Date/Time : 11 Sep 2024 16:30
Venue : S17 #06-11 (SR6)
Speaker : Yang Liu, National University of Singapore

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs without any fine-tuning? In-context operator learning and the corresponding model In-Context Operator Networks (ICON) represent an initial exploration of these questions. We present a detailed methodology for solving PDE problems with ICON, and show how a single ICON model can make forward and reverse predictions for different equations with different strides, provided with appropriately designed data prompts. We show positive evidence for the questions above, through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. In particular, we show that an ICON model trained on conservation laws with cubic flux functions can generalize well to some other flux functions of more general forms, without fine-tuning. We will also discuss different theoretical interpretations of in-context learning.

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Yuxuan Wang, Tsinghua University

First examined by Riemann in 1860 in one-dimensional space, the Riemann problem has since been extensively studied and advanced by numerous scholars. Our talk will investigate the interaction of two approaching shocks in the context of spherical symmetry in three-dimensional space, presuming the fluid to be ideally barotropic. We will outline the proof of the local existence, uniqueness, and higher regularity of the solution to the shock interaction problem.

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Date/Time : 09 Sep 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Ichiro Shimada, Hiroshima University

The Ap\’ery-Fermi pencil is a certain one-dimensional family of complex affine surfaces related to a Fermi surface in solid state physics. This pencil is also connected to Ap\’ery’s proof of the irrationality of the zeta value at s=3.

A general member of the Ap\’ery-Fermi pencil is birational to a K3 surface with Picard number 19. We calculate the automorphism group of this K3 surface by means of Borcherds’ method.

I also present some general algorithms that may be useful in computational algebraic geometry.

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Date/Time : 09 Sep 2024 16:00
Venue : S17 #05-11 (SR5)
Speaker : Jean-Bernard Lasserre, LAAS-CNRS

We briefly describe the Christoffel-Darboux (CD) kernel and the Christoffel function (CF). While well-known tools in orthogonal polynomials and approximation theory, it is only recently that CF’s properties have been shown to be very useful and powerful (as well as easy to implement) to help solve some problems

— in data analysis and datra mining where the underlying (empirical) measure is usually discrete, e.g. for outlier detection, support inference, density approximation, and

— in approximation of discontinuous functions.

Finally we will also briefly describe some (in the author’s opinion surprising) links of the CF with seemingly less disconnected areas (Pell equation, equilibrium measure, convex optimization and positivity certificates).

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Date/Time : 05 Sep 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Ning Li, Nankai University

The notion of transfer is a way to relate representation theory of different real forms of a complex semisimple Lie group. In this talk, I will explicitly describe the transfer of \theta(1) of Sp(2n,R) in the case of non-stable range via the study of K-types. As a corollary, we get certain compatibility of local theta correspondence and the transfer functor, and every unitary irreducible constituent of degenerate principal series of maximal Gelfand-Kirillov dimension is a derived functor module, which coincide with the results of Matumoto-Trapa for the case of Sp(p,q), SO*(2n), and U(m,n).

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Date/Time : 04 Sep 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Peng Fei, National University of Singapore

A quasi-kernel of a digraph is an independent set Q such that every vertex can reach Q in at most two steps. The 48-year-old small QK conjecture says that every sink-free digraph contains a quasi-kernel of size at most n/2. Spiro posed the large QK conjecture, which states that every digraph contains a quasi-kernel Q such that |N^-[Q]| ≥ n/2, and showed that it follows from the small QK conjecture. We show that the converse is true with a weaker constant. We also show that every sink-free digraph contains a quasi-kernel of size at most (1−1/k)n, where k is the digraph’s kernel-perfect number, generalizing a result by Kostochka, Luo and Shan. Joint work with Ai and Liu.

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Date/Time : 04 Sep 2024 16:45
Venue : S17 #04-04 (SR3)
Speaker : Chong Chi Tat, National University of Singapore

Let $\alpha$ be an admissible ordinal. A set $A\subset \alpha$ is of minimal $\alpha$-degree if it is non -recursive and every set of lower $\alpha$-degree is recursive. In 1956 Spector introduced the technique of forcing with perfect sets to prove the existence of a minimal $\omega$-degree. Sacks posed the problem of the existence of a minimal $\alpha$-degree for all admissible $\alpha$. This problem, poised in the early 1970’s, remains unsolved today. The best result to-date is that of Shore (1972) in which he showed that there is a positive solution if $\alpha$ is $\Sigma_2$-admissible. There has been no progress made on this problem since. In this talk we present a recent result on the minimal $\alpha$-degree problem for singular cardinals of uncountable cofinality which are not $\Sigma_2$-admissible.

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Date/Time : 04 Sep 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wilderich Tuschmann, Karlsruhe Institute of Technology

Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, of special holonomy, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also study this question for the corresponding moduli spaces of metrics,i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure ‘how many’ different metrics and geometries the given manifold actually does exhibit.

The history of the subject as a whole indeed goes back more than a century, and since H. Weyl’s early result on the connectedness of the space of positive Gaussian curvature metrics on the two-sphere and the foundings of Teichmüller, infinite- dimensional manifold and Lie group theory, uniformization and geometrization, the study of spaces of metrics and their corresponding moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike.

In my talk, I will provide a gentle introduction to the subject with a focus on lower curvature bounds and present recent results and open questions about the global topological properties of moduli spaces of nonnegatively curved Riemannian metrics on manifolds, and as well, if time permits, also corresponding facts and questions for the moduli of nonnegative curvature metrics on RCD spaces.

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Date/Time : 04 Sep 2024 16:30
Venue : S17 #06-11 (SR6)
Speaker : Lin Bo, National University of Singapore

This talk focuses on the development of time-dependent regularized 13-moment equations for general elastic collision models within the linear regime. The detailed derivation demonstrates that these proposed equations exhibit super-Burnett order for small Knudsen numbers and feature a symmetric structure. A novel modification to stable boundary conditions is introduced to remove boundary layers. The effectiveness of the model is validated through numerical simulations of channel flows.

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