Joint BCAM-UPV/EHU Analysis and PDE seminar: The role of the Algebraic Multiplicity in Topological Degree Theory

LOCATION: BCAM Seminar Room and Online

Abstract
In the foundational 1934 paper of the degree for compact perturbation of the identities (Leray & Schauder degree), the authors established a hidden relation between the degree and the Algebraic multiplicity of linear compact operators. This relation motived Esquinas and L opez-G omez to generalize in 1988 the concept of the algebraic multiplicity. In this talk, we will summarize the concept of the generalized algebraic multiplicity of Esquinas and L opez-G omez through an early hidden relationship found by the speaker [3] of this concept and the notion of intersection multiplicity of algebraic varieties. Later, this construction will be used to relate, in an analogy to the formula of Leray & Schauder, the topological degree for Fredholm operators of Fitzpatrick, Pejsachowicz and Rabier (that include almost all the topological degrees known up to date) and the generalized algebraic multiplicity [1]. As a result of our  ndings [2], we will be able to prove an analogue of the axiomatization & uniqueness theorems of Fuhrer and Amann & Weiss to cover the degree of Fredholm operators of Fitzpatrick, Pejsachowicz and Rabier. Finally, we will show, though a nonlinear PDE example, how the developed geometrical techniques are used in applications.

[1] J. L opez-G omez and J. C. Sampedro, Algebraic multiplicity and topological degree for Fredholm operators, Nonl. Anal. 201 (2020), 112019, 28 pp.
[2] J. L opez-G omez and J. C. Sampedro, Axiomatization of the degree for Fredholm operators of Fitzpatrick, Pejsachowiz and Rabier, Fixed Points and App., in press.
[3] J. L opez-G omez and J. C. Sampedro, New analytical and geometrical aspects of the algebraic multiplicity, J. Math. Anal. Appns. 504 (2021), 125375, 21 pp.



Link to the session: 
https://us06web.zoom.us/j/99649860282?pwd=SE0vemtYMFlwbFBNTXQyOTBONG0vZz09

More info at https://sites.google.com/view/apdebilbao/home