# Pure and Applied Analysis, and Mathematical Physics

## Research topics

**Pure analysis**

- Functional analysis
- Operator theory
- Real Analysis
- Several complex variables and analytic spaces

**Applied analysis**

- Calculus of variations and optimal control; optimization
- Difference and functional equations
- Linear and multi-linear algebra; matrix theory
- Numerical analysis
- Ordinary differential equations
- Operator theory
- Partial differential equations
- Probability theory and stochastic processes

**Mathematical physics**

- Differential geometry
- Functional analysis
- Global analysis; analysis on manifolds
- Information and communication circuits
- Mechanics of particles and systems
- Quantum theory

## Related website

- Institute for Quantum Science and Technology (IQST)

## Research Spotlight: Dr. Carlo Maria Scandolo

My research interests are in the area of quantum information theory, which is a branch of mathematical physics. Quantum information is a very successful research area, at the intersection of math, physics, and computer science, which studies the quantum world (the microscopic world of atoms and molecules) from the angle of information theory. Quantum information results have produced a considerable scientific and technological revolution, with quantum technologies that are becoming concrete and commercially available (e.g. quantum communication, cryptography, computing, etc.). The reason for such a revolution stems from the fact that the quantum world provides us with new *resources* allowing us to do things that were unthinkable before. My research in quantum information theory branches in two directions: one aims at exploring the role and quantification of quantum resources, the other at answering the deepest and most fundamental questions of Nature (e.g. why the world is quantum) with mathematical rigour.

More specifically, some of my current research projects include:

**Understanding the value of quantum channels as resources.**Quantum channels represent ways to transmit quantum information in space and time. As such, they are ubiquitous in quantum information and essential for several protocols, e.g. for implementations of quantum communication and quantum computing. In particular, I want to understand how known quantum resources, such as quantum entanglement, interplay with quantum channels, and how channels themselves can be considered as resources. One of the goals in this respect is to quantify how much resource (e.g. entanglement) is consumed or generated by a quantum channel. This analysis is based on the mathematical framework of*resource theories*, which describes all physical situations subject to external constraints or limitations.

**Thermodynamics in the quantum regime**. In many thermodynamic situations we are interested only in knowing if a specific thermodynamic transformation is possible between different thermodynamic states. Resource theories are again extremely useful in this respect, as they can be used to explain thermodynamics in the quantum regime. This has become one of the hottest quantum research areas, with far-reaching applications to nanotechnologies. Now we can take a step further, and study how different thermodynamic transformations can be transformed between each other. This entails treating channels implementing thermodynamic transformations as resources, a situation that can be fully captured by the framework of resource theories for quantum channels.

**General probabilistic theories and quantum foundations.**Some fundamental questions about Nature can be answered in full generality by phrasing them in the mathematical framework of*general probabilistic theories*(GPTs), which is built from the marriage of probability theory with category theory. GPTs can be used to study the structural properties of all conceivable physical theories admitting probabilistic processes. I am particularly interested in analyzing the role and consequences of the fundamental principle of Causality, which stipulates that information propagates from the past to the future. The role of Causality is crucial in mathematical physics, as modifications of the principle of Causality have been proposed in the quest for a theory of quantum gravity. One of the aspects I am particularly interested in are the implications of Causality for the emergence of our everyday world, which is classical and objective, out of a more fundamental physical theory.

I am always looking for outstanding and highly motivated students and postdocs who have similar research interests, to explore these fascinating research topics (and even more) together! Check my departmental page for contact information.