Date/Time : 16 Nov 2023 10:00
Venue : via Zoom
Speaker : Yang Liu, University of California, Los Angeles

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https://nus-sg.zoom.us/j/83242625129?pwd=ZmRzME93QndWbTNZQUdPUVdIS080Zz09

Meeting ID: 832 4262 5129
Passcode: 157423

In this talk, I will first briefly introduce my work in Ph.D. years on physics-informed deep generative models. The general motivation is to exploit the flexibility and high-dimensional representability of deep generative models, and improve the generative models with physical and mathematical knowledge. I will present some numerical and theoretical results on PDE uncertainty quantification, optimal transport learning, particle dynamic inference, etc.

The main part of the talk will be on my recent work in UCLA. Realizing the limitations of task-specific models, I proposed a new paradigm for scientific machine learning, namely “In-Context Operator Networks” (ICON). A distinguishing feature of ICON is its ability to learn operators from data prompts during the inference phase, without weight adjustments. Much like how a single Large Language Model can solve a variety of natural language processing tasks specified by the language prompt, a single ICON model, trained as a generalist operator learner rather than being tuned to approximate a specific operator, can tackle a wide range of tasks involving different operators. I will show how a single ICON model manages 19 distinct problem types, encompassing forward and inverse ODE, PDE, and mean-field control (MFC) problems, each spanning a continuous spectrum of operators. I will also show the great accuracy of ICON on forward and backward predictions of conservation laws with discontinuities. The learning capability is generalized to operators beyond the training distribution without any fine-tuning.

Reference:
https://www.pnas.org/doi/10.1073/pnas.2310142120
https://arxiv.org/pdf/2308.05061.pdf

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Date/Time : 15 Nov 2023 10:00
Venue : via Zoom
Speaker : Alex Cohen, Massachusetts Institute of Technology

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

Heilbronn’s triangle problem asks, how small is the smallest triangle formed by a set of points? Suppose n points are placed in the unit square and Delta is the smallest area triangle formed by three of these points. It is not too hard to see that Delta < C n^{-1}. Komlos, Pintz, and Szemeredi proved in 1981 that Delta << C n^{-8/7} by building on an ingenious method of Roth. We improve the bound to Delta < C n^{-8/7-1/2000}. The improvement comes from establishing new connections between Heilbronn’s problem and projection theory.

All joint with Dmitrii Zakharov and Cosmin Pohoata.

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Date/Time : 15 Nov 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Zou Guijin, Institute of High Performance Computing

At the nano-bio interface, mechanics can play important roles in the development of health-related technologies. In this talk, we will highlight two examples of such applications. First, we will examine how the mechanical properties of flexible nanomaterials affect their encapsulation in lipid vesicles. Through molecular dynamics simulations complemented by theoretical analysis, we have mapped out the phase diagram for these systems, revealing how nanomaterial stiffness and size relative to the vesicle dictate encapsulation outcomes. These results enhance our understanding of cell-nanomaterial interactions such as nano-toxicity, help guide the design of artificial cells and biohybrid microrobots. In the second example, we will introduce the emerging field of membrane targeting nanomedicine (MTN), an area where our mechanistic understanding is still at a primitive stage. Collaborating with experimental groups, we demonstrate how atomistic modelling and simulation facilitate the identification of membrane targeting antimicrobials (MTAs) capable of selectively penetrating and aggregating in bacterial lipid membranes, causing membrane permeability/rupture. These studies suggest a general simulation-assisted platform to accelerate discovery and innovation in nanomedicine.

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Date/Time : 14 Nov 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Liu Shixiao,

Last semester I talked about the upper density Mathias forcing which we prove to be proper. This time we take a look at several similar forcings, each of which collapses the continuum to omega and thus fails to be proper.

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Date/Time : 09 Nov 2023 16:00
Venue : S17 #06-11 (SR6)
Speaker : Andreas Wieser, Hebrew University of Jerusalem

In the 80’s, Livingston asked whether there exist Seifert surfaces F,F’ for the same knot so that, after pushing their interiors into the four-ball, the surfaces are not topologically ambient isotopic. The question has recently been resolved by Hayden, Miller, Kim, Park, and Sundberg whose example satisfies that the interiors of the surfaces are disjoint. In joint work with Menny Aka, Peter Feller, and Alison B. Miller we classify all examples that can be recognized algebraically. This is based on an exact number-theoretic characterization of pairs of Seifert forms that can arise from Seifert surfaces with disjoint interiors. Here, the characterization is in terms of the Gauss composition of binary quadratic forms. In the second half of the talk we will also discuss asymptotic independence questions related to the above setting (based on work with Aka and Einsiedler) as well as open questions. No preliminaries are assumed. The first half of the talk will be focused on geometry and the second will be number-theoretical in nature (bordering on homogeneous dynamics).

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Date/Time : 09 Nov 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Mengqi, National University of Singapore

Recent studies revealed that viscoelastic (polymeric) pipe flows can be linearly unstable, in contrast to the stability of the Newtonian pipe flows. The new linear instability was found to be a center mode with the eigenfunction concentrating in the center part of the pipe (Garg et al. 2018). We carried out linear stability analysis and transient energy growth analysis to further explore the linear dynamics of the center mode. It is found that an Orr mechanism(along with the critical-layer mechanism) is at play in the conformation tensor field, contributing to the total energy growth.

From a nonlinear point of view, the linear instability implies a possibility of supercritical bifurcation in viscoelastic pipe flows, again, in contrast to the subcritical Newtonian pipe flows. We applied a multiple-scale expansion method to study the bifurcation of the viscoelastic pipe flows. It is found that both supercritical and subcritical bifurcations exist in this flow. We calculated the boundary between the two bifurcations in terms of viscosity ratio (related to the polymer concentration) and our theoretical prediction gave a value about 0.785 (below which, the flow is supercritical; vice versa),qualitatively close to the experimental observation around 0.855. A scaling for the Landau coefficient has also been found.

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Date/Time : 08 Nov 2023 14:00
Venue : S17 #05-11 (SR5)
Speaker : Xiaoqian Xu, Duke Kunshan University

In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of application is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation, such as the famous Keller-Segel equation for chemotaxis. I will also discuss about the examples of such flows in this talk.

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Date/Time : 07 Nov 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Ye Jinhe , University of Oxford

A difference field is a field equipped with a given automorphism and a difference variety is the natural analogue of an algebraic varieties in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields.

For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to Lang-Weil estimate for difference varieties in finite difference fields. The proof uses pseudofinite difference fields, where the automorphism is the nonstandard Frobenius. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.

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Date/Time : 01 Nov 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Lin Ting, Peking University

In this talk, I will introduce the construction of the $C^r$ finite element space in any dimension. We show that $C^r$ finite element in $\mathbb R^d$ can be constructed, with the local shape function space $\mathcal P_{k}$ ($k\ge 2^d r+1$). Moreover, we show that the assumptions in our construction are sharp. This is based on joint work with Jun Hu (PKU), Qingyu Wu (PKU), and Beihui Yuan (YMSC).

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