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.

Add to calendar :

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.

Add to calendar :

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).

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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

Add to calendar :

Date/Time : 21 Jan 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Ara Basmajian, City University of New York

There are many topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), the Goldman bracket, complementary component types of a curve, mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we focus on the so called inf invariant (shortest length metric) associated to a filling curve, its relationship with the geometric self-intersection number, and its relation to the optimal metric that is tailored to produce the minimum length. While clearly the geometric self-intersection number is a type invariant, we address whether the inf invariant can distinguish between curves that have the same self-intersection. This is joint work with Sayantika Mondal.

Add to calendar :

Date/Time : 21 Jan 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Yunpeng Shi, University of California, Davis

In this talk, we explore two classical image processing tasks motivated by cryo-electron microscopy imaging: tomographic image denoising and rigid image registration. Both tasks inherently involve operations of 2D rotations, where leveraging specific transforms can significantly enhance the speed and robustness of the algorithms.

In the first part of the talk, I will introduce an unsupervised image denoiser based on estimating the covariance matrix of clean images. A key insight is that if the image manifold is invariant under global in-plane rotations, this symmetry can be exploited to accelerate computations and reduce dimensionality. I will discuss recent advances in fast expansion into steerable bases that allow us to efficiently utilize this rotational symmetry, leading to a thousand fold improvement in the speed of covariance estimation over existing methods. This technique has been successfully applied to joint deconvolution and denoising of large-scale, real-world cryo-EM images.

In the second part, I will present a fast algorithm for aligning images using optimal transport. Our method leverages the sliced Wasserstein distance by computing the 1D Wasserstein distance between radial line projections of input images. By applying a special transform to the frequency domain, we achieve efficient alignment of two L by L images in O(L^2 log L) operations — matching the complexity of alignment using Euclidean distance. I will demonstrate the robustness of our method to translations and deformations.

Finally, I will comment on the practical limitations of both methods and discuss open questions that remain, highlighting areas for future exploration.

Add to calendar :

Date/Time : 21 Jan 2025 14:00
Venue : S17 #06-11 (SR6)
Speaker : Johannes Lederer, University of Hamburg

Artificial intelligence is arguably the hottest topic in all of science. Computer science and engineering currently set the agenda in this field, sidelining mathematics to a large extent. This talk, however, will highlight that mathematics has a lot to offer. We will introduce mathematical guarantees that provide deep insights into the inner workings of AI, and we will show how statistical principles can make AI more efficient. More generally, we will discuss the role of mathematics, especially statistics, in AI and data science.

Add to calendar :