Date/Time : 18 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Ruixiang Zhang, University of California, Berkeley

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\”{o}rmander asked if a more general class of operators (known as H\”{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\”{o}rmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\”{o}rmander’s question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\”{o}rmander type operator satisfying Bourgain’s condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain’s condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

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Date/Time : 17 Jan 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yifeng Liu, IASM, Zhejiang University

BSD conjecture is one of the most influential problems in number theory, which concerns arithmetics of elliptic curves. It has been generalized to motives over number fields by Beilinson, Bloch and Kato in the 1980s. In this talk, we will survey recent progress of these conjectures for higher dimensional varieties.

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Date/Time : 17 Jan 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruixiang Zhang, University of California, Berkeley

Fourier restriction type problems form a class of important problems in harmonic analysis. They are also related to PDE, analytic number theory, geometric measure theory and mathematical physics. In the past 15 years, much progress has been made on these problems. In this talk, we will introduce Fourier restriction type problems and survey some recent results, and then conclude by mentioning some personal favorite future directions.

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Date/Time : 17 Jan 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Sam Stechmann, University of Wisconsin–Madison

Sea ice floats on the surface of the Arctic Ocean, and plays an important role in Earth’s climate. Traditionally, sea ice has been modeled as a material, as a continuum, with various rheological models proposed. From a more detailed perspective, sea ice is a collection of many individual blocks of ice, called “floes.” In this talk, I will discuss approaches for floe-based (i.e., particle-based) models of sea ice, including multi-scale models to bridge the gap between floe-based and continuum-based models.

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Date/Time : 17 Jan 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Makoto Tatsuta, National Institute of Informatics, Japan

An inductive definition is a way to define a predicate by an expression which may contain the predicate itself. The predicate is interpreted by the least fixed point of the defining equation. Inductive definitions are important in computer science, since they can define useful recursive data structures such as lists and trees.  Inductive definitions are important also in mathematical logic, since they increase the proof theoretic strength. Martin-Lof’s system of inductive definitions given in 1971 is one of the most popular system of inductive definitions. In 2006 Brotherston proposed an alternative formalization of inductive definitions, called a cyclic proof system. In general, for proof search, a cyclic proof system can find an induction formula in a more efficient way than Martin-Lof’s system, since a cyclic proof system does not have to choose fixed induction formulas in advance. The equivalence of the provability of Martin-Lof’s system for inductive definitions and that of the cyclic proof system was conjectured in 2006. The speaker and Berardi solved it in 2017.

This talk will explain this problem.

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Date/Time : 17 Jan 2024 16:00
Venue : S17 #06-11 (SR6)
Speaker : Tan Sheng Li, East China Normal University

Foliations are generalizations of fibrations on an algebraic surface. Some invariants of fibrations are generalized to foliations, e.g., Kodaira dimensions, geometric genus, Chern numbers, volume and slope. Fibrations with minimal slope must be hyperelliptic, which are double covers of ruled surfaces. We have no slope inequality for foliations, so it is interesting to construct some examples with small slopes. A nature way is to consider double covers of some transcendental foliations, we need to generalize the theory of double covers of fibrations to foliations. We will present an example whose slope is 12/7. This is a joint work with Jun LU and Shi XU.

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Date/Time : 15 Jan 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Liu Jialin, Alibaba Group (U.S.) Inc, Damo Academy

AI and data science have demonstrated remarkable promise in enhancing optimization algorithms. Utilizing AI/ML techniques, we aim to complement and potentially enhance traditional optimization methods, offering improvements in aspects like computational speed and solution quality. Despite these advancements, a deep, systematic understanding of these methods remains underdeveloped. In this talk, the speaker will demonstrate through concrete examples how mathematical tools, particularly optimization theory, can be used to unravel the mysteries of these ‘black boxes’. The speaker will also discuss the development of interpretable and reliable AI models for optimization, grounded in these mathematical principles.

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Date/Time : 11 Jan 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Yair Shenfeld, Brown University

It is classical that the ball is the unique shape which minimizes surface area for a given fixed volume. But what if we want to minimize surface area, where in addition to fixing the volume, we also fix the average width of the shape? This question turns out to be a very special case of long-standing conjectures at the heart of convex geometry. In the first part of the talk I will explain these conjectures and the significant progress that we made towards their resolutions. In the second part of the talk I will explain how these geometric problems can provide information on problems in combinatorics, which led us to the discovery of some surprising results in the theory of partially ordered sets.

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Date/Time : 11 Jan 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Mathias Weller, Technische Universität Berlin

In this talk, I will give a high-level introduction of some important computational problems in “Phylogenetics”, a branch of Biology that works with evolutionary histories of “taxa” (species, genes, samples, …). We will see some landmark results related to parameterized algorithms that analyze such histories.

In the second part of the talk, I would like to speak about a recently introduced layout-parameter for phylogenetic networks (basically rooted directed acyclic graphs with leaf-labels) called “scanwidth”, which has close ties to the treewidth of the underlying undirected graph. I will give examples for algorithmic results parameterized by the scanwidth and talk about efforts to efficiently compute the parameter itself.

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