the sense that ridge points are local maxima constrained in some subspace. in this talk we give nonparametric confidence regions for $r$-dimensional ridges of probability density functions on ${r}^d$ , where $1 \leq r \< d$. we view ridges as the intersections of level sets of some special functions. the vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. two types of confidence regions for density ridges will be presented: one is based on the asymptotic distribution of the maximal vertical deviation, which is established by utilizing the extreme value theory of nonstationary $\chi^2$-fields indexed by manifolds; and the other is a bootstrap approach (including multiplier bootstrap and empirical bootstrap), the theoretical validity of which leverages the recent study in the literature on the gaussian approximation of suprema of empirical processes.

khovanov homology is a powerful invariant of knots which has many connections to other areas of mathematics, such as representation theory and gauge theory. however, it is strangely difficult to extract topological information from this invariant. after discussing this invariant, we will show that it is able to detect the figure-eight knot.

as in complex analysis, liouville theorems in pdes assert that any solution of a specific equation is trivial. the meaning of
``trivial'' depends on the context. in this lecture, we will discuss liouville theorems for superlinear parabolic pdes. an overview, a typical application, and recent results will be presented.

can one emulate the simple random walk without actually doing anything random? this talk will be about a deterministic version of random walk called rotor walk, and we will measure its performance in emulating the simple random walk with respect to different parameters, e.g., the shape of the trajectory, number of returns to the origin, etc. in particular, we will see that the number of returns to the origin for the rotor walk can be made equal to the same number for the simple random walk. this resolves a conjecture of florescu, ganguly, levine, and peres (2014).

we describe an ongoing program to resolve certain problems at the interface of diophantine approximation and homogenous dynamics. highlights include computing the hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, thereby resolving a conjecture of kadyrov-kleinbock-lindenstrauss-margulis (2014) as well as answering a question of bugeaud-cheung-chevallier (2016). as a corollary of the dani correspondence principle, this implies that the set of divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two lyapunov exponents with opposite signs has equal hausdorff and packing dimensions. other applications include dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions. this is joint work with david simmons, lior fishman, and mariusz urbanski. the reduction of various problems to questions about certain combinatorial objects that we call templates along with a variant of schmidt's game allows us to answer some of these problems, while leaving plenty that remain open. the talk will be accessible to 2022年亚洲世界杯预选赛 and faculty interested in some convex combination of homogeneous dynamics, diophantine approximation and geometric measure theory.

in perelman's well-known paper, he claimed the unqueness of the 3d \verb=\=kappa-noncollpased steady grs (without giving any proof or ideas) and conjectured that 3d noncompact \verb=\=kappa-solution with positive sectional curvature must be the ricci flow generated the bryant soliton. perelman's claim and conjecture has been proved by simon brendle in 2012 and 2018, respectively. the classification of 3d \verb=\=kappa-noncollpased steady grs plays an important role in his proof of the conjecture. brendle's work is based on the observation that 3d \verb=\=kappa-noncollpased steady grs must be asymptotically cylindercial. in higher dimensions, the asymptotical geometry of steady grs is much more complicated. in this talk, we will talk about some recent progress on the asymptotical geometry of 4d steady grs. this is a joint work with prof. bennett chow.

a triangulation of a topological space is a homeomorphism from this space to a simplicial complex. a famous problem in topology is whether all manifolds are triangulable. surprisingly, the answer is no when dimension is at least 4. in this talk, i will explain the beautiful work of galewski-stern and matumoto, which provides an obstruction theory for triangulating manifolds. i will also explain manolescu's disproof of triangulation conjecture in all dimensions greater than 4.

the growth of bacterial colony exhibits striking complex patterns and robust scaling laws. understanding the principles that underlie such growth has far-reaching consequences in biological and health sciences. in this work, we develop a mechanical theory of cell-cell and cell-environmental interactions and construct a hybrid three-dimensional computational model for the growth of {\it e.~coli} colony on a hard agar surface. our model consists of microscopic descriptions of the growth, division, and movement of individual cells, and macroscopic diffusion equations for the nutrients. the cell movement is driven by the cellular mechanical interactions. our unique treatment of the force arising from the liquid-air surface tension is applicable to both the monolayer (discrete) and multilayer (continuum) growth regimes. our large-scale simulations and analysis predict the linear growth of the colony in both the radial and vertical directions, conforming the experimental observations. this work is the first step toward detailed computational modeling of bacterial growth with mechanical and biochemical interactions. this is joint work with mya warren, hui sun, yue yan, jonas cremer, and terence hwa.

motivated by quantum information theory, we study log-sobolev inequalities for matrix valued functions on finite graphs and compact lie groups. using tools from noncommutative geometry we find a surprising link between graph laplacians and sublaplacians on the orthogonal group. simple combinatorial tools, namely the existence of a spanning tree, then allows us to find concrete lower bounds for spectral gaps. joint work with h. li and n. laracuente.

in this talk, we will be discussing the discrete fourier transform in one
and two dimensions, including some of the transform's properties, as well
as various strategies for efficiently evaluating the discrete fourier transform
via several fast fourier transform algorithms. the algorithms discussed
will include the cooley-turkey algorithm, the radix-2 decimation in time
strategy, the split-radix algorithm, the mixed-radix algorithm, the prime
factor algorithm, and rader's algorithm. finally, we will discuss various
applications of the fast fourier transform, as well as considerations for
using it in practical applications.

recently there has been a lot of work on the construction of the k-moduli space, i.e. a good moduli space parametrizing k-polystable fano varieties. it is conjectured that this moduli space is projective and the polarization is given by a natural line bundle, the chow-mumford (cm) line bundle. in this talk, i will present a recent joint work with chenyang xu where we show the cm line bundle is ample on any proper subspace parametrizing reduced uniformly k-stable fano varieties, which conjecturally should be the entire k-moduli. as an application, we prove that the moduli space parametrizing smoothable k-polystable fano varieties is projective.