Date/Time : 17 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jit Wu, Harvard University

Let K be a number field, S a finite set of places containing all Archimedean places and f a rational map defined over K of degree d >_ 2. Given a non-preperiodic point a and non-exceptional b, both in P^1(K), Silverman has shown that there are only finitely many points among {a,f(a),f^2(a),…} that are S-integral relative to b. More recently, Hsia and Silverman showed an upper bound of the form O(4^|S|) on the number of such points as S varies. In this talk we will improve this upper bound to |S|^(1+o(1)) and show it is possible to get O(|S|) for certain rational maps.

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Date/Time : 17 Aug 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Jia Jia, Tsinghua University

TopoloAn automorphism of a compact complex space is called wild if there is no non-trivial proper invariant analytic subset. In this talk, I will show that a compact complex surface admitting a wild automorphism is either a complex torus or an Inoue surface of certain type, and this wild automorphism has zero entropy. This is based on a joint work with Long Wang.gy and geometry

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Date/Time : 17 Aug 2023 13:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Fei, Shanghai Jiaotong University.

This paper concerns the existence of global weak solutions {\it \`a la Leray} for compressible Navier–Stokes–Fourier system with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

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Date/Time : 16 Aug 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Chengbo Wang, Zhejiang University

In this talk, I will present a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable C^1 coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy’s inequality, fractional chain rule and fractional Leibniz rule) which are adapted for the problem, the well-posed result is proved by iteration. In addition, our argument yields almost global existence for n=3 and global existence for dimension 4 and higher, when the initial data are small, spherically symmetric with almost critical Sobolev regularity.

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Date/Time : 16 Aug 2023 16:10
Venue : S17 #04-04 (SR3)
Speaker : Naqing Xie, Fudan University

We discuss a uniqueness result of the electro-vacuum Einstein-Maxwell equations near the null infinity. This talk is based on a recent joint work with Xiaokai He (Hunan First Normal, Changsha) and Xiaoning Wu (AMSS, CAS, Beijing).

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Date/Time : 16 Aug 2023 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yifan Jing, University of Oxford

Let G be a connected compact group equipped with the normalised Haar measure μ. Our first result shows that given α, β > 0, there is a constant c = c(α, β) > 0 such that for any compact sets A, B ⊆ G with αμ(B)≥μ(A)≥μ(B) and μ(A)+μ(B)≤1−β, there exist b_1, …, b_c∈ B such that μ(A · {b_1,…,b_c}) ≥ μ(A) + μ(B). A special case of this, that is, when G is the d-dimensional torus confirms a recent conjecture of Bollobas, Leader and Tiba.

We also prove a quantitatively stronger version of such a result in the discrete setting of d -dimensional Euclidean space. Thus, given d ∈ N, we show that there exists c = c(d) > 0 such that for any finite, non-empty set A ⊆ R^d which is not contained in a translate of a hyperplane, one can find a_1,…,a_c∈ A satisfying
|A + {a_1, . . . , a_c}| ≥ (d + 1)|A| − O_d(1). The main term here is optimal and recovers the bounds given by Freiman’s lemma up to the O_d(1) error term.

(Joint with Akshat Mudgal)

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Date/Time : 08 Aug 2023 15:00
Venue : S17 #05-12 (SR4)
Speaker : Barbara Wohlmuth, Technical University of Munich

In this talk, we discuss several aspects of time‐fractional differential operators. The non‐local nature of the operator is challenging from a theoretical and algorithmic point of view but allows us to account for memory effects in the mathematical model. We consider a wide class of time‐fractional PDE models, including gradient flow, viscoelasticity, random field generators, and polymeric fluids. Of special interest is the equivalence of the time fractional gradient flow model with an integer order system in a higher dimension. Exploiting this equivalence allows us to construct efficient algorithms for time fractional PDEs. Replacing the kernel by a finite sum of exponentials transforms the non‐local PDE into an integer‐order system for the modes. To balance the discretization error in time and space with the kernel approximation error, the number of modes has to grow logarithmically. However, we observe numerically that the number of modes is very moderate if the finite sum of exponentials is based on a rational approximation of the kernel function. In this talk, we present theoretical existence results, algorithmic aspects and show several applications in which fractional operators play an important role. Numerical examples illustrate the large flexibility of the proposed techniques.

Joint work with Marvin Fritz (RICAM, Austria), Brendan Keith (Brown University, USA), Ustim Khristenko (Safran, France) and Endre Süli (University of Oxford, UK).

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Date/Time : 03 Aug 2023 14:30
Venue : S17 #04-05 (SR2)
Speaker : Dinesh Thakur, University of Rochester

Without assuming any background, we will explain how some basic techniques and results from computer science help us in understanding the nature of some naturally occurring numbers.

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Date/Time : 03 Aug 2023 16:00
Venue : S17 #04-05 (SR2)
Speaker : Wong Tian An, University of Michigan, Deerfield

Following Langlands’ original Beyond Endoscopy proposal, Arthur has outlined two refinements of the stable trace formula: the r-trace formula and the primitive trace formula. Spectrally, the first should detect specific functorial transfers from smaller groups, while the second should consist of certain primitive automorphic forms. In this talk, I will give a friendly introduction to these ideas. Then I will present a framework for establishing each of them separately, and report on some progress along these lines.

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