Institution: CSIC
Position: Científico Titular
Office: 406
Phone: +34 912999 786
Email: keith.rogers@icmat.es
About:
Keith grew up on the west coast of Scotland and completed his bachelor's degree in Mathematics at the University of Edinburgh in 1999. After obtaining a master's (Part III) at Cambridge University he completed his PhD in 2004 at the University of New South Wales in Sydney. In 2011 Keith was awarded a Starting Grant by the European Research Council and in 2019 he formed part of a team of four that was awarded an Advanced Grant.
Keith's speciality is Fourier Analysis; the art of recomposing a signal after decomposing it into different frequencies. This technique is fundamental in quantum mechanics, and so much of Keith's research has concerned the Schrödinger and wave equations. However, Fourier Analysis can also be employed to attack more basic questions, such as his extension of the Fundamental Theorem of Calculus in three or more dimensions. This joint work with Javier Parcet was published in the Proceedings of the National Academy of Sciences USA.
Fourier Analysis also finds application in a number of emerging technologies and Keith has redirected part of his energy in this direction. He recently solved an outstanding conjecture, made by Uhlmann in the International Congress of Mathematicians in 1998, proving that an imaging technique, known as Electrical Impedance Tomography, can be expected to produce faithful images of relatively rough (Lipschitz) objects. This joint work with Pedro Caro was published in the Forum of Mathematics Pi.
Keith also studies the fundamentals of Fourier Analysis itself. Whether Fourier Series and Integrals converge to their original signal or not is closely related to whether the Fourier Transform can be meaningfully restricted to a sphere (Stein's restriction conjecture). In collaborations with Nets Katz, published in Geometric and Functional Analysis, and Jonathan Hickman, published in the Cambridge Journal of Mathematics, the state of the art for this outstanding problem was improved.
