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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Date/Time : 21 Jan 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Hui Gao, Southern University of Science and Technology

Using the Breuil–Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.

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Date/Time : 21 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yusuf Baris Kartal, University of Edinburgh

Morse theory provides a method for studying the topology of smooth manifolds through the critical points of a function and its gradient flow. In the 1980s, Andreas Floer applied ideas from Morse theory to the infinite dimensional setting, producing an invariant in symplectic topology that has since become a central tool in this field, and later understood to be closely related to enumerative geometry and mirror symmetry. Inspired by Floer’s insight, Cohen-Jones-Segal proposed a framework to obtain finer information from the data arising in Morse and Floer theory. However, this setup is not a priori compatible with the compact group actions, which limits their usefulness in the study of equivariant topology and geometry. In this talk, I will explain how, in joint work with Laurent Cote, we extend the setup and the construction of Cohen-Jones-Segal to study equivariant invariants. I will then discuss applications to topology and symplectic geometry, and explain how tools from homotopy theory can be used to relate symplectic invariants to the topology of the underlying manifold.

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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.

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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.

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