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

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.

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.

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.

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. 

Time:    2022.05.26 Thu    16:00-17:00

Venue:  BIMSA 1110

Zoom:    361 038 6975 （PW: BIMSA）

Video

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).

Time:    2022.05.05 Thu    09:00-10:30

Venue:  BIMSA 1120

Zoom:    388 528 9728 （PW: BIMSA）

Video

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. 

Time:    2022.04.28 Thu    17:00-18:30

Venue:  BIMSA 1110

Zoom:    388 528 9728 （PW: BIMSA）

         Video