Date/Time : 05 Feb 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Frank Stephan, National University of Singapore

The talk presents joint work with David Belanger (Singapore), C.T.~Chong (Singapore), Rupert Hoelzl (Munich), Gordon Hoi (Singapore), Sanjay Jain (Singapore), Dilip Raghavan (Singapore), Frank Stephan (Singapore), Daniel Turetsky (Wellington) and Jing Zhang (Toronto) published in four papers. The first one dealt with comparing the growth behaviour of functions or families of functions partial-recursive relative to an oracle and classified oracles with respect to this measure. The second, third and fourth papers are about reverse mathematics and study weakly represented families, that are families of functions computed relative to some oracle in the second order part of the model (all functions in the family use the same oracle). These families are used as a tool to define notions like dominated (every weakly represented family is dominated by some function in the model), hyperimmune, biimmune and so on. Furthermore, one can such families also use as a hypothesis space for classes to be learnt – this extends the study to inductive inference. This is a survey talk with sample proofs.

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Date/Time : 04 Feb 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : David Hansen, National University of Singapore

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on “everything” related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring – the stable Bernstein center – which should be intimately related with the local Langlands correspondence. However, while it is easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue’s theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I’ll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G.

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Date/Time : 03 Feb 2025 16:30
Venue : S16 #03-04
Speaker : Lin Bo, Huairou Laboratory

In this talk, I will introduce a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging technique to solve the drift-diffusion model for semiconductor devices. The proposed scheme combines the robustness of the Scharfetter-Gummel method, which is a classical discretization for this model, with the high-order accuracy and hp-flexibility by the HDG scheme. Gradient-based indicators are introduced to utilize the hp-adaptivity. Numerical examples suggest that the proposed scheme does not exhibit oscillations even when applied to heavily doped and sharp PN-junctions.

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Date/Time : 27 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Wil Schilders, President of ICIAM, Eindhoven University of Technology & TU Munich-Institute for Advanced Study

Scientific machine learning (SciML) has been taking the academic world by storm as an interesting blend of traditional scientific modeling with machine learning (ML) methodologies like deep learning. While traditional machine learning methodologies have difficulties with scientific issues like interpretability, and enforcing physical constraints, the blend of ML with numerical analysis and differential equations has evolved into a novel field of research which overcome these problems while adding the data-driven automatic learning features of modern machine learning. Many successes have already been demonstrated, with tools like physics-informed neural networks, universal differential equations, deep backward stochastic differential equation solvers for high dimensional partial differential equations, and neural surrogates showcasing how deep learning can greatly improve scientific modeling practice. Consequently, SciML holds promise for versatile application across a wide spectrum of scientific disciplines, ranging from the investigation of subatomic particles to the comprehension of macroscopic systems like economies and climates.

However, despite notable strides in enhancing the speed and accuracy of these methodologies, their utility in practical and specifically industrial settings remain constrained. Many domains within the scientific community still lack comprehensive validation and robustness testing of SciML approaches. This limitation is particularly pronounced when confronted with complex, real-world datasets emanating from interactions between machinery and environmental sensors as usually addressed in industry. Still if appropriately addressed, SciML with its promise to accelerate innovations and scientific discoveries by orders of magnitudes, offers unique opportunities to address the insatiable desire for faster and more accurate predictions in many fields.

This presentation is dedicated to exploring recent advancements in the implementation of SciML techniques. We will discuss how methodologies can be refined to ensure their practical viability and scalability, particularly in industrial sectors where digital and physical components converge.

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Date/Time : 24 Jan 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Zhihan Wang, University of Chicago

I’ll discuss an upcoming work joint with Otis Chodosh, Christos Mantoulidis and Felix Schulze, in which we prove generic regularity of area-minimizing hypersurfaces in ambient dimension 11, extending the previous result of my co-authors in dimension 9 & 10.

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Date/Time : 24 Jan 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Wu Xinming, Fudan University

In this talk, we propose a finite element method for solving the coupled model of classical drift-diffusion equations and Schrödinger-Poisson equations in simulating a resonant tunneling diode (RTD). The coupling coefficient $\Theta$ and the quantum electron density $n^{\rm qu}$ are defined by weighted integrals of $|\psi_p|^2$ where $\psi_p$ is the wave function of Schrödinger equation for each momentum $p\in\mathbb{R}$. We first prove a growth rate of $|\psi_p|$ as $|p|\to +\infty$. This enables us to truncate the momentum space $\mathbb{R}$ to a bounded interval $[-P,P]$, which corresponds to an energy interval $[0,E_P]$. Based on a posteriori error estimate for computing $\Theta$ and $n^{\rm qu}$, we propose an energy-adaptive algorithm. The algorithm reduces the number of nodes greatly by capturing resonance peaks accurately and adjusting the partition of $[0,E_P]$ adaptively. Numerical experiment for an RTD shows that, based on energy-adaptive grids, the Gummel method for solving the coupled model is robust and can converge in several iterative steps.

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Date/Time : 24 Jan 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Douglas Zhou, Shanghai Jiao Tong University

Grasping causal connectivity within a network is essential for uncovering its functional dynamics. However, the inferred causal connections are fundamentally influenced by the choice of causality measure, which may not always correspond to the network’s actual structural connectivity. The relationship between causal and structural connectivity, particularly how different causality measures influence the inferred causal links, warrants further exploration. In this talk, we first theoretically establish a mapping between structural and causal connectivity based on voltage signals from simulated Hodgkin-Huxley and integrate-and-fire neuronal networks. To extend our results to practical cases, we then examine nonlinear networks with pulse signal outputs, such as spiking neural networks, employing four prevalent causality measures: time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy in the context of pairwise treatment. We conduct a theoretical analysis to elucidate the interconnections among these measures when applied to pulse signals. Through case studies involving both a simulated Hodgkin-Huxley network and an empirical mouse brain network, we validate the quantitative relationships between these causality measures. Our findings demonstrate a strong correspondence between the causal connectivity derived from these measures and the actual structural connectivity, thus establishing a direct link between them. We underscore that structural connectivity in networks with pulse outputs can be reconstructed pairwise, circumventing the need for global information from all network nodes and effectively avoiding the curse of dimensionality. Our approach provides a robust and practical methodology for reconstructing networks based on pulse outputs, offering significant implications for understanding and mapping neural circuitry.

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Date/Time : 23 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Liu Beibei, Ohio State University

The critical exponent is an important numerical invariant of discrete isometry groups acting on symmetric spaces of non-compact type. In this talk, I will explain how the numerical invariant is closely related to geometry, dynamics, and representation of the group action in rank 1 symmetric spaces. On the higher rank side, I will focus on the comparison of critical exponent with other quantities such as Benoist limit cone and cubic differential under the deformation of surface group representations in the Lie group PSL(n, R).

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Date/Time : 22 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhihan Wang, University of Chicago

A central question in geometric flow is to understand the change of geometry and topology after passing through singularities.

In this lecture, I will focus on mean curvature flow, the negative gradient flow of area functional, and explain how the local dynamics influence the shape of the flow near a singularity, as well as how the geometry and topology of the flow change after passing through a singularity with generic dynamics.

This talk is based on the joint work with Ao Sun and Jinxin Xue.

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