Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>

Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

A Note on Real Operator Monotone Functions

In this paper, we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

Fractal simplices

There are three constructions of which I know that yield higher dimensional analogues of Sierpinski’s triangle. The most obvious is to remove the open convex hull of the midpoints of the edges of the [Formula: see text]-simplex. The complement is a union of simplices. Continue the removal recursively in each of the remaining sub-simplices. The result is an uncountably infinite figure in [Formula: see text]-dimensional space that is Cantor-like in a manner analogous to the Sierpinski triangle. A countable analogue is obtained by means of playing the chaos game in the [Formula: see text]-simplex. In this “game” one chooses a random [Formula: see text]-ary sequence; starting from the initial point (that is identified with a vertex of the simplex), one continues to plot points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the figure described above. Similarly, coloring the multinomial coefficients black or white according to their parity results in a similar figure, when the [Formula: see text]-dimensional analogue of the Pascal triangle is rescaled and embedded in space.