### My Research

My research interests include Automatic Continuity, Several Complex
Variables, H^{p} spaces, and Reproducing Kernel Hilbert Spaces.
In my research, I studied certain interpolation problems
for analytic two-by-two matrix-valued functions on the unit disc.

The problem of designing controllers that are robust with respect to uncertainty leads to questions that are in the areas of Operator Theory and Several Complex Variables. One direction is the engineering problem of mu-synthesis, which has led to the study of certain inhomogeneous domains. Two such domains, the symmetrised bidisc and the tetrablock, are central in my research.

The mu-synthesis problem involves the construction of holomorphic matrix valued functions on the disc, subject to interpolation conditions and a boundedness condition. Using the properties of the tetrablock, I obtained a new solvability criterion for one such interpolation problem. The criterion uses the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method for the construction of the required interpolating functions.

The research was joint work with Dr Zinaida Lykova and Prof. Nicholas Young. More information on the research can be found in our paper "A rich structure related to the construction of analytic matrix functions".

### My Publications

D. C. Brown, Z. A. Lykova and N. J. Young, A rich structure related to the construction of analytic matrix functions, J. Funct. Anal. 272 (4) (2017), 1704-1754.

D. C. Brown, A rich structure related to the construction of holomorphic matrix functions, Thesis for admission to the degree of Doctor of Philosophy, Newcastle University, (2016). Link.

### Talks given

- 4 October 2016 "A rich structure related to the construction of holomorphic matrix functions" Newcastle University School of Mathematics and Statistics Analysis Seminar.
- 18 July 2016 "A criterion for the solvability of a mu-synthesis problem" IWOTA Washington University in St Louis, USA.
- 8 April 2016 "A criterion for the solvability of a mu-synthesis problem" YFAW Queen's University Belfast.
- 2 December 2015 "An interpolation problem related to the tetrablock" Newcastle University Postgraduate forum.
- 12 June 2015 "The rich structure related to the construction of holomorphic matrix functions: tetrablock" YFAW Imperial College London.
- 18 February 2015 "The rich structure related to the construction of holomorphic matrix functions: tetrablock" Newcastle University Postgraduate forum.
- 24 April 2014 "The rich structure related to the construction of holomorphic matrix functions" YFAW Lancaster University.
- 26 March 2014 "The rich structure related to the construction of holomorphic matrix functions" Newcastle University Postgraduate forum.

### Conferences attended

- 18-22 April 2017 "LMS Invited Lecture Series 2017: Function Theory by Hilbert Space Methods" Newcastle University.
- 18-22 July 2016 "The International Workshop on Operator Theory and Applications" Washington University in St Louis, USA.
- 10 June 2016 "Interactions of Operator Theory with Quantum Processes" Newcastle University.
- 6-8 April 2016 "The Young Functional Analysts' Workshop" Queen's University Belfast.
- 9-10 October 2015 "North British Functional Analysis Seminar" Newcastle University.
- 10-12 June 2015 "The Young Functional Analysts' Workshop" Imperial College London.
- 23-25 March 2015 "Spring School on Several Complex Variables" Newcastle University.
- 21-25 July 2014 "Function theory in several complex variables in relation to modelling uncertainty" ICMS Edinburgh.
- 23-25 April 2014 "The Young Functional Analysts' Workshop" Lancaster University.