Events of the Week

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Recent progress in multiplexed tissue imaging is advancing the study of tumor microenvironments to enhance our understanding of treatment response and disease progression. Despite its popularity, there are significant challenges in data analysis, including high computational demands that limit feasibility for large-scale applications and the lack of a principled strategy for integrative analysis across images. To overcome these challenges, we introduce a spatial topic model designed to decode high-level spatial architecture across multiplexed tissue images. Our method integrates both cell type and spatial information within a topic modelling framework, originally developed for natural language processing and adapted for computer vision. We benchmarked its performance through various case studies using different single-cell spatial transcriptomic and proteomic imaging platforms across different tissue types. We show that our method runs significant faster on large-scale image datasets, along with high precision and interpretability. It consistently identifies biologically and clinically significant spatial “topics”, such as tertiary lymphoid structures.

In this talk, we discuss the properties of homological invariants of random graphs under the Erdős–Rényi models. In particular, we focus on the law of large numbers for regularity, depth, v-numbers and other invariants of edge ideals, path ideals and cover ideals of Erdős–Rényi random graphs. (This is a joint work with Arindam Banerjee, Ritam Halder and Pritam Roy).

This is an introductory talk on the story of mathematical research on waves, aimed at graduate students from all areas and undergraduate students. This will serve as both a colloquium talk and a precursor to a self-contained mini course that will be given by the speaker in our “Integrability and Beyond!!!” Seminar, starting on October 20.

Focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schrödinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporal decay and are connected to the third Painlevé equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy—allowing for arbitrarily many simultaneous flows. Time permitting, we will report on recent work regarding their space-time asymptotic behavior under an arbitrary flow from the hierarchy.

This mini-course consists of a set of 5 90-minute long lectures that are aimed at graduate students from all areas.
Knowledge of complex analysis and differential equations should be enough to follow the lectures. There will be exercises.
Lecture 1; Integrable wave models: linearizing a flow
Introduces the scattering transform for the defocusing nonlinear Schrödinger equation and describes the analogy with Fourier transform methods for linear problems. Formulates a Riemann-Hilbert problem for the associated inverse-scattering transform.

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Solitons from Lecture 0 come into the picture, and consequently Riemann-Hilbert problems with pole singularities. A nonlinear version of the superposition principle present for linear homogeneous problems is introduced.

Let Gr(k,n) be the Grassmannian of k-planes in C^n. The standard torus action on C^n induces a torus action on Gr(k,n), whose orbits encode interesting combinatorial and geometric invariants. I will discuss explicit degenerations of the “generic” torus orbit closure into a union of Richardson varieties, and a further degeneration in the union of Schubert varieties. These give new proofs of cohomological formulas of Berget-Fink and Klyachko, respectively. I will also mention various extensions and open directions.

Describes the so-called “dressing” method: deriving differential equations satisfied by quantities extracted by the solution of a Riemann-Hilbert problem. Introduces a unified framework to capture arbitrary singularities in Riemann-Hilbert problems arising from direct scattering transform and its applications.

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I became aware of the extent of the attacks on the publication and citation processes though Taylor & Francis, the publisher of Quality Engineering for which I am the editor.

All areas of science are being affected, from mathematics to medicine. The following are some of the primary concerns: authorships being sold, fake papers produced by paper mills, sham reviews, unethical behavior of guest editors for special issues, bribes offered to editors, the rise of predatory journals and conferences, citation cartels, plagiarism, misuse of AI, and the fabrication of data and images.

These and other issues, illustrated with numerous examples, will be discussed in this presentation. A related issue is the proliferation of junk science.

Efforts to protect the scientific literature will be discussed, such as the contributions by individual sleuths and the use of the STM Integrity Hub that was established by the major academic publishers. There will also be some discussion of the article retraction process. Over 10,000 scientific papers were retracted in 2023, a record number.

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