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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Date/Time : 22 Jan 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Gaofeng Huang, University of Bern

In this talk we survey a few results in the study of large holomorphic automorphism groups. First we give a short historical development of the so-called Andersen–Lempert theory, namely the study of large holomorphic automorphism groups. This field nowadays focuses on the so-called density property and a criterion for discovering more examples, many of which are smooth affine algebraic varieties.

In the second part we will consider the symplectic setting, namely complex symplectic manifolds with large group of holomorphic symplectic automorphisms. An example is the even dimensional Euclidean space endowed with the standard complex symplectic form. As a new example, the Calogero–Moser space, also admits the symplectic density property, which in turn leads to a strong approximation result for symplectic automorphisms in the Carleman sense.

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Date/Time : 22 Jan 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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