the concept of disjointness of dynamical systems (both topological and
measure-theoretic) was introduced by furstenberg in the 60s and has
since then become a fundamental tool in dynamics. in this talk, i will
discuss disjointness of topological systems of discrete groups. more
precisely, generalizing a theorem of furstenberg (who proved the result
for the group of integers), we show that for any discrete group $g$, the
bernoulli shift $2^g$ is disjoint from any minimal dynamical system. this
result, together with techniques of furstenberg, some tools from the
theory of strongly irreducible subshifts, and baire category methods,
allows us to answer several open questions in topological dynamics: we
solve the so-called ``ellis problem'' for discrete groups and characterize
the underlying topological space for the universal minimal flow of
discrete groups. this is joint work with eli glasner, benjamin weiss,
and andy zucker.

the kronecker coefficients are the notoriously elusive structure constants for the decomposition into irreducibles of the tensor product of irreducible representations of the symmetric group. in recent joint work with marni mishna and mercedes rosas, we study the piecewise quasipolynomial nature of the kronecker function using tools from polyhedral geometry. in this talk i will describe this approach, which begins with jacobi's definition of the schur function as a quotient of alternants, an idea originally exploited by rosas in her thesis. i will then illustrate its power by focusing on the first nontrivial case, showing how we derive, for this case, new exact formulas and an upper bound for the kronecker coefficients (the {\em atomic} kronecker coefficient) as well as other properties. the polyhedral geometry gives a curious digraph whose nodes are certain monomials in the alternant. an additional advantage of this approach is that asymptotic estimates for dilations can be computed using techniques of analytic combinatorics in several variables.

in 1908, paul koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. the weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 in to the hyperbolic 3-space. we show that there are close relationships among the koebe conjecture, the weyl problem and the work of alexandrov and thurston on convex surfaces. this is a joint work with tianqi wu.

in this talk, i will focus on principal component analysis (pca) in the presence of missing data. under a homogeneous and independent missingness mechanism, we showed that the leading eigenspaces of a hadamard-reweighted sample covariance matrix achieves the (nearly) minimax optimal rate with a phase transition. if the true leading eigenspaces satisfy an incoherence assumption, we can embrace much more flexible missingness mechanisms: we derived the statistical rate of this reweighted-covariance-based estimator under arbitrary deterministic observation regime. then we proposed to use this estimator to initialize a tuning-free iterative algorithm called primepca to further enhance the statistical accuracy. we showed that under the noiseless setting, primepca achieves exact recovery of the true leading eigenspaces with geometric convergence, provided that the initializer is close to the truth. simulation study shows that primepca performs similarly as softimpute with oracle tuning within a wide range of heterogeneity levels of observation probabilities and signal-to-noise ratios.

i will review and present some new results on construction, long time behaviour and coercive inequalities for markov semigroups on nilpotent lie groups.

the hall-magnetohydrodynamics (mhd) system is obtained from the ideal mhd system by incorporating a quadratic second-order correction, called the hall current term, that takes into account the motion of electrons relative to positive ions. in recent work with sung-jin oh, we investigated the cauchy problem in the irresistive case. we first study the linearized systems around a special class of stationary magnetic fields with certain symmetries, and obtain ill- and well-posedness results, depending on the profile of the magnetic field. we then pass from linear to nonlinear results: near a non-zero constant magnetic field, the system is well-posed but it is ill-posed (in the strongest sense of hadamard) near the trivial magnetic field. we are mainly guided by the behavior of bicharacteristics for the principal symbol. the key tools are: dispersive smoothing in the well-posedness case and construction of degenerating wave packets together with a systematic use of a generalization of the energy identity in the ill-posedness case.

information geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. instead of using metric for measuring distances on such manifolds, these applications often use “divergence functions” for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance kullback-leibler divergence, bregman divergence, f-divergence, etc. divergence functions are tied to generalized entropy (for instance, tsallis entropy, renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. it turns out that divergence functions enjoy pleasant geometric properties – they induce what is called “statistical structure” on a manifold m: a riemannian metric g together with a pair of affine connections d, d*, such that d and d* are both codazzi coupled to g while being conjugate to each other. divergence functions also induce a natural symplectic structure on the product manifold mxm for which m with statistical structure is a lagrange submanifold. in joint work with m. leok, we shown how divergence functions allow us to decouple hamiltonian and lagrangian dynamics in geometric mechanics. we recently characterize (para-) holomorphicity of d, d* in the (para-)hermitian setting, and show that statistical structures can be enhanced to (para-)hermitian and (para-)kahler manifolds. the surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages.

there are several major obstructions that arise in the attempt
to formulate an analog of the maeda conjecture in the setting of drinfeld
modular forms. i will report on an undergoing joint work with g. boeckle
and p. graef wherein we identify various hecke stable filtrations on the
spaces of cuspidal drinfeld modular forms of full level and a given weight
by identifying them with spaces of positive characteristic valued,
$sl_2(\mathbb{f}_q[\theta])$-invariant, harmonic cocycles on the oriented
edges of the bruhat-tits tree for $pgl_2 / \mathbb{f}_q((1/\theta))$ made
possible by an isomorphism of tietelbaum. i will compare with the
classical situation along the way and provide some of the relevant
background on drinfeld modular forms in a pre-talk aimed at graduate
2022年亚洲世界杯预选赛 and postdocs.

we discuss some of the recent work on discrete conformal geometry of polyhedral surfaces. the following topics will be addressed: the relationship among discrete conformal geometry, convex surfaces in hyperbolic 3-space, and the koebe circle domain conjecture, and the convergence of the discrete uniformization metrics to the ponicare metric. this is a joint work with d. gu, j. sun, and t. wu.

vinogradov's three prime theorem states that every sufficiently large odd number $n$ can be written in terms of the sum
of three odd primes. we will give a very light sketch of the proof of vinogradov's three prime theorem and give a comparison to
the previous talk on roth's theorem to arithmetic progressions.