Yangtze Analysis Seminar

Organizers: Li Gao and Maofa Wang ( Wuhan University )

Time and Location: April 5-6, at Wuhan University. The conference will be one and half day, starting from Saturday 5th morning and ending at Sunday 6th noon.  The participants can arrive at Wuhan in the afternoon/evening of Friday April 4 and leave in the afternoon of Sunday 6th.

Registration: Click here. We expect to arrange the local hotel lodging for most of the partipants.  To have your lodging reserved by us, please register by March 8th. Transportation are at your own expense.

If you have any questions and inquiries, please send to gao.li@whu.edu.cn 

Confirmed Invited Speakers: 

• Qingquan Deng (Central China Normal University)

• De Huang (Peking University)

• Guixiang Hong (Harbin Institute of Technology)

• Yong Jiao (Central South University)

• Hui Li* (Central South University)

• Chieu-Minh Tran (National University of Singapore)

• Jinsong Wu (BIMSA)

• Quanhua Xu (University of Franche-Comté & Harbin Institute of Technology)

• Gan Yao* (Harbin Institute of Technology)

• Lei Yu (Nankai University)

* are graduate students.

Schedule

Location: Leijun Technology Building (雷军科技楼) 644 

Saturday, April 5 

9:00-9:05 Opening 

9:05-9:50 Quanhua Xu 

10:00-10:45 Chieu-Minh Tran 

10:45-11:15 Tea break 

11:15-12:00 Lei Yu 

12:00-2:00 Lunch Break

2:00-2:45 Guixiang Hong 

2:55-3:25 Hui Li 

3:25-3:55 Tea break 

3:55-4:40 Jinsong Wu 

4:50-5:20 Gan Yao 

Sunday, April 6 

9:00-9:45 Qingquan Deng 

9:55-10:40 De Huang 

10:40-11:10 Tea break 

11:10-11:55 Yong Jiao 

11:55-12:00 Closing 

Titles and abstracts

Fourier multipliers on twisted crossed products

Quanhua Xu

Harbin Institute of Technology 

Abstract: Given a twisted W*-dynamic system (M,α,σ,Γ), let M⋊_(α,σ)Γ be the associated twisted crossed product. In this talk, we first consider permanent properties under twisted crossed product like injectivity and w*-amenability. The second part of the talk is devoted to the study of Fourier multipliers on M⋊α,σ Γ. Wegive criteria of completely bounded Lp Fourier multipliers and establish a link between Fourier and Schur multipliers. This talk is based a joint work with Xiao Xiong and Kai Zeng. 

Toward a nonabelian Brunn-Minkowski theory

 Chieu-Minh Tran

 National University of Singapore 

Abstract: The Brunn-Minkowski theorem tells us that if λ is the Lebesgue measure on R^d and A,B ⊆ R^d are compact with positive measure, then 

λ(A+B)^(1/d) ≥ λ(A)^(1/d)  +λ(B)^(1/d) ,

and the equality holds if and only if A and B are homothetic convex sets. In this talk I will discuss progress in obtaining a counterpart to this inequality for locally compact groups equipped with Haar measures. I will also explain the motivation from additive com binatorics and the role that model theory (from logic) plays in this story. (Based on joint work with Yifan Jing, Simon Machado, and Ruxiang Zhang) 

Anti-contractive inequalities and information theory

 Lei Yu

 Nankai University

Abstract: Hypercontractive inequalities are a family of classic functional inequalities that concern how slowly the norms of functions contract under the action of Markov operators. In this talk, we introduce a new family of inequalities, called anti contractive inequalities, which concern how fast the norms of functions contract under the action of Markov operators. We relate this kind of inequalities to the soft-covering problem in infor mation theory. Using this relation, we derive a sharp dimension free version of anti-contractivity inequalities. 

Navier-Stokes equations on the generalized phase spaces 

Guixiang Hong 

Harbin Institute of Technology

 Abstract: The noncommutative partial differential equations, such as the time-dependent Bogoliubov-de Gennes (BdG) equa tion, sometimes referred to as the generalized Hartree–Fock equa tion or the Hartree–Fock–Bogoliubov equation, have arisen nat urally in the theory of effective evolution equations. Recently, there have appeared a lot of works in this direction such as non commutative Burgers equations, quantum Laplace equations, quantum Fokke-Planck equations, noncommutative Schrodinger equations, and quantum wave equations etc.. However, the Lp theory has not been implemented successfully in most of the previously mentioned work; this is due to the fact that the Lp theory is the core of noncommutative analysis which has been neglected by the PDE community. In this talk, I shall present the study of the Navier-Stokes equation on the phase spaces by developing the noncommutative analysis techniques. This is based on joint work with Deyu Chen, Liang Wang and Wenhua Wang.

Continuous asymmetric Doob inequalities in noncommutative symmetric spaces

Hui Li 

Central South University

Abstract: Let (M,τ) be a noncommutative probability space equipped with a filtration (M_t)t∈[0,1] whose union is w∗-dense in M, and let (E_t)t∈[0,1] be the associated conditional expectations. In this talk, we will present that if the symmetric space E ∈ Int[Lp, Lq] with 1 < p ≤ q < 2 and E is 2(1 − θ)-convex and w-concave with 0 < w < 2, then the following holds: 

||(E_t(x))_(t∈[0,1])||_(E(M; ℓ^θ_∞)) ≤ C_(E,θ) ||x||_(H^c_E) , x ∈ H^c_E(M)

provided 1 −p/2 < θ < 1. Similar result holds for x ∈ Hr E(M). Moreover, if E ∈ Int[Lp,Lq] with 1 < p ≤ q < 2 and E is w concave with 2 < w < 2p/(2−p), then for each x ∈ E(M) there exist y, z ∈ E(M) such that x = y +z and

||(E_t(y))_(t∈[0,1]) ||_(E(M;ℓ^c_∞)) + ||(E_t(z))_(t∈[0,1]) ||_(E(M; ℓ^r_∞)) ≤ c_E ||x||_(E(M)).

These results can be considered as continuous analogues of those due to Randrianantoanina et al. One of the key ingredients in our proof is a new decomposition theorem of E(M)-modules for general symmetric space E, which extends the known result of Junge and Sherman. 

Bimodule Quantum Markov Semigroup

Jinsong Wu

Beijing Institute of Mathematical Sciences and Applications

Abstract: We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. Building on the structure of quantum sym metries, we introduce the concepts of bimodule equilibrium and bimodule detailed balance conditions, which generalize the clas sical notions of equilibrium and detailed balance. We demon strate that the evolution of densities governed by the bimodule quantum Markov semigroup is the bimodule gradient flow for the relative entropy with respect to quantum symmetries. Fur thermore, we establish a bimodule Poincar´e inequality for irre ducible inclusions. 

Proper cocycles, measure equivalence and Lp-Fourier multipliers 

Gan Yao 

Harbin Institute of Technology 

Abstract: We establish a new transference method of com pletely bounded Lp-Fourier multipliers for proper cocycles for probability measure preserving group actions on standard prob ability spaces. This generalizes the previous results by Haagerup and Jolissaint which only deals with the case p = ∞. In partic ular, this gives a natural transference method of Fourier multi pliers between groups with measure equivalence, which directly implies and notably generalizes the main result of Hong-Wang Wang on the pointwise convergence of noncommutative Fourier series on amenable groups. As a second application, this the ory also yields a transference method of Lp-Fourier multipliers from lattices in a linear Lie group to the whole group, which strengthens the previous results for Schur multipliers obtained by Haagerup and Lafforgue-de la Salle. As an example, we give a reasonable analog of Hilbert transform on SL2(R) using the transference method and compute the decay of its Lie derivations. 

The maximal estimates and point-wise convergence for Schr¨ odinger operators with potentials 

Qingquan Deng 

Central China Normal University 

Abstract: Denote by H the one dimensional Schr¨ odinger operator

H=−∆+V,

where the potential V is either a real-valued function in weighted space L^(1,γ). In this work, we focus on the results related to point wise convergence for Schr¨ odinger groups e^(−itH). The following statements are established. 

• Pointwise convergence of e^(−itH). For s ≥ 1 4 and f ∈ H^s(R), it holds that 

lim_{t→0} e^(−itH)f(x) = f(x), a.e. 

• The rate for point-wise convergence of e−itH. Assume that s ≥ 1/4 , λ ≥ 0 and 1/4 ≤ s+λ < 1/2 . One has for f ∈ H^(s+λ)(R),

e^(−itH)f(x) − f(x) = o(t^(λ/2)), a.e. as t → 0. 

• Point-wise convergence of e^(−itH) along curves. Assume that γ(x,t) satisfies H¨ older condition of order σ ∈ [1/2 ,1] in t and Bilipschitz condition in x. For s ≥ 1/4 and f ∈ H^s(R), it holds that 

lim_{t→0} e^(−itH)P_c(H)f(γ(x,t)) = f(x), a.e. 

The proofs are based on the H^s − L^p estimates for the maxi mal operators corresponding to e^(−itH). To do so, we shall make singular-regular decomposition of distorted plane waves to rep resent the multiplier m(H) point-wisely in terms of the distorted Fourier transform, and then the TT∗ argument, as well as the theory of functions spaces related to H will be applied. 

Multiscale self-similar finite-time blowups and traveling wave solutions of incompressible Euler equations 

De Huang 

Peking University 

Abstract: It remains an open problem whether the 3D in compressible Euler equations can develop finite-time singularity from smooth initial data in the whole space. In this talk, I will introduce some most recent results on finite-time blowups with self-similar features, divided into three parts. In the first part, I will explain what self-similar finite-time blowups are and how to construct and prove them in general. In the second part, I will talk about recent numerical findings on potential self-similar finite-time blowups of the 3D Euler equations with multi-scale features, which are closely related to traveling wave solutions. In the last part, I will introduce the rigorous construction of the Sadovskii vortex patch, a traveling vortex-patch solution of the 2D Euler equation that has been conjectured for decades. 

Some progress on weighted inequalities 

Yong Jiao 

Central South University 

Abstract: TBD 

Talk Slides

Quanhua Xu, Fourier multipliers on twisted crossed products. [Sildes]

Chieu-Minh Tran, Toward a nonabelian Brunn-Minkowski theory. [Sildes]

Lei Yu, Anti-contractive inequalities and information theory. [Sildes]

Guixiang Hong, Navier-Stokes equations on the generalized phase spaces. [Sildes]

Hui Li, Continuous asymmetric Doob inequalities in noncommutative symmetric spaces. [Sildes]

Jinsong Wu, Bimodule Quantum Markov Semigroup. [Sildes]

Gan Yao, Proper cocycles, measure equivalence and Lp-Fourier multipliers. [Sildes] 

Qingquan Deng, The maximal estimates and point-wise convergence for Schr¨ odinger operators with potentials. [Sildes] 

De Huang, Multiscale self-similar finite-time blowups and traveling wave solutions of incompressible Euler equations. [Sildes] 

Yong Jiao, Some progress on weighted inequalities. [Sildes] 

Gallary

Conference Venue:  雷军科技楼，Room 644

Lodging information:  All conference lodging will be reserved at Junyi Dynasty Hotel (君宜王朝大酒店)

Address: No.87, Luoyu Road, Hongshan, Wuhan (武汉市洪山区珞喻路87号)

Phone: +86 027 8768 7777 

Transportation to Hotel

From Tianhe Airport: 

By Taxi or E-hailing：100-150 yuan, ~1 hour (if no traffic). 

By Metro Line 2: 1.5 hour.

From Wuhan Railway Station: 

By Taxi or E-hailing：30-50 yuan, ~30 mins (if no traffic). 

By Metro Line 4+Line 2: 50 mins.

When buying tickets, note that there are also two other rail stations: Wuchang Station and Haikou Station.

Supports from

• School of Mathematics and Statistics, Wuhan University  (武汉大学数学与统计学院）

• Tianyuan Mathematical Center at Central China  (天元数学中部中心）

• National Natural Science Foundation of China  (国家自然科学基金委)