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INFORMATION ON DOCTORAL THESIS
1. Full name: Vo Thi Nhu Quynh 2. Sex: Female
3. Date of birth: 28/09/1979 4. Place of birth: Nghệ An
5. Admission of decision number: 459/ĐT-SĐH. Dated 16 November 2001
6. Changes in academic process:
7. Official thesis title: The squaring operation in study of the cohomology of the Steenrod algebra and the Lannes-Zarati homomorphism
8. Major: Algebra and Number Theory 9. Code: 62 46 05 01
10. Supervisors: Prof. Dr Sc. Nguyễn Hữu Việt Hưng
11. Summary of the new findings of the thesis:
1. Prove that the critical element of lowest homological degree in the cohomology of the Steenrod algebra is not in the image of Singer’s fifth transfer. (Here, an element in the cohomology of the Steenrod algebra is called critical if it is annihilated by the classical squaring operation and its stem satisfies some arithmetical condition.) This result somehow supports Singer's conjecture on the injectivity of the algebraic transfer.
2. Prove that the squaring operation Sq on the dual of the minimal generators of the Dickson algebra (regarded as a module over the Steenrod algebra) is an isomorphism on its image. We also show certain degrees in which the squaring operation vanishes.
3. Based on the results on the squaring operation, we show that the Lannes-Zarati homomorphism vanishes on (1) every element in any finite Sq -family except possibly the family's initial element, and (2) almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.
12. Practical applicability, if any:
13. Further research directions, if any:
1. Study Singer’s conjecture stating that the algebraic transfer is a monomorphism.
2. Study the conjecture of Nguyễn H. V. Hưng predicting that the squaring operation is an eventual isomorphism on the cohomology of the Steenrod algebra.
14. Thesis-related publications:
1. Võ T. N. Quỳnh (2007), “On behavior of the fifth algebraic transfer”, in Proceedings of the School and Conference in Algebraic Topology, 11, Geom. Topol. Publ., Conventry, pp. 309 - 326.
2. Nguyễn H. V. Hưng, Võ T. N. Quỳnh (2009), “The squaring operation on A-generators of the Dickson algebra”, Proc. Japan Acad., 85(6), Ser. A, pp. 67 – 70.
3. Nguyễn H. V. Hưng, Võ T. N. Quỳnh (2009), “The image of Singer’s fourth transfer”, C. R. Acad. Sci. Paris, Serie I, 347, pp. 1415 – 1418.
4. Nguyễn H. V. Hưng, Võ T. N. Quỳnh (2010), “The squaring operation on A- generators of the Dickson algebra”, Math. Proc. Camb. Phil. Soc., 148, pp. 267 – 288.
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Date: 31 March 2010
Signature:
Full name: Vo Thi Nhu Quynh
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