Differential Geometry Seminar (2026)

Geometric Means that preserve sparsity

This talk starts from a simple question: how can one define a geometric mean for sparse positive definite matrices without destroying their zero pattern?
For the arrowhead pattern, this leads naturally to the five-dimensional Vinberg cone, a basic non-symmetric homogeneous cone.
I will present two intrinsic means on this cone: a Cholesky-Vinberg mean built from triangular Cholesky factors, and a Log-Vinberg mean built from global clan (Vinberg algebra) coordinates.
The first is tied to a flat affine geometry with torsion, while the second belongs to a torsion-free flat geometry. I will also explain why these means differ from the classical Riemannian midpoint and why this difference is a genuinely non-symmetric phenomenon.
I will conclude my presentation by an application to quantum information theory.

Relaxation to an Ideal Chern Band through Coupling to a Markovian Bath

In condensed matter physics, the problem of electrons hopping on a two-dimensional lattice naturally leads to the notion of a Chern band. Mathematically, a Chern band is described by a smooth map from the two-torus---which labels the crystalline momenta which, mathematically, are the unitary characters of the lattice---to the Grassmannian of $r$-dimensional subspaces in $\mathbb{C}^N$. The geometry of these maps plays an important role in physical responses of materials and in properties of interactions, especially if the band has no energy dispersion. In particular, the so-called ideal bands are quite remarkable from the physical point of view as they reproduce physical properties of electrons under a uniform magnetic field. Ideal bands---also called K\"ahler bands---correspond to the case in which the map is holomorphic. Wirtinger's inequality in K\"ahler geometry allows one to test whether the band is ideal or not. In this talk, I will propose a microscopic mechanism by which generic Chern bands relax toward ideal bands. We consider coupling interacting electrons to a Caldeira-Leggett like Ohmic bosonic bath. Under the Born-Markov and Hartree-Fock approximations, Slater determinant states of a Chern band evolve toward Slater determinant states corresponding to an ideal Chern band. The effective dynamics corresponds to a metriplectic flow in the space of maps from the 2-torus to the Grassmannian, which is closely related to the harmonic map heat-flow. Our proposal is tested by performing numerical simulation of a massive Dirac model, showing how Wirtinger's inequality is approached in the long-time limit. Our proposal provides a concrete dissipative route to realize ideal Chern bands.

Japanese page only

Japanese page only