2022-11-28 Mon
The Extended Persistent Homology Transform for Manifolds with Boundary
AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its ApplicationsSpeaker: Vanessa Robins Australian National University
Time: 11:00-12:30
Venue: Online
Zoom: 787 662 9899（PW: BIMSA）
Abstract
The Persistent Homology Transform (PHT) is a topological transform introduced by Turner, Mukherjee and Boyer in 2014. Its input is a shape embedded in Euclidean space; then to each unit vector the transform assigns the persistence module ofthe height function over that shape with respect to that direction. The PHT is injective on piecewise-linear subsets of Euclidean space, and it has been demonstrably useful in diverse applications as it provides a landmark-free method for quantifying the distance between shapes. One shortcoming is that shapes with different essential homology (i.e., Betti numbers) have an infinite distance between them. The theory of extended persistence for Morse functions on a manifold was developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes. By using extended persistence modules of height functionsover a shape, we obtain the extended persistent homology transform (XPHT) which provides a finite distance between shapes even when they have different Betti numbers. It may seem that the XPHT requires significant additional computational effort, but recent work by Katharine Turner and myself shows that when A is a compact manifold with boundary X, embedded in Euclidean space, the XPHT of A can be derivedfrom the PHT of X. James Morgan has implemented the required algorithms for 2-dimensional binary images as an R-package. This talk will provide an outline of our results and an illustration of their application to shape clustering.
Speaker Intro
Vanessa Robins is an associate professor in the Research School of Physics at the Australian National University. She develops theory and algorithmsfor the quantification of shape in data. Her major contributions include fundamental mathematical results for persistent homology, algorithm and software development for computing topological information from digital images, and their application to the characterisationof porous and granular materials.
Organizers：
Jie Wu, Guo-Wei Wei, Stephen Shing-Toung Yau, Haibao Duan, Yong Lin, Jianzhong Pan, Fei Han, Kelin Xia, Chao Zhou
2022-11-28 Mon
Local theta correspondence via C*-agebras of groups
Automorphic Theory SeminarSpeaker: Haluk Sengun University of Sheffield
Time: 19:00-20:30
Venue: Online
Zoom: 293 812 9202（PW: BIMSA）
Abstract
Local theta correspondence sets up a bijection between certain sets of admissible irreps of a pair of reductive groups G,H which sit as each others' centralizers in a larger symplectic group. The local correspondences then bundle up to set up a bijection between certain sets of automorphic representations of G and H. As a result, local theta correspondence is of importance in both representation theory and in the theory of automorphic forms. In joint work with Bram Mesland (Leiden), we have used Rieffel's theory of induction for representations of C*-algebras to prove that in many cases, local theta lifting is functorial and is continuous with respect to weak containment. In the talk, I will explain our approach and time permitting, will discuss further applications. Some of the results I will discuss can be found in the preprint arXiv:2207.13484.
Speaker Intro
Haluk Sengun is a number theorist based at the University of Sheffield, UK. He obtained his PhD at the University of Wisconsin-Madison and has held postdoctoral positions in Essen, Bonn, Barcelona and Warwick. He is interested arithmetic aspects of automorphic forms, cohomology of arithmetic groups and computational approaches. His recent work focuses on bringing in tools from the theory of C*-algebras and noncommutative geometry to his areas of interest listed above.
Organizers：Taiwang Deng, Bin Xu, Jun Yu
2022-11-29 Tue
Opinion Dynamics on Sheaves
AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its ApplicationsSpeaker: Robert Ghrist
Time: 09:00-10:30
Venue: Online
Zoom: 787 662 9899（PW: BIMSA）
Abstract
There is a long history of networked dynamical systems that models the spread of opinions over social networks, with the graph Laplacian playing a lead role. One of the difficulties in modelling opinion dynamics is the presence of polarization: not everyone comes to consensus. This talk will describe work with Jakob Hansen introducing a new model for opinion dynamics using sheaves of vector spaces over social networks. The graph Laplacian is enriched to a Hodge Laplacian, and the resulting dynamics on discourse sheaves can lead to some very interesting and perhaps more realistic outcomes. Extensions of these ideas to sheaves of lattices (in joint work with Hans Riess) will also be surveyed.
Speaker Intro
Robert Ghrist is the Andrea Mitchell PIK Professor of Mathematics and Electrical & Systems Engineering at the University of Pennsylvania. After earning a BS in Mechanical Engineering (University of Toledo, 1991), and the MS and PhD in Applied Mathematics (Cornell University, 1994, 1995), he held positions in Mathematics departments at the University of Texas (Austin), Georgia Tech, and the University of Illinois (Urbana-Champaign). He has been at Penn since 2008. Ghrist is a recognized leader in the field of Applied Algebraic Topology, with publications detailing topological methods for sensor networks, robotics, signal processing, data analysis, optimization, and more. He is the author of a leading textbook on the subject (Elementary Applied Topology, 2014), and has managed numerous large DoD grants from AFOSR, ASDRE, DARPA, and ONR. His research has been recognized with the NSF CAREER, NSF PECASE, SciAm50, and Vannevar Bush Faculty Fellow awards. Ghrist has been an invited speaker at two International Congresses of Mathematicians: once (Madrid 2006) for research and once (Seoul, 2014) for education. Ghrist is a dedicated expositor and communicator of Mathematics, with teaching awards that include the MAA James Crawford Prize, Penn's Lindback Award, and the S. Reid Warren award in Engineering at Penn. Ghrist is the author, designer, and animator of popular YouTube video texts (featureing the Calculus BLUE Project), as well as an online course on Coursera featured in the New York Times, BoingBoing, and Gizmodo.
Organizers：
Jie Wu, Guo-Wei Wei, Stephen Shing-Toung Yau, Haibao Duan, Yong Lin, Jianzhong Pan, Fei Han, Kelin Xia, Chao Zhou
2022-11-29 Tue
Symmetric Mass Generation
Category and Topological Order SeminarSpeaker: Yizhuang You UCSD
Time: 09:30-11:00
Venue: Online
Tencent: 607 3645 0351
Abstract
Symmetric mass generation (SMG) is a novel mechanism for massless fermions to acquire a mass via a strong-coupling non-perturbative interaction effect. In contrast to the conventional Higgs mechanism for fermion mass generation, the SMG mechanism does not condense any fermion bilinear coupling and preserves the full symmetry. It is connected to a broad range of topics, including anomaly cancellation, topological phase classification, and chiral fermion regularization. In this talk, I will introduce SMG through toy models, and review the current understanding of the SMG transition. I will also mention recent numerical efforts to investigate the SMG phenomenon. I will conclude the talk with remarks on future directions.
Speaker Intro
尤亦庄，加利福尼亚大学圣迭戈分校助理教授。研究领域为理论凝聚态物理，强关联系统，量子拓扑物态，多体局域化与热化，纠缠，张量网络，以及全息原理与机器学习。
Organizers：Hao Zheng
2022-11-29 Tue
A proof of Kudla-Rapoport conjecture for Kramer models at ramified primes
BIMSA-YMSC Tsinghua Number Theory SeminarSpeaker: Qiao He University of Wisconsin-Madison
Time: 10:30-11:30
Venue: Online
Zoom: 293 812 9202（PW: BIMSA）
Abstract
n this talk, I will first talk about the Kudla-Rapoport conjecture, which suggests a precise identity between arithmetic intersection numbers of special cycles on Rapoport-Zink space and derived local densities of hermitian forms. Then I will discuss how to modify the original conjecture over ramified primes and how to prove the modified conjecture. On the geometric side, we completely avoid explicit calculation of intersection number and the use of Tate’s conjecture. On the analytic side, the key input is a surprisingly simple formula for derived primitive local density. This talk is based on joint work with Chao Li, Yousheng Shi and Tonghai Yang.
Organizers：Hansheng Diao, Yueke Hu, Emmanuel Lecouturier, Cezar Lupu
2022-11-30 Wed
The Loop Hecke Algebra and Charge Conserving Yang–Baxter Operators
BIMSA-Tsinghua Quantum Symmetry SeminarSpeaker: Eric Rowell Texas A&M University
Time: 09:30-11:00
Venue: Online
Zoom: 537 192 5549（PW: BIMSA）
Abstract
The Loop Braid Group 𝐿𝐵𝑛 is the motion group of 𝑛 free loops in 𝑆3, with generators the “leapfrog” motions and the symmetric exchanges. In recent work with Celeste Damiani and Paul Martin, we defined a family of finite dimensional quotients 𝐿𝐻𝑛 of 𝐿𝐵𝑛 via certain quadratic relations. The structure of these (non-semisimple) Loop Hecke Algebras is partially understood through (conjecturally faithful) representations obtained from a loop braided vector space, i.e. a pair (𝑆, 𝑅) of Yang–Baxter operators satisfying the appropriate mixed relations. One salient feature of the matrix 𝑅 is that it is charge conserving. This begs the question: can we classify charge conserving Yang–Baxter operators? Recently with Martin, we have found such a classification with a concise combinatorial description. In this talk, I will give an overview of these two projects. Time permitting we will circle back to the question of 𝐿𝐵𝑛 representations.
Organizers：Zhengwei Liu, Sebastien Palcoux, Yilong Wang, Jinsong Wu
2022-11-30 Wed
Differential calculus in vector bundles: a (graded) algebraic approach
BIMSA-BIT Differential Geometry SeminarSpeaker: Jacob Kryczka BIMSA
Time: 15:15-16:30
Venue: Online
Zoom: 559 700 6085（PW: BIMSA）
Abstract
We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras. In particular we will describe a special class of short exact sequences of such functors and discuss how they may be interpreted as higher order Atiyah sequences. Time permitting, we will interpret the corresponding notion of the Atiyah class in this setting.
Organizers：Kotaro Kawai, Sebastian Heller, Lynn Heller, Chao Qian
2022-11-30 Wed
A number theory problem arising in quantum information theory
Geometry and Gravitation SeminarSpeaker: Ingemar Bengtsson University of Stockholm
Time: 17:15-18:30
Venue: Online
Zoom: 839 2831 2262（PW: BIMSA）
Abstract
A maximal regular simplex inscribed in the set of quantum states has some engineering applications --- if it exists. Attempts to prove that it does, in all finite dimensional Hilbert spaces, have revealed an unexpected connection to an open problem in algebraic number theory. The whole story is quite new, and it it may have ramifications that we have not thought of yet.
Speaker Intro
"My name is Ingemar Bengtsson, and I have been a lecturer at Fysikum in Stockholm since '93 (and a professor since '00). My previous 'career' was at Chalmers, CERN and Imperial College. The research areas that I like the best usually have something to do with geometry. General relativity is a favourite. Most of my work there is on black holes. My strongest prejudice is that the world has four dimensions; this is the direction in which I look for clues about quantum gravity. Then I work on quantum information theory, since the geometry of the space of quantum states is wonderful and rather mysterious. What I find fascinating about relativity and quantum mechanics---as it happens, the two deepest theories we have---is that their basic equations have been around for a century, and yet they keep springing conceptual surprises on us. I am looking for the next surprise there, but I do keep a weather eye open on other subjects as well."
Organizers：Lars Andersson
2022-12-01 Thu
基于深度学习的地震勘探速度建模
BIMSA-Tsinghua Seminar on Machine Learning and Differential EquationsSpeaker: Jianwei Ma Peking University
Time: 10:00-11:30
Venue: Online
Zoom: 537 192 5549（PW: BIMSA）
Speaker Intro
马坚伟，1998年本科毕业于大连理工大学工程力学系，2002年博士毕业于清华大学固体力学专业。在国外从事博士后和访问学者研究多年，先后受聘于清华大学航天航空学院、哈尔滨工业大学数学学院和人工智能研究院、北京大学地空学院和人工智能研究院。曾获国家杰出青年科学基金项目资助、国家高层次人才计划（领军人才），先后主持国家重点研发计划项目2项、国家自然科学基金重点项目1项。担任中国地球物理学会智能地球物理专业委员会主任、中国地质学会数据驱动与地学发展专业委员会副主任、IEEE Transactions on Geoscience and Remote Sensing期刊副主编。从事勘探地球物理学、应用数学、人工智能的交叉学科研究，发表论文在谷歌学术被引约5800次，其中在Reviews of Geophysics, IEEE Signal Processing Magazine发表综述论文。
Organizers：Fansheng Xiong, Wuyue Yang, Wen-An Yong, Yi Zhu
2022-12-01 Thu
Maximum entropy model and its application
Internal Seminar on Machine Learning and Differential EquationsSpeaker: Wuyue Yang BIMSA
Time: 14:00-16:00
Venue: Online
Zoom: 537 192 5549（PW: BIMSA）
Abstract
Entropy is an important concept in thermodynamics and statistical mechanics. Maximum entropy model (MaxEnt) is a method for solving probabilistic models. In this talk, I will introduce Maximum Entropy models and its applications in biology and other fields.
Speaker Intro
杨武岳，毕业于清华大学，理学博士、助理研究员，就职于北京雁栖湖应用数学研究院。长期从事生物信息、人工智能及其应用等研究。在Journal of Computational Physics，The Journal of Chemical Physics，Chemistry - A European Journal，统计与信息论坛，数学的实践与认识等国内外期刊发表学术论文10余篇。曾作为封面文章发表在The Journal of Chemical Physics期刊上。参与国家级、省级科研项目。多次获得学术学业奖学金及荣誉，如清华大学未来学者奖学金、国家奖学金、北京市优秀毕业生等。
Organizers：Wuyue Yang, Fansheng Xiong, Xiaopei Jiao
2022-12-02 Fri
Mathematics of topological quantum computing
YMSC-BIMSA Quantum Information SeminarSpeaker: Xingshan Cui Purdue University
Time: 09:30-11:00
Venue: 清华大学近春园西楼一层报告厅
Tencent: 337 8937 1456
Abstract
Topological quantum computing (TQC) is among the best approaches to building a large-scale fault-tolerant quantum computer. The quantum media for TQC are topological phases of matter that harbor non-Abelian anyons and quantum gates are implemented by braiding of anyons. The mathematics of topological phases of matter is described by modular tensor categories or equivalently by topological quantum field theories. We give a review on the rich interactions between TQC and the various subjects mentioned above. We illustrate the concept of TQC with an important class of anyons, namely, metaplectic anyons, and show that braidings of anyons assisted by certain topologically protected measurements is universal for quantum computing. The interest in metaplectic anyons arises from the potential physical realization in fractional quantum Hall systems. Some related questions on universality of anyons will also be addressed.
Organizers：Zhengwei Liu
2022-12-02 Fri
Suspension Homotopy of 4-manifolds And the Second 2-local Cohomotopy Sets
Math and Biology SeminarSpeaker: Pengcheng Li Southern University of Science and Technology
Time: 10:30-11:30
Venue: Online
Zoom: 293 812 9202（PW: BIMSA）
Abstract
In this talk we study the homotopy type of the (double) suspension of an orientable, closed, connected $4$-manifold $M$, whose integral homology can have $2$-torsion. This gives a complete solution to a prior research problem of So and Theriault. Moreover, the decomposition results are applied to give a characterization of the second $2$-local cohomotopy set $\pi^2(M;\mathbb{Z}_{(2)})$, which is the set of homotopy classes of based maps from $M$ into the $2$-local sphere $S^2_{(2)}$.
Organizers：Jie Wu, Jingyan Li, Xiang Liu, Fedor Pavutnitskiy
