On December 8, 2023, the Vietnam Institute of Advanced Study in Mathematics (VIASM) organized the Workshop on New Trends in Algebraic Topology at the International Center for Interdisciplinary Science and Education (ICISE). The workshop was a part of the Winter School on Algebraic Topology and Its Applications, within the framework of the National Priority Program for the Development of Mathematics for the period 2021 - 2030. In addition to the excellent organizational support from Quy Nhon University, the Winter School received sponsorship from the French National Center for Scientific Research (CNRS) through the International Research Laboratory Vietnam - France on Mathematics and its Applications (IRL-FVMA), as well as the personal research fund of participating scientists.
The presentations in the workshop highlighted new research directions in algebraic topology, particularly focusing on the theory of stable homotopy groups and invariant theory. The workshop provided an opportunity for students and young mathematicians to listen to and engage in direct exchanges with experts in the field of Algebraic Topology
Speakers and participants at the workshop
The workshop program took place over 1 day with 6 presentations from the following experts: Dr. Lorenzo Guerra (University of Roma Tor Vergata, Italy), Prof. Huynh Mui (University of Natural Sciences - Hanoi National University), Prof. Jean Lannes (University of Paris 7, France), Assoc. Prof. Dr. Nguyen Dang Ho Hai (University of Sciences, Hue University), Dr. Nguyen Duc Nga (Phenikaa University), and Prof. Sarah Whitehouse (University of Sheffield, England).
Dr. Lorenzo Guerra presented at the workshop
Dr. Lorenzo Guerra presented a novel approach to the stable homotopy of spaces D_k(X) = (E(Σ_k) × (X^k)) / Σ_k with coefficients in the field F_2, emphasizing the geometric significance in relation to natural structures, specifically Hopf ring structures. Consequently, the author achieves a characteristic of the stable homotopy of E_∞-algebras and the space of infinite loops generated from a topological space X.
Prof.Huỳnh Mùi delivered a presentation at the workshop
Prof. Huynh Mui presented on the modular exotic invariants of a pair of linear groups, particularly emphasizing the relationship with the Lyndon-Hochschild-Serre spectral sequence.
Prof.Jean Lannes reported at the workshop
Prof. Jean Lannes discussed recent advances in research on the Quillen congruence relating the stable homotopy of a finite group to the stable homotopy of its elementary abelian subgroups. The main result by Prof. Lannes indicates that the Quillen congruence is an isomorphism with a wide class of groups, including symmetric groups, alternating groups, and finite Coxeter groups.
Assoc Prof. Nguyễn Đặng Hồ Hải presented at the workshop
In his presentation, Assoc. Prof. Nguyen Dang Ho Hai examined the action of the general linear group GL_n(F_q) on the quotient ring Q := S/I, where S is the polynomial ring over F_q in n variables x_1, ..., x_n, and I is the ideal generated by the powers q^m of the variables x_i. For each parabolic subgroup P of GL_n(F_q), a conjecture by Lewis, Reiner, and Stanton (2017) predicts an explicit formula for the Poincaré series of the graded vector space of P-invariants in the quotient ring Q. This report provides a proof of this conjecture for the Borel subgroup.
Msc. Nguyễn Đức Ngà delivered his report
Msc. Nguyen Duc Nga reported on the Margolis homology of Mui algebras with differentiation given by Milnor operators. The main result is an explicit formula for the Margolis homology groups. Margolis homology provides us with information for studying Morava K-theory..
Prof. Sarah Whitehouse presented at the workshop
Prof. Sarah Whitehouse presented on suspension models related to spectral sequences and utilizes them to explore stable homotopy theory.