we present our construction of novel continuous and discrete type ii variational principles for hamiltonian systems on cotangent bundles of lie groups, which allows for type ii boundary conditions, i.e., fixed initial position and terminal momenta boundary conditions. the motivation for these boundary conditions arises from the adjoint sensitivity method, which is ubiquitous in dynamically-constrained optimization and optimal control problems. traditionally, such type ii variational principles are only defined locally. however, for dynamics on the cotangent bundle of a lie group, left-trivialization allows us to define this variational principle globally. our discrete variational principle leads to an intrinsic, symplectic, and momentum-preserving integrator for lie group hamiltonian systems that allows for type ii boundary conditions and maximally degenerate hamiltonians. we show how this method can be used to exactly compute sensitivities for optimization problems subject to dynamics on a lie group. we conclude with numerical examples of optimal control problems on so(3) and a discussion of future applications of this method. 

voiculescu developed two main candidates for the analogs of entropy in free probability: the microstates and non-microstates free entropies. in 2003, biane-capitaine-guionnet established a relationship between the two: the microstates free entropy is always bounded above by the non-microstates free entropy. in this talk, we discuss the conditional versions of these notions of free entropy. then, by connecting each of the free entropies with the asymptotics of their classical counterparts, we provide an elementary proof of the result of biane-capitaine-guionnet. this is joint work with david jekel.

stein's restriction conjecture is one of the central topics in fourier analysis. it is closely related to other areas of math, for example, number theory, pdes. in this talk, i will discuss a recent improvement of this conjecture in r^3, based on the joint work with hong wang. our proof is built upon the framework of polynomial partitioning, and, among other things, it uses the refined decoupling theorem, a two-ends kakeya estimate. the two-ends estimate captures some information from the method of induction on scales.

the eigencurve is an important object in the study of p-adic families of modular forms, yet many of its geometric structures remain mysterious. i will give an elementary introduction to the theory of p-adic modular forms and the construction of eigencurve. then we will take a glance at this beautiful conjecture.

in 2016 abert, gelander, and nikolov made what they called a provocative conjecture: for lattices in higher-rank simple lie groups, the minimum size of a generating set (rank) is sublinear in the volume. i will discuss our solution to this conjecture. it is a corollary of our main result, where we establish "fixed price one" for a more general class of "higher rank" groups. no familiarity of fixed price or cost is required for the talk. joint work with mikolaj fraczyk and amanda wilkens.

the analysis of non-markovian, self-interacting diffusions, which has motivations from probability, physics, biology, etc., is intimately connected with that of an associated stochastic interface. in this talk, we will look at dynamical fluctuations of this interface, and derive a kpz-type stochastic pde as a scaling limit. unlike the usual kpz equation, the geometry of the underlying manifold plays an important role in the analysis of this spde. deriving the spde from the diffusion model is based on a novel "local-to-global" stochastic homogenization principle. studying the spde itself amounts to relatively modern ideas in stochastic analysis coupled with analysis of pseudo-differential operators on manifolds. based on joint work with amir dembo.

 we prove that a sequence of multi-scaled stationary distributions of reflected brownian motions (rbms) has a product-form limit. each component in the limit is an exponential distribution. the multi-scaling corresponds to the "multi-scale heavy traffic" recently advanced in dai, glynn and xu (2023) for generalized jackson networks. the proof utilizes the basic adjoint relationship (bar) first introduced in harrison and williams (1987) that characterizes the stationary distribution of an rbm. this is joint work with jin guang and xinyun chen at cuhk-shenzhen,  and peter glynn at stanford.

this talk is based on a joint work with xiaojun huang entitled “bergman metrics as pull-backs of the fubini-study metric”. we study domains in cn or stein manifolds m such that their bergman metrics have constant holomorphic sectional curvature κ. we prove a uniformization theorem when κ < 0 through the calabi rigidity theorem and holomorphic extension theorems. we also discuss the case when κ ≥ 0. we provide several interesting examples of the existence of such m. under certain conditions on m, we prove that the bergman metric of m can not have non-negative constant holomorphic sectional curvatures.

in 1976, ken ribet used modular techniques to prove an important relationship between class groups of cyclotomic fields and special values of the zeta function.  ribet’s method was generalized to prove the iwasawa main conjecture for odd primes p by mazur-wiles over q and by wiles over arbitrary totally real fields.  

central to ribet’s technique is the construction of a nontrivial extension of one galois character by another, given a galois representation satisfying certain properties.  throughout the literature, when working integrally at p, one finds the assumption that the two characters are not congruent mod p.  for instance, in wiles’ proof of the main conjecture, it is assumed that p is odd precisely because the relevant characters might be congruent modulo 2, though they are necessarily distinct modulo any odd prime.

in this talk i will present a proof of ribet’s lemma in the case that the characters are residually indistinguishable.  as arithmetic applications, one obtains a proof of the iwasawa main conjecture for totally real fields at p=2.  moreover, we complete the proof of the brumer-stark conjecture by handling the localization at p=2, building on joint work with mahesh kakde for odd p.  our results yield the full equivariant tamagawa number conjecture for the minus part of the tate motive associated to a cm abelian extension of a totally real field, which has many important corollaries.

this is joint work with mahesh kakde, jesse silliman, and jiuya wang.

zeta functions are central objects of study in number theory, and can often be found wherever there is galois theory. in this talk, we will discuss the ihara zeta function of an undirected graph and compare it to the dedekind zeta function. then we will talk about incidence algebras and use them to describe a categorification of both types of zeta functions.

in this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of l-functions.  the goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of hilbert's 12th problem.  meanwhile, there is an abundance of conjectures on the special values of l-functions at certain integer points.  of these, stark's conjecture has special relevance toward explicit class field theory.  i will describe two recent joint results with mahesh kakde on these topics.  the first is a proof of the brumer-stark conjecture. this conjecture states the existence of certain canonical elements in cm abelian extensions of totally real fields.  the second is a proof of an exact formula for brumer-stark units that has been developed over the last 15 years.  we show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.