Sometimes I forget which things I’m currently reading (i.e. dipping in and out of). So, here are a few notes, mainly to myself and mainly about books and more obscure sources than the usual current research papers.
A couple of things on category theory: Category Theory for the Sciences by Spivak and Sets for Mathematics by Lawvere and Rosebrugh. (Also Mathematical Physics by Geroch, but that is more of a broad coverage of essential mathematics using category theory than a book introducing/studying category theory itself.) Really enjoying both. Would like to code up some of the content of Spivak to illustrate the main ideas.
A few things on mathematical biology/physiology etc (mainly for work/background I should know but have either forgotten or not learned). Mathematical Physiology by Keener and Sneyd (the latter being my old PhD supervisor). Free Energy Transduction and Biochemical Cycle Kinetics by Hill (as well as the older, longer version). An underrated book, I need to summarise the best bits at some point. Basic Principles of Membrane Transport by Schultz. Another great classic, helped me a lot during my PhD. Both a bit old but the main thing that seems to have changed is that we have actually identified a lot of the proteins behind the mechanisms originally predicted on based on coarse information and largely theoretical modelling!
Stochastic Modelling for Systems Biology by Wilkinson, Chemical Biophysics by Qian and Beard and Stochastic Process for Physics and Chemistry by van Kampen. Good complements to the above books, generally more focused on stochastic aspects, but still similar concepts. See also the papers Entropy Production in Mesoscopic Stochastic Thermodynamics: Nonequilibrium Kinetic Cycles Driven by Chemical Potentials, Temperatures, and Mechanical Forces by Qian et al. as well as Contact Geometry of Mesoscopic Thermodynamics and Dynamics by Grmela. Also, the book Statistical Thermodynamics of Nonequilibrium processes by Keizer. Should summarise the various key concepts and how to think about ‘mesoscopic’ processes in biology.
A few references on mechanics: some point particle stuff (want to use in some applications), also differential geometry, symmetry etc. Introduction to Physical Modelling by Wellstead (mainly interested in the ‘mobility analogy’). The Variational Principles of Mechanics by Lanczos (a classic!). Analytical Dynamics by Udwadia and Kalaba. Nonholonomic Mechanics and Control by Bloch et al. First Steps in Differential Geometry: Riemannian, Contact, Symplectic by McInerney. Discrete Differential Geometry: An Applied Introduction by Grinspun et al. Foundations of Mechanics by Abraham and Marsden. Introduction to Mechanics and Symmetry by Ratiu and Marsden. Mathematical Foundations of Elasticity by Marsden and Hughes. Also the paper: ‘On the Nature of Constraints for Continua Undergoing Dissipative Processes’ by Rajagopal and Srinivasa.
Dynamical systems (research and teaching – solution and analysis methods): Numerical Continuation Methods for Dynamical Systems by Krauskopf, Osinga and Galan-Vioque. Recipes for Continuation by Dankowicz and Schilder. Stability, Instability and Chaos by Glendinning. Nonlinear Systems by Drazin. Elements of Applied Bifurcation Theory by Kuznetsov. Applications of Lie Groups to Differential Equations by Olver. Scaling by Barenblatt. Renormalization Methods: A Guide For Beginners by McComb. Multiple Time Scale Dynamics by Kuehn.
Measure, Integral and Probability by Capinski and Kopp, Integral, Measure and Derivative by Shilov and Gurevich and Hilbert Space Methods in Probability and Statistical Inference by Small and McLeish (see also Functional Analysis by Muscat). Probability via Expectation By Whittle. Functional Analysis for Probability and Stochastic Processes: An Introduction by Bobrowski. Trying to decide on my preferred abstract framework for thinking about these topics. Each presents a slightly different perspective, each has its strengths and weaknesses. Will have to write a ‘compare and contrast’ to help me decide. I’ve pretty well decided on the functional analysis point of view. Update: see also Differential Geometry and Statistics by Amari and Differential Geometry and Statistics by Murray and Rice. So basically: functional analysis + differential geometry seems to be the way to go. Same as for mechanics.
Related to the above, a few books (and a paper or two) on inverse problems, parameter estimation, Bayesian inference and numerical approximation. Data Assimilation: A Mathematical Introduction by Law, Stuart and Zygalakis. Inverse Problems: A Bayesian Perspective by Stuart. Mapping Of Probabilities by Tarantola (as well as his classic book Inverse Problem Theory). Statistical and Computational Inverse Problems by Kaipio and Somersalo. PTLoS by Jaynes (Ch. 18; I keep reinventing something similar to this but don’t quite understand it. I think it might correspond to reinventing the functional analysis approach?). Data Analysis and Approximate Models by Davies. Moore, Kearfott and Cloud Introduction to Interval Analysis. Measuring Statistical Evidence Using Relative Belief by Evans. Theoretical Numerical Analysis: A Functional Analysis Framework by Atkinson and Han. Moore and Cloud Computational Functional Analysis. Discrete and Continuous Boundary Problems by Atkinson. Fletcher Computational Galerkin Methods. Functional Data Analysis by Ramsay Silverman.
Teaching PDEs: Partial Differential Equations for Scientists and Engineers by Farlow. Applied Mathematics by Logan. Partial Differential Equations by Evans. Advanced Engineering Mathematics by Greenberg. Green’s functions and boundary value problems by Stakgold. Principles and Techniques of Applied Mathematics by Friedman. Partial Differential Equations of Applied Mathematics by Zauderer. A First Course in Continuum Mechanics by Gonzalez and Stuart. Physical Foundations of Continuum Mechanics By Murdoch. Nonlinear Partial Differential Equations by Debnath. Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods by Tang.Methods of Mathematical Physics II by Courant and Hilbert. Ames Nonlinear PDEs in Engineering.Ern and Guermond Theory and Practice of Finite Elements. Still need to find a book I really like that balances mathematical, numerical and physical concepts at the right level. The short article Generalized Solutions by Tao is nice.