Stochastic thermodynamics
We learn various thermodynamic processes in freshman thermodynamics courses. In order to be able to calculate the work done, heat dissipated, entropy generated or the temperature change during a process we need two key assumptions: the system is composed of extremely large number of particles (thermodynamic limit) and that the process occur extremely slowly (quasi-static limit).
However, most real life processes are not quasi-static. Small-scale engines occur in our body. Bio-molecules like Kinesin and Dyenin transport materials and tiny organelles use ATP molecules produce energy in our cells. These systems are far from satisfying the above assumptions. The work, entropy or heat generated during these processes are not definite numbers, but fluctuating quantities that depend on the initial configuration of the system and the exact trajectory of the system. The second law of thermodynamics is not a fundamental law but an emergent property and can be very easily violated by small systems.
Theoretical investigations of such systems have led to important theorems such as the Jarzynski equality, Crooks fluctuation theorem, Evans-Searles fluctuation theorem and the Gallavotti-Cohen fluctuation theorem.
Read some reviews here, here, or here.
Quantum thermodynamics and quantum statistics
How does quantum mechanics describe familiar concepts like temperature, pressure, work and heat? We can clearly distinguish between work and heat generated by an engine or a generator. How can we make such a distinction in a quantum system evolving with time? How do the fluctuation theorems above extend to quantum systems? Read this excellent review.
Thermodynamics at the nanoscale
Current flows through a conductor connecting two objects at different electric potentials. Heat flows through a conductor connecting two objects at different temperatures. However, there is an important difference: all charge carriers are driven due to the electric field in the conductor, while heat flow is a stochastic process. Heat flows in both directions even when the temperatures of the two objects are equal, leading to small fluctuations in temperature. At the nano-scale, such fluctuations play an important role in determining heat flows. Do macroscopic laws like the Fourier’s law of conduction or Joule’s law of heating hold at the nano-scale? New technologies for constructing molecular junctions and nano-structures have allowed us to investigate such questions. Read more in this colloquium.
Theory of properties of solids
Did you know that no more than two electrons in any piece of solid have the same momentum? (Since all electrons in the universe interact with each other no matter how far apart they are, no more than two electrons in the universe have the same momentum state.)
Electrons, phonons (quantized vibrations in the crystalline solid) and their interactions explain almost all properties of solids: specific heat, electrical and thermal conductivity, color and other optical properties and even superconductivity.
However, unanswered questions remain, especially in the theory of amorphous solids. For instance, low temperature specific heats of amorphous solids do not agree with theoretical prediction and mysterious tunneling two level systems need to be introduced to explain it. (This is an old review on this topic, you can find many more recent developments.) Solids such as some Cuprate compounds show high temperature superconductivity, which is one of the biggest unsolved problems in condensed matter theory.
Theory of phenomena in semiconductors and their applications
Optical and electrical properties of semiconductors can be varied easily, for example by introducing dopants at different concentrations. They are used to convert light into a voltage difference (photo-voltaics and light detectors), to convert current into light (LED and semiconductor lasers) or as photo-resistors (alter resistance depending on light intensity e.g. Cadmium Sulfide).
There is an enormous effort to increase efficiency of solar cells, by improving light absorption, efficiency of carrier transport and collection of charges. Surface of silicon can be nano-patterned to absorb more light (black silicon) or different kinds of solar cells can be placed above one another (tandem solar cells).
Methods of statistical physics such as the Boltzmann equation and non-equilibrium Green's functions are indispensable in predicting the kinetic behavior of semiconductors. (This is an excellent resource.)
Differential geometry: Manifolds, dynamical systems on manifolds, Hamiltonian dynamics on manifolds. Lie groups and Lie algebras.
Analysis
Real analysis: metric spaces, measure theory, function spaces
Functional analysis: Topological vector spaces, theory of distributions, Fourier analysis, Hilbert spaces, operators on Hilbert spaces, spectral theorem and functional calculi.
Operator theory: Operator algebras- C*-algebras and von Neumann algebras. (learning)
Foundations of relativistic statistical mechanics: Dynamical systems described by cotangent bundles on manifolds.
Foundations of quantum mechanics: Rigorous mathematical formulation of quantum mechanics using functional analysis and operator theory.
Mathematics of quantum fields: Axiomatic quantum field theory, Wightman axioms, etc. (learning)
Foundations of statistical mechanics: Nonlinear dynamics and chaos, ergodic theory, spin-statistics theorem. (learning)
Networks
Cascades on networks, voter models, spreading of ideas, opinions and innovations
Dynamics of social networks
Inference problems
Bayesian inference, inferring networks
Unity of entropy
Jaynes' principle and maximum entropy principle
Maximum caliber principle