For ($n\times n$)-matrices $A$ and $B$, define ${\displaystyle \eta (A,B)=\sum _S\det (A[S])\det (B[S^{\text{'}}]),}$ where the summation is over all subsets of $\{1,\dots ,n\}$, $S^{\text{'}}$ is the complement of $S$, and $A[S]$ is the principal submatrix of $A$ with rows and columns indexed by $S$. We prove that if $A\ge 0$ and $B$ is Hermitian, then

(1) the polynomial $\eta (\mathrm{zA},-B)$ has all real roots;

(2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of $B$ if $A>0$; and

(3) for $1\le i\le n$, the roots of $\eta (\mathrm{zA}[\{i{\}}^{\text{'}}],-B[\{i{\}}^{\text{'}}])$ interlace those of $\eta (\mathrm{zA},-B)$.

Assertions (1)–(3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard, Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants

Let $F$ be a totally real number field, and let $\mathfrak{p}$ be a finite prime of $F$ such that $\mathfrak{p}$ splits completely in the finite abelian extension $H$ of $F$. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a $\mathfrak{p}$-unit $u$ in $H$ with absolute values at the places above $\mathfrak{p}$ specified in terms of the values at zero of the partial zeta functions associated to $H/F$. This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of $u$ in $F_{\mathfrak{p}}^{\times }/\stackrel{\wedge }{E}$, where $F_{\mathfrak{p}}$ denotes the completion of $F$ at $\mathfrak{p}$ and $\stackrel{\wedge }{E}$ denotes the topological closure of the group of totally positive units $E$ of $F$. We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of $u$ in $F_{\mathfrak{p}}^{\times }$

We prove Bourgain's return times theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost-everywhere convergence for the signed average analogue of Bourgain's averages

By counting triangles and applying the octahedral axiom, we find a direct way to prove the formula of Toën in [T, Section 5.1] for a triangulated category satisfying the (left) homological finiteness condition

This is a continuation of [N2], where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution.

The purpose of this article is to describe the remainder. We first construct a deformation of the nilpotent orbit closure in a canonical manner, according to Brieskorn and Slodowy (see [S]), and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers.

Moreover, by using this construction, one can generalize the main result of [N2] to arbitrary Richardson orbits whose Springer maps have degree greater than $1$. New Mukai flops, different from those of types $A$, $D$, and $E_6$, appear in the birational geometry for such orbits