## Mid-Term Report Summary - MASTRUMAT (The Mathematics of the Structure of Matter)

The goal of the project "The Mathematics of the Structure of Matter" is to understand the structure of matter from a mathematical analysis of the underlying quantum theory. The system range from ordinary matter described by atoms and molecules to exotic forms of matter e.g., systems exhibiting superconductivity or superfluidity.

In the first period of the project progress has been made in several directions.

The celebrated 1947 theory of superfluidity of Bogolubov has been formulated as an exact mathematical variational model and analyzed in detail. This work is not directly concerned with the accuracy of the approximation, but studies the Bogolubov Theory itself. Among the many results achieved is the mathematical fact that the model, indeed, is well-posed and has solutions. Of more physical interest it is shown that the model exhibits a phase transition between a condensed and a non-condensed phase and the exact asymptotic behavior of the critical temperature in the dilute limit has been derived. It agrees remarkably well with published numerical calculations for the Bose gas performed in the full quantum many-body context. Closely related to this work is the derivation of a new optimal characterization of the applicability of the Bogolubov diagonalization method in a general mathematical setting.

For the corresponding theory for superconductivity of Bardeen, Cooper and Schrieffer the critical temperature for the transition between a superconducting and normal state has been analyzed. It has been shown that in an appropriate asymptotic limit the critical temperature is correctly described by the Ginzburg-Landau model.

In the study of interacting many-body quantum systems the project has led to the development of a new general technique, in terms of Lieb-Thirring inequalities, for estimating the thermodynamic pressure of a variety of systems including Bose gases, Fermi gases and gases of particles with more exotic statistics such as anyons.

A different direction of research concerns the interaction of matter with electromagnetic fields. In this context work has been done in relating the structure of matter to the geometry of an external magnetic field. In particular, it has been shown how linked field lines will lead to the occurrence of particular zero-energy bound states.

This is of interest both to physics because of the applications it may have to the stability of matter and to pure mathematics because it gives a new exact formula relating the geometry of the magnetic field (in terms of linking number) and the spectrum of the corresponding quantum operator.

A very fundamental and very challenging problem is to understand the relation between quantum and classical physics. In this relation the project has achieved the solution to an old conjecture on understanding the classical entropy of quantum states. As a byproduct it has been possible to calculate the minimal output entropy of a family of important quantum communication channels, i.e., the universal quantum cloning channels.

Finally, work has been done in understanding the size of atoms. Using a very simple idea based on the famous Thomas-Fermi Theory it has been possible to give estimates on the size of Alkali atoms which is in good agreement with empirical values.

The ideas are based on earlier mathematical work of the project PI.

In the first period of the project progress has been made in several directions.

The celebrated 1947 theory of superfluidity of Bogolubov has been formulated as an exact mathematical variational model and analyzed in detail. This work is not directly concerned with the accuracy of the approximation, but studies the Bogolubov Theory itself. Among the many results achieved is the mathematical fact that the model, indeed, is well-posed and has solutions. Of more physical interest it is shown that the model exhibits a phase transition between a condensed and a non-condensed phase and the exact asymptotic behavior of the critical temperature in the dilute limit has been derived. It agrees remarkably well with published numerical calculations for the Bose gas performed in the full quantum many-body context. Closely related to this work is the derivation of a new optimal characterization of the applicability of the Bogolubov diagonalization method in a general mathematical setting.

For the corresponding theory for superconductivity of Bardeen, Cooper and Schrieffer the critical temperature for the transition between a superconducting and normal state has been analyzed. It has been shown that in an appropriate asymptotic limit the critical temperature is correctly described by the Ginzburg-Landau model.

In the study of interacting many-body quantum systems the project has led to the development of a new general technique, in terms of Lieb-Thirring inequalities, for estimating the thermodynamic pressure of a variety of systems including Bose gases, Fermi gases and gases of particles with more exotic statistics such as anyons.

A different direction of research concerns the interaction of matter with electromagnetic fields. In this context work has been done in relating the structure of matter to the geometry of an external magnetic field. In particular, it has been shown how linked field lines will lead to the occurrence of particular zero-energy bound states.

This is of interest both to physics because of the applications it may have to the stability of matter and to pure mathematics because it gives a new exact formula relating the geometry of the magnetic field (in terms of linking number) and the spectrum of the corresponding quantum operator.

A very fundamental and very challenging problem is to understand the relation between quantum and classical physics. In this relation the project has achieved the solution to an old conjecture on understanding the classical entropy of quantum states. As a byproduct it has been possible to calculate the minimal output entropy of a family of important quantum communication channels, i.e., the universal quantum cloning channels.

Finally, work has been done in understanding the size of atoms. Using a very simple idea based on the famous Thomas-Fermi Theory it has been possible to give estimates on the size of Alkali atoms which is in good agreement with empirical values.

The ideas are based on earlier mathematical work of the project PI.