Date/Time : 22 Dec 2023 10:00
Venue : via Zoom
Speaker : Li Bowen, Duke University

Zoom link: https://nus-sg.zoom.us/j/81215174559?pwd=T21GRHJBSTRIMHhta0hVUlRrQVQvdz09

Meeting ID: 812 1517 4559
Passcode: 938817

In this talk, we discuss some recent results for photonic and quantum interacting systems. We first consider the scattering resonances in the wave interaction with nanoparticles of high refractive indices and their applications in imaging and material science. We will establish the existence and asymptotic behaviors of dielectric resonances and show that near a resonant frequency, a singular particle exhibits resonant magnetic dipole characteristics. We then discuss the topological properties of scattering resonances for the nanophotonic crystals. In particular, we prove the genericity of the Dirac dispersion cone for honeycomb lattice and the existence of embedded eigenvalue with Fano-type asymmetric peaks in transmission spectra. Further, we focus on two prototypical quantum interacting systems that are closely related to quantum optimal transport: the ground-state energy problem and open quantum dynamics. We will present an SDP relaxation hierarchy with optimized clusters for the ground-state energy of the quantum many-body Hamiltonian that admits a quantum embedding interpretation. For open quantum dynamics, we introduce quantum Beckner’s inequalities to quantify its long-time behavior with respect to convex entropies. We will discuss the properties of the Beckner constant in detail and provide a uniform lower bound in terms of the spectral gap. Moreover, we define a new class of quantum Wasserstein distances such that the quantum dynamic is a gradient flow and its convergence can be characterized by entropic Ricci curvature lower bound associated with the convex entropy. Some ongoing works and open questions will also be discussed.

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Date/Time : 22 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Gang Bao, Zhejiang University

Inverse problems are concerned with determining model parameters from the observed data, which arise in diverse application areas, such as nondestructive testing, seismic imaging, optical imaging, materials characterization, and medical imaging. A well-known representative inverse problem is electrical impedance tomography (EIT), also known as Calderón’s problem, which determines the electrical conductivity from the voltage to current map on the boundary. In general, inverse problems are known to be severely ill-posed, in particular, small perturbations in the measured data may lead to large errors in the reconstructions. Stable numerical solution of inverse problems remains a big challenge in computational inverse problems. In this talk, the speaker will discuss some new directions for solving inverse problems. Of particular interest are recursive linearization methods based on multiple frequency data, deep learning methods, and optimal transportation approaches. These methods are shown to be suitable for solving severely ill-posed problems. Numerical results will be presented to validate the effectiveness and feasibility of the methods.

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Date/Time : 19 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhuan Koh, The national research institute for mathematics and computer science (Netherlands)

Linear programs (LPs) can be solved in polynomial time using the ellipsoid algorithm or interior point methods (IPMs). However, it is a major open question whether LPs can be solved in strongly polynomial time. Given an LP with m inequalities and n variables, a strongly polynomial algorithm must compute an optimal solution using poly(m,n) elementary arithmetic operations (+,-,*,/,<?). In particular, the number of such operations cannot depend on the size of coefficients describing the LP. This question was first asked by Megiddo in 1983, and listed as one of Smale’s problems for mathematics in the 21st century. In this talk, I will cover recent progress on this open problem. By combining circuits from combinatorial optimization and IPMs from continuous optimization, we provide an affirmative answer to this question when every inequality has at most two variables.

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Date/Time : 14 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Wang Ligong, Northwestern Polytechnical University

Let \mu(G) denote the spectral radius of a graph G. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdos-Sos Conjecture that any tree of order t is contained in a graph of average degree greater than t − 2. Let S_{n,k} be the graph obtained by joining every vertex of a complete graph on k vertices to every vertex of an independent set of order n−k, and let S_{n,k}^+ be the graph obtained from S_{n,k} by adding a single edge joining two vertices of the independent set of S_{n,k}. In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with \mu(G) \geq \mu(S_{n,k}^+) contains all trees of order 2k + 3, unless G = S_{n,k}^+. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for k \geq 8. If a graph G with sufficiently large order n satisfies \mu(G) ≥ \mu(S_{n,k}) and G \neq S_{n,k}, then G contains all trees of order 2k + 3 with diameter at most four, except for the tree obtained from a star on k + 2 vertices by subdividing each of its k + 1 edges once. We also introduce some new results about the above conjecture. This is a joint work with Xiangxiang Liu and Hajo Broersma.

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Date/Time : 14 Dec 2023 15:40
Venue : S17 #04-06 (SR1)
Speaker : Bai Xueli, Northwestern Polytechnical University

As a first attempt to understand the effects of temporal periodicity on dynamics of general periodic parabolic reaction-diffusion systems, we analyze the asymptotic behaviour of the principal eigenvalue for linear cooperative/competition periodic-parabolic systems with small/large diffusion rates and then discuss local and global stability of related periodic-parabolic Lotka Volterra competition diffusion system. This is a joint work with Xiaoqing He and Wei-Ming Ni.

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Date/Time : 14 Dec 2023 16:20
Venue : S17 #04-06 (SR1)
Speaker : Xiao Manyu, Northwestern Polytechnical University

The common issue of surrogate models is to make good use of sampling data. In theory, the higher the fidelity of sampling data obtained, the more accurate the approximation model built. However, in practical engineering problems, high-fidelity data may be less available, and such data may also be computationally expensive. On
the contrary, we often obtain low-fidelity data under certain simplifications. Although low-fidelity data is less accurate, such data still contains much information about the real system. So, combining both high- and low-fidelity data in the construction of a surrogate model may lead to better representation of the physical phenomena.

In this work, the original principle of Multi-Fidelity Proper Orthogonal Decomposition will consist in projecting the high-dimensional system of equations into a low-dimensional space, with reduced basis vectors. These vectors are updated with incremental “on-the-fly” merges between costly high-fidelity simulation snapshots and more rapid lower-fidelity vectors. Successful achievements of this work will lead to potential benefits in the fields of large-scale mechanical engineering.

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Date/Time : 13 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Xin Sun, Peking University and University of Pennsylvania

Smirnov’s proof of Cardy’s formula for percolation on the triangular lattice leads to a discrete approximation of conformal maps, which we call the Cardy-Smirnov embedding. Under this embedding, Holden and I proved that the uniform triangulation converge to a continuum random geometry called pure Liouville quantum gravity. There is a variant of the Gaussian free field governing the random geometry, which is an important example of conformal field theory called Liouville CFT. A key motivation for understanding Liouville quantum gravity rigorously is its application to the evaluation of scaling exponents and dimensions for 2D critical systems such as percolation. Recently, with Nolin, Qian and Zhuang, we used this idea and the integrable structure of Liouville CFT to derive a scaling exponent for planar percolation called the backbone exponent, which was unknown for several decades.

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Date/Time : 12 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Maxim Raginsky, University of Illinois at Urbana-Champaign

The majorizing measure theorem of Fernique and Talagrand is a fundamental result in the theory of random processes. It relates the boundedness of random processes indexed by elements of a metric space to complexity measures arising from certain multiscale combinatorial structures, such as packing and covering trees. In this talk, based on joint work with Yifeng Chu, I will build on the ideas first outlined in a little-noticed preprint of Andreas Maurer to present an information-theoretic perspective on the majorizing measure theorem, according to which the boundedness of random processes is phrased in terms of the existence of efficient variable-length codes for the elements of the indexing metric space. I will close by discussing the relevance of these results to the analysis of statistical performance of machine learning algorithms through the lens of data compression and the Minimum Description Length principle.

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Date/Time : 12 Dec 2023 16:30
Venue : S17 #04-04 (SR3)
Speaker : Lucas Mann, University of Muenster

Based on classical constructions in algebraic topology and algebraic geometry, we introduce abstract 6-functor formalisms, which are a form of “abstract cohomology theories”. It turns out that the abstract theory of 6-functor formalisms is surprisingly rich and has applications in non-geometric fields as well. As an example we provide new tools for studying smooth representations of locally profinite groups — in particular in natural characteristic, where many of the classical representation theoretic methods fail. An application of the theory is a conceptual construction of an anti-involution on the (derived) Iwahori-Hecke algebra, which previously could only be constructed on the associated Ext-algebra using explicit computations.

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