Workshop on August 21, 2015

Talks and discussions on Friday afternoon, in room B11, 227 Nguyễn Văn Cừ, District 5.

13:00-13:40

Title: On combinatorial forms

Speaker: Huỳnh Quang Vũ (HCMUS)

Abstract: Differential forms on smooth manifolds is a rather common object in mathematics. It is well-known that there is a homology built from differential forms that is isomorphic to the usual singular homology. In the 1990s R. Forman developed a combinatorial analog - the "combinatorial forms" on simplicial complexes. We will discuss this theory, particularly relations between closed/exact forms in the case of 1-forms. (Joint work with Phan Van Phuong.)

13:40-14:20

Title: A brief introduction to algorithmic graph theory and related issues

Speaker: Nguyễn Phúc Sơn (University of Economics and Law, VNUHCM)

Abstract: Graph theory was invented to model binary relations, thus, it has had numerous usages in various applications. However, taking computational complexity into account, most graph theory problems don’t have efficient solutions (NP hard), so it leads to the needs of designing approximation algorithms. In this talk, I will discuss two major techniques coming from linear algebra (spectral methods) and convex programming. At present, the interactions of these two techniques are being gradually understood through the lens of real algebraic geometry, in particular via the nullstellensatz, the positivstellensatz and the Lasserre hierachies. As regards topology, classically, a graph can be considered as a 1-skeleton of higher dimensional cell complexes. Indeed, this view is useful in applications since cell complexes model n-ary relations (n>1). Recently, there have been a lot of efforts (especially by statisticians) trying to transfer graph theory techniques to topology. I will quickly mention a few interesting questions in this setting.

14:30-15:10

Title: The persistent homology of crossed modules

Speaker: Lê Văn Luyện (HCMUS)

Abstract: In 1946 J.H.C Whitehead introduced an algebraic model for spaces X which have trivial homotopy groups $\pi_n(X)$ for $n\geq 3$. This model is called a crossed module. Two crossed modules are defined to be quasi-isomorphic if there is a sequence of quasi-isomorphisms connecting them.
In this talk, we introduce a notion of persistent homology for crossed modules of prime-power order. We also give an algorithm for computing this notion. The persistent homology is an invariant of quasi-isomorphism classes of crossed modules. In many cases, this invariant could be used to show that certain crossed modules are not quasi-isomorphic.

15:10-15:50

Title: An algorithm for cohomology of certain crystallographic groups

Speaker: Bùi Anh Tuấn (HCMUS)

Abstract: This paper contributes to the computation of integral cohomology rings of crystallographic groups. We introduce the class of cubical crystallographic groups and, for this class, provide an algorithm for computing cohomology rings.

16:00-16:40

Title: Hierarchical clustering methods on Asymmetric Networks

Speaker: Facundo Mémoli (Ohio State University, USA)