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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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.

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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 : 20 Jan 2025 09:00
Venue : Zoom link TBA
Speaker : Yangyang Li, The University of Chicago

Minimal surfaces, which generalize geodesics to higher dimensions, naturally model various physical phenomena and are fundamental tools in geometric analysis. In this talk, I will survey recent advances on the existence, regularity, and Morse-theoretic study of minimal (hyper)surfaces. Highlights include our recent results on the enumeration of minimal surfaces, the generic regularity of minimal hypersurfaces, and their anisotropic counterparts. I will also discuss how these findings illuminate the interplay between the space of minimal hypersurfaces and the ambient Riemannian manifold.

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Date/Time : 17 Jan 2025 15:30
Venue : S17 #04-04 (SR3)
Speaker : Tang Lan, Central China Normal University

We mainly study regularity theory for the Monge-Ampère equations. Firstly, we establish the optimal global C^(1,α) regularity results of generalized solutions to the degenerate Monge-Ampère equations; secondly, we give the free boundary regularity for a class of obstacle problem.

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Date/Time : 17 Jan 2025 15:30
Venue : S17 #04-05 (SR2)
Speaker : Hao Wu, University of Zurich

Dynamical systems are deterministic systems. However, in chaotic systems where the entropy is positive, the ergodic sums often behave similarly to the sums of independent random variables and satisfy the spatial central limit theorems (CLT).

In zero-entropy systems, the spatial CLT often fails due to the lack of (fast) mixing properties. But in some cases, such as irrational rotations of bounded type, we can retrieve the central limit theorem if we study single orbit statistics. Here we fix the starting point but randomise time, hence the word “temporal” in the title. In this talk, I will present an ongoing joint work with Bromberg and Ulcigrai, where we make effective the method in their previous work, coding and Markov chains, to obtain a temporal central limit theorem for a broader class of observables over bounded irrational rotations.

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Date/Time : 16 Jan 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Weisheng Wu, Xiamen University

Geodesic flows in (and beyond) nonpositive curvature are important examples of nonuniformly hyperbolic dynamical systems. We discuss some ergodic properties of geodesic flows including the ergodicity of Liouville measure and the uniqueness of measure of maximal entropy, and the applications to the problem of counting closed geodesics.

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