Inequalities and limits of weighted spectral geometric mean

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between $left(B^{ts/2}A^{(1-t)s}B^{ts/2} right)^{1/s}$ and the $t$-spectral mean $A atural_t B :=(A^{-1}sharp B)^{t}A(A^{-1}sharp B)^{t}$ of two positive semidefinite matrices $A$ and $B$, where $Asharp B$ is the geometric mean, and the $t$-spectral mean is the dominant one. The limit involving $t$-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature.

Large $ Y_{k,b} $-tilings and Hamilton $ ell $-cycles in $k$-uniform hypergraphs

Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $binom{n}{3} - binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$ vertex-disjoint copies of $Y_{3,2}$, for $mle n/7$. The bound on the number of edges is asymptotically best possible. This can be viewed as a generalization of the ErdH{o}s Matching Conjecture.We then use this result together with the absorbing method to determine the asymptotically best possible minimum $(k-3)$-degree threshold for $ell$-Hamiltonicity in $k$-graphs, where $kge 7$ is odd and $ell=(k-1)/2$. Moreover, we give related results on $ Y_{k,b} $-tilings and Hamilton $ ell $-cycles with $ d $-degree for some other $ k,ell,d $.

Curvature of matrix and reductive Lie groups

In this paper, we give a simple formula for sectional curvatures on the general linear group, which is also valid for many other matrix groups. Similar formula is given for a reductive Lie group. We also discuss the relation between commuting matrices and zero sectional curvature.

On two geometric means and sum of adjoint orbits

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Ptak, originally for positive definite matrices. The relation between $t$-metric geometric mean and $t$-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices $X$ and $Y$, Sos matrix exponential formula asserts that there are unitary matrices $U$ and $V$ such that $$e^{X/2}e^Ye^{X/2} = e^{UXU^*+VYV^*}.$$ In other words, the Hermitian matrix $log (e^{X/2}e^Ye^{X/2})$ lies in the sum of the unitary orbits of $X$ and $Y$. Sos result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.

Hamiltonicity in Cherry-quasirandom 3-graphs

We show that for any fixed $alpha>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $alpha binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and Levy.