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

This ERC project has made significant progress on four different topics of the mathematical descip tion of the structure of matter.

The first topic has been models of superconductivity and superfluidity.The famous Bogolubov ap proximation of superfluidity for bosponic systems has been reformulated as a variational theory and this reformulation has been analyzed for its general mathematical properties such as existence and uniqueness/non-uniqueness of minimizers as well as general properties of the minimizers.

These properties which may be rigorously established in the Bogolubov variational model agree well with what is expected but not rigorously established in a full many-body formulation. In particular, it is possible to establish Bose-Einstein condensation in the Bogolubov model, something which is still a challenging mathematical problem in the many-body formulation of translation invariant systems in the continuum. In the dilute limit the Bogolubov variational model gives the correct leading order be havior of the free energy (something which, in fact, fails for the original Bogolubov approximation). The next to leading order term in the Bogolubov model is different from what is expected in many- body theory but is correct in the limit of soft potentials.

Finally, in the dilute limit the Bogolubov model gives a two term asymptotic for the critical temperat ure which agrees well with numerical calculations. In two dimensions the Bogolubov model also pre dicts Bose-Einstein condensation, although this is known, from the Mermin-Wagner Theorem, not to be the case in the many-body formulation. If we however interpret condensation in the sense of the existence of a quasi-condensate the critical temperature as calculated in the Bogolubov model in the dilute limit agrees well with what is expected for the Berezinskii1Kosterlitz1Thouless transition temperature.The validity of the Bogolubov model as an approximation to many-body theory in the di lute limit was also studied in the project and established for soft potentials.

For models of superconductivity the project established that the transition temperature calculated in the Bardeen-Cooper-schrieffer model agrees in the appropriate limit to what is found in the Gin zburg-Landau model, which for fermions is the analogue of the Bogolubov model for bosons.

The second topic of the project studied the possible zero modes of charged relativistic particles inter acting with magnetic fields. These zero modes are of utmost importance in the analysis of stability of charged particles. In particular, the project studied singular magnetic fields similar to the celebrated Aharonov-Bohm solenoids but now supported on knots and links. It was shown that an appropriately defined spectral flow could be calculated explicitly from geometric and topological data of the links. In fact, the spectral flow depends on the writhes of the knots and the linking numbers of the links.

The spectral flow counts (with multiplicity) the number of eigenvalues crossing zero under certain variations of the field. In this sense it indirectly implies the existence of zero modes.

The third topic of the project was to understand the Born-Oppenheimer energy curves of diatomic molecules. The PI conjectured that in the limit of infinitely large atoms the Born-Oppenheimer curve would converge to a curve with a short range asymptotics given by Thomas-Fermi theory.This was subsequently established in a Hartree-Fock type model. Numerical calculations show amazing agree ment even for usual size diatomic molecules.

Finally, the last project dealt with analyzing the minimal output entropy of certain quantum channels such as the universal quantum cloning channels. Establishing the minimal output entropy and analyz ing it in the semi-classical limit solved a long standing conjecture for the classical entropy of quantum states.

The first topic has been models of superconductivity and superfluidity.The famous Bogolubov ap proximation of superfluidity for bosponic systems has been reformulated as a variational theory and this reformulation has been analyzed for its general mathematical properties such as existence and uniqueness/non-uniqueness of minimizers as well as general properties of the minimizers.

These properties which may be rigorously established in the Bogolubov variational model agree well with what is expected but not rigorously established in a full many-body formulation. In particular, it is possible to establish Bose-Einstein condensation in the Bogolubov model, something which is still a challenging mathematical problem in the many-body formulation of translation invariant systems in the continuum. In the dilute limit the Bogolubov variational model gives the correct leading order be havior of the free energy (something which, in fact, fails for the original Bogolubov approximation). The next to leading order term in the Bogolubov model is different from what is expected in many- body theory but is correct in the limit of soft potentials.

Finally, in the dilute limit the Bogolubov model gives a two term asymptotic for the critical temperat ure which agrees well with numerical calculations. In two dimensions the Bogolubov model also pre dicts Bose-Einstein condensation, although this is known, from the Mermin-Wagner Theorem, not to be the case in the many-body formulation. If we however interpret condensation in the sense of the existence of a quasi-condensate the critical temperature as calculated in the Bogolubov model in the dilute limit agrees well with what is expected for the Berezinskii1Kosterlitz1Thouless transition temperature.The validity of the Bogolubov model as an approximation to many-body theory in the di lute limit was also studied in the project and established for soft potentials.

For models of superconductivity the project established that the transition temperature calculated in the Bardeen-Cooper-schrieffer model agrees in the appropriate limit to what is found in the Gin zburg-Landau model, which for fermions is the analogue of the Bogolubov model for bosons.

The second topic of the project studied the possible zero modes of charged relativistic particles inter acting with magnetic fields. These zero modes are of utmost importance in the analysis of stability of charged particles. In particular, the project studied singular magnetic fields similar to the celebrated Aharonov-Bohm solenoids but now supported on knots and links. It was shown that an appropriately defined spectral flow could be calculated explicitly from geometric and topological data of the links. In fact, the spectral flow depends on the writhes of the knots and the linking numbers of the links.

The spectral flow counts (with multiplicity) the number of eigenvalues crossing zero under certain variations of the field. In this sense it indirectly implies the existence of zero modes.

The third topic of the project was to understand the Born-Oppenheimer energy curves of diatomic molecules. The PI conjectured that in the limit of infinitely large atoms the Born-Oppenheimer curve would converge to a curve with a short range asymptotics given by Thomas-Fermi theory.This was subsequently established in a Hartree-Fock type model. Numerical calculations show amazing agree ment even for usual size diatomic molecules.

Finally, the last project dealt with analyzing the minimal output entropy of certain quantum channels such as the universal quantum cloning channels. Establishing the minimal output entropy and analyz ing it in the semi-classical limit solved a long standing conjecture for the classical entropy of quantum states.