# AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its Applications

**Aim and scope**

In the area of topology, there has been a lot of exciting progress in recent years. Topological tools and ideas have been used in arithmetic geometry, for instance, topological cyclic homology, and in low dimensional topology, like constructions of new knot and link invariants using homotopy theory and so on. On the other hand, topology has found exciting applications in science and technology. In mathematical physics, topics like topological quantum field theory, quantum anomaly cancellation, topological T-duality, topological insulators, topological order, and others, have made tremendous progress. In modern technology, topology has also played prominent roles. In particular, TDA (topological data analysis) has demonstrated great potential for big data and has been widely used in the study of robotics, materials, chemistry, biology, drug design and discovery. More and more topological tools and ideas have been used in machine learning and deep learning models. The aim of the seminar is to invite top experts and young researchers to introduce and share their progress in the study of topology and its applications. We expect that the seminar will bring potential collaborations and broaden the horizon of young students.

**Advisory board**

Guowei WEI (Michigan), Stephan YAU (Tsinghua)

**Organizers**

Haibao DUAN (AMSS), Yong LIN (YMSC), Jianzhong PAN (AMSS), Jie WU (BISMA)

**Associate organizers**

Fei HAN (NUS), Kelin XIA (NTU), Chao ZHOU (NUS)

**Format**

hybrid (face to face and online). The seminar will be held on every other Thursday from March 10, 2022.

**Cohomological Physicist**

**Speaker: James Stasheff**

**Time: **22:00-23:00 2022/12/15

**Zoom: ** 787 662 9899 （PW: BIMSA）

**Abtract：**

This is a mini-autobiography with emphasis on those who have most influenced my mathematical career and whence came the accidental flashes of insight that drew me.

**Speaker Intro.**

Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Phil. from Oxford under the supervision of Ioan James and a Ph.D. later the same year from Princeton under the supervision of John Coleman Moore.

From 1961 to 1962, Stasheff was a C.L.E. Moore instructor at the Massachusetts Institute of Technology. Then in 1962 joined the faculty of University of Notre Dame as an assistant professor; he was promoted to full professor there in 1968. He visited Princeton University from 1968 to 1969 and then stayed there the next year as a Sloan Fellow. Then in 1970 he moved to Temple University, where he held a position until 1978. In 1976, he joined the UNC faculty. He has also visited the Institute for Advanced Study, Lehigh University, Rutgers University, and the University of Pennsylvania.

Stasheff was an editor of the Transactions of the American Mathematical Society from 1978 to 1981, and managing editor from 1979 to 1981. He has been married since 1959 and has two children.

**Opinion Dynamics on Sheaves**

**Speaker: Robert Ghrist**

**Time: **9:00-10:30 2022/11/29

**Venue: **BIMSA 1120

**Zoom: **787 662 9899 （PW: BIMSA）

**Abtract：**

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.

**The Extended Persistent Homology Transform for Manifolds with Boundary**

**Speaker: Vanessa Robins**

**Time: **11:00-12:30 2022/11/28

**Venue: **BIMSA 1120

**Zoom: **787 662 9899 （PW: BIMSA）

**Abtract：**

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.

**Leibenson's equation on Riemannian manifolds**

**Speaker: Alexander Grigoryan**

**Time: ** 2022.09.29 Thu 17:00-18:30

**Venue:** BIMSA 1120

**Zoom: **518 868 7656 （PW: BIMSA）

**Abtract：**

We consider on arbitrary Riemannian manifolds the Leibenson equation:

$$\partial _{t}u=\Delta _{p}u^{q}.$$

This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium. Here $u(x,t)$ is the fraction of the volume that the liquid takes in porous medium at time $t$ at point $x$. The parameter $p$ characterizes the turbulence of a flow, while $q$ describes the compressibility of the liquid.

We prove that if $p>2$ and $1/(p-1)<q\leq 1$ then solutions of this equation have finite propagation speed. On complete manifolds with non-negative Ricci curvature, we obtain a sharp propagation rate that matches that of Barenblatt solution.

**Speaker Intro:**

Professor Alexander Grigor'yan received his PhD from Lomonosov Moscow University in 1982. Since then, he worked at the State University of Volgograd and the Institute of Control Sciences in Moscow with receiving his habilitation in 1989. He has been a Humboldt Fellow at Bielefeld University in 1992-93 and a Guest Scholar at Harvard University in 1993/94. After working from lecturer to professor at Imperial College London in 1994-2005, he became a professor at Bielefeld University from 2005.

Professor Grigor'yan won a gold medal at International Mathematical Olympiad when he was 17, awarded by the Prize of the Moscow Mathematical Society in 1988, and received the Whitehead Prize of London Mathematical Society in 1997. He was an invited speaker of European Congress of Mathematicians in Barcelona in 2000.

Professor Grigor'yan is one of the top experts in geometric analysis on Riemannian manifolds, metric spaces and graphs, with making various important contributions on the topics.

**Depth in arrangements: Dehn--Sommerville--Euler relations with applications**

**Speaker: Herbert Edelsbrunner**

**Time: ** 2022.05.26 Thu 16:00-17:00

**Venue:** BIMSA 1110

**Zoom: **361 038 6975 （PW: BIMSA）

**Abtract：**

The depth of a cell in an arrangement of $n$ (non-vertical) great-spheres in $S^d$ is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn--Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.

This is work with Ranita Biswas, Sebastiano Cultrera, and Morteza Saghafian.

**Speaker Intro:**

Herbert Edelsbrunner is one of pioneers for topological and geometric approaches to data science, by his excellent research on the topic during the decades, from his PhD thesis on computational geometry in 1982 and assistant professorship in Information Processing at the Graz University of Technology during 1981-1985, to his professorship positions at UIUC, Duke and ISTA. Professor Edelsbrunner received Alan T. Waterman Award, he is members of the Academia Europaea, the American Academy of Arts and Sciences and the German Academy of Sciences Leopoldina. Professor Edelsbrunner is widely recognized as one of founding fathers of the new-born area of topological data analysis (TDA).

**Identifying Nonlinear Dynamics with **

**High Confidence from Sparse Time Series Data**

**Speaker: Konstantin Mischaikow**

**Time: ** 2022.05.05 Thu 09:00-10:30

**Venue:** BIMSA 1120

**Zoom: **388 528 9728 （PW: BIMSA）

**Abtract：**

I will introduce a novel 5 step procedure that given time series data generated by a stationary deterministic nonlinear dynamical system provides a lower bound on the probability that the system generates specific local and/or global dynamic behavior. More precisely, the time series data is used to define a Gaussian process (GP). The mean of this GP provides a surrogate model. The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the probability that these topological invariants (and hence the characterized dynamics) apply to the unknown dynamical system (a random path of the GP). The focus of this talk is on explaining the ideas, thus I restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, chaotic dynamics, and bistability.

This is based on joint work with B. Batko, M. Gameiro, Y. Hung, W. Kalies, E. Vieira, and C. Thieme.

**Speaker Intro:**

Prof. Konstantin Mischaikow received his Phd from University of Wisconsin-Madison in 1985. He has worked as assistant professor and associated professor at Michigan State University and Georgia Tech. He is the director of CDSNS at Georgia Tech for eight years. Currently, he is the distinguished professor at Rutgers University. Prof. Konstantin Mischaikow is a world-leading expert in computational homology, Conley index, and their applications in dynamic systems. He has published more than 300 papers with a total citation > 9000.

**Parametric toric topology of complex Grassmann manifolds**

**Speaker: Victor Buchstaber**

**Time: **2022.04.28 Thu 17:00-18:30

**Venue: ** BIMSA 1110

**Zoom: **388 528 9728 （PW: BIMSA）

**Abtract：**

**Profile:**

Prof Victor Matveevich Buchstaber is a famous expert in algebraic topology, homotopy theory, and mathematical physics. He received his PhD in 1970 under the supervision of Sergei Novikov and Dr. Sci. in Physical and Mathematical Sciences in 1984 from Moscow State University. In 1974 he was an invited speaker in the International Congress of Mathematicians in Vancouver (Canada). He became a corresponding member of the Russian Academy of Sciences In 2006 . Now he is a principle researcher in the Department of Geometry and Topology of the Steklov Mathematical Institute. Prof Buchstaber is a leading expert of the new-born area of toric topology.

**Topological Methods for Deep Learning **

**Speaker: Gunnar Carlsson (Stanford University) **

**Time: **11:00-12:00, Apr. 21, 2022

**Venue: **1110

**ZOOM:** 388 528 9728, **密码：**BIMSA

**Abstract: **

Machine learning using neural networks is a very powerful methodology which has demonstrated utility in many different situations. In this talk I will show how work in the mathematical discipline called topological data analysis can be used to (1) lessen the amount of data needed in order to be able to learn and (2) make the computations more transparent. We will work primarily with image and video data.

**Speaker Introduction: **

Gunnar Carlsson received his Ph.D. from Stanford University in 1973, and has taught at University of Chicago, University of California (San Diego), Princeton University, and since 1991 at Stanford University. His early work was in algebraic topology and homotopy theory, and includes proofs of Segal's Burnside ring conjecture, Sullivan's fixed point conjecture, and many cases of the Novikov's higher signature conjecture. Since the late 1990's, he has also worked on topological approaches to data analysis, machine learning, and deep learning. He was a founder of the data analytics company Ayasdi Inc., and is a founder of the Deep Learning startup BlueLightAI INc.

**How Math and AI are revolutionizing biosciences **

**Speaker: Guo-Wei Wei (Michigan State University)**

**Time: **9:00-10:30, Mar. 24, 2022

**Venue: **1110

**ZOOM:** 388 528 9728, **密码：**BIMSA

**Abstract: **

Mathematics underpins fundamental theories in physics such as quantum mechanics, general relativity, and quantum field theory. Nonetheless, its success in modern biology, namely cellular biology, molecular biology, biochemistry, genomics, and genetics, has been quite limited. Artificial intelligence (AI) has fundamentally changed the landscape of science, technology, industry, and social media in the past few years and holds a great future for discovering the rules of life. However, AI-based biological discovery encounters challenges arising from the structural complexity of macromolecules, the high dimensionality of biological variability, the multiscale entanglement of molecules, cells, tissues, organs, and organisms, the nonlinearity of genotype, phenotype, and environment coupling, and the excessiveness of genomic, transcriptomic, proteomic, and metabolomic data. We tackle these challenges mathematically. Our work focuses on reducing the complexity, dimensionality, entanglement, and nonlinearity of biological data. We have introduced evolutionary de Rham-Hodge, persistent cohomology, persistent Laplacian, and persistent sheaf theories to model complex, heterogeneous, multiscale biological systems and thus significantly enhance AI's ability to handle biological data. Using our mathematical AI approaches, my team has been the top winner in D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design and discovery for years. By further integrating with millions of genomes isolated from patients, we reveal the mechanisms of SARS-CoV-2 evolution and transmission and accurately forecast emerging SARS-CoV-2 variants.

**Geometry and Topology of Data**

**Speaker: Prof. Jürgen Jost (MPI-Leipzig)**

**Time:** 17:00, Mar. 10, 2022

**Venue: **1110

**ZOOM: **388 528 9728, **PW:** BIMSA

**Abstract:**

Data sets are often equipped with distances between data points, and thereby constitute a discrete metric space. We develop general notions of curvature that capture local and global properties of such spaces and relate them to topological concepts such as hyperconvexity. This also leads to a new interpretation of TDA.