Date/Time : 21 Feb 2025 13:00
Venue : S17 #04-04 (SR3)
Speaker : Weikun He, AMSS, Chinese Academy of Sciences

On homogeneous spaces such as SL_2(R)/SL_2(Z), there are interesting random walks which are driven by group action. We focus on asymptotic behaviour of such random walk, especially on the property of equidistribution in law. In this talk, I will present a method developed in a joint work with Timothée Bénard to estimate the rate of convergence of such equidistribution. It involves ideas from fractal geometry and it constitutes a crucial ingredient in the study of Diophantine approximation on fractal sets, presented in the YMLS talk.

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Meng Chen, Fudan University

In this lecture, I will provide an outline to prove the following theorem: the inequality K^3 ≥ 4/3 p_g – 10/3 holds for any complex 3-folds of general type. Based on this inequality, I will explain an effective method to study moduli spaces of canonical 3-folds with boundary volumes.

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Date/Time : 19 Feb 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Makoto Tatsuta, National Institute of Informatics

Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the validity such as decidability and complexity for separation logic with numbers have not been fully studied yet. This paper presents the translation of Pi-0-1 formulas in Peano arithmetic to formulas in a small fragment of separation logic with numbers, which consists only of the intuitionistic points-to predicate, 0 and the successor function.
Then this paper proves that a formula in Peano arithmetic is valid in the standard model if and only if its translation in this fragment is valid in the standard interpretation. As a corollary, this paper also gives a perspective proof for the undecidability of the validity in this fragment. Since Pi-0-1 formulas can describe consistency of logical systems and non-termination of computations, this result also shows that these properties discussed in Peano arithmetic can also be discussed in such a small fragment of separation logic with numbers.

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : He Weikun, AMSS, Chinese Academy of Sciences

Diophantine approximation studies how well a real number can be approximated by rational numbers. Following a question asked by Mahler in 1984, we are interested in Diophantine properties of points in the middle third Cantor set or more generally fractal sets. I will survey briefly this topic and present a recent joint work with Timothée Bénard and Han Zhang, where we obtained a Khintchine dichotomy for the middle third Cantor set.

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Date/Time : 19 Feb 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course explores the fundamental aspects of local theta correspondence. The series consists of 5 lectures covering the following topics:

1. Models of the Heisenberg-Weil representation and the Metaplectic group
2. Dual pairs and general properties of theta correspondences
3. Theta correspondences over real fields
4. Theta correspondences over finite fields and p-adic fields
5. Geometric aspects of theta correspondence

This mini-course is designed for senior undergraduate students and graduate students.

This mini-course will live stream. Here is the zoom ID:
Meeting ID: 848 8485 0855
Passcode: 226224

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Yue Xie, University of Hong Kong

The two-metric projection method is a simple yet elegant algorithm proposed by Bertsekas to address bound/box-constrained optimization problems. The algorithm’s low per-iteration cost and potential for using Hessian information make it a favorable computation method for this problem class. Inspired by this algorithm, we propose a two-metric adaptive projection method for solving the $\ell$_1-norm minimization problem that inherit these advantages. We demonstrate that the method is theoretically sound – it has global convergence. Furthermore, it is capable of manifold identification and has superlinear convergence rate under the error bound condition and strict complementarity. Therefore, given sparsity in the solution, the method enjoys superfast convergence in iteration while maintaining scalability, making it desirable for large-scale problems. We also equip the algorithm with competitive complexity to solve nonconvex problems. Numerical experiments are conducted to illustrate the advantages of this algorithm implied by the theory compared over other competitive methods, especially in large-scale scenarios. Compared with the two-metric projection method, our algorithm directly solves the l1-norm minimization problem without resorting to the intermediate reformulation as a bound-constrained problem, so it circumvents the issue of numerical instability.

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Date/Time : 18 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Bao Huanchen, National University of Singapore

Let G be a connected reductive group over an algebraically closed field, e.g. the general linear group. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. Let K be the fixed point subgroup of an involution of G. For example, when G is the general linear group, then K could be the group of orthogonal matrices.

In this talk, I shall discuss the construction of integral models of the symmetric pair (G, K) and the symmetric space G/K. I will also mention various applications of our new constructions, e.g., Frobenius splittings, canonical bases, spherical embeddings, etc.. This is based on several joint works with Jinfeng Song (NUS).

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Date/Time : 18 Feb 2025 10:00
Venue : S17 Level 3 (IMS executive seminar room)
Speaker : Qiang Du, Columbia University

Recent applications and theoretical developments of models of integral equations using nonlocal operators have shown promise as effective alternatives to local models, especially in the presence of singularities and anomalies. These models also serve as continuum limits for large-scale discrete models used in data learning and network analysis. We present these models on bounded spatial domains and discuss related modeling, analysis, and computational issues.

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Date/Time : 17 Feb 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Li Mengni, Southeast University

In this talk we focus on the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible magnetohydrodynamics (MHD) system. By bridging together geometric methods and weighted energy estimates, we establish a set of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields that are detected by faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

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