One sharp matrix bound controls Schwarz-Pick at n points
For self-maps of the disk the factor is n; for degree-d Blaschke products it drops to min(n,d).
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Complex Variables
Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
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Verblunsky coefficients dictate bidisc core spectra
For homogeneous generators all core singular values are alpha_n; Mahler measure and a monotonicity counterexample follow.
· “Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules”
Vanishing disk-area integrals force f=0
Answers the area-integral problem: bounded solutions vanish for every α; smooth unbounded ones persist for α<1.
· “Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman”
Chebyshev polynomials on any analytic arc get exact large-degree limit
A 1969 prediction, revised by Christiansen–Simon–Zinchenko, is proven for every analytic Jordan arc.
Adjacent binomial samples interlace in a sharp 0.732 strip
Even entire functions keep the quotient pencil real-rooted up to this bound; one step beyond, degrees 3 and 5 fail.
· “Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples”
Reinhardt domains produce uncountable families with fixed Bergman curvature
For every m and n at least 2 an R-parameter family of mutually inequivalent domains has Bergman metric locally isometric to m times the Fub
· “Abundance of Bergman metrics with constant positive holomorphic sectional curvature”
Complete Crouzeix conjecture holds for every 3×3 matrix
Square-function inequalities settle the matrix-valued Crouzeix bound with the optimal constant 2 in dimension three.
· “Square Functions and the Complete Crouzeix Conjecture in Dimension Three”
Cœuré–Loeb fibration embeds in a Stein space
The classic non-Stein bundle is the complement of an analytic set, so it is not a counterexample to open immersion.
· “The C{oe}ur\'e-Loeb example as an open subset of a Stein space”
Equilibrium measure fixes a generic polynomial map
For most regular endomorphisms of C^k, equal measures force equal maps—up to iteration in C^2.
· “Measure rigidity for regular polynomial endomorphisms and applications”
J-equation breakdown set equals J-non-ample locus
Under J-bigness and a boundary cone condition, the degeneracy locus is a finite union of prime divisors and one current realizes it.
· “The J-equation on K\"ahler manifolds under a smooth boundary cone condition”
A singular continuity theorem ties alpha and delta invariants
The equality also pins the critical exponent for positively curved KE metrics to the normalized volume.
· “Demailly-Koll\'ar continuity on klt pairs, and applications to alpha and delta invariants”
Burkholder function is quasiconvex on symmetric matrices
Sharp weak-type and Lp bounds for the Beurling–Ahlfors transform follow from one integral inequality.
· “Quasiconvexity of the Burkholder function on symmetric matrices”
Orbit-invariant functions extend iff a block kernel is positive
A function constant on Blaschke orbits extends to H-infinity exactly when a block kernel is positive semidefinite.
· “Dynamic Nevanlinna-Pick Theory, Covariance Dilations, and Non-commutative Varieties”
The central charge is the only invariant of local conformal anomalies
It proves the same number governs conformal field theory anomalies and the Brownian loop measure.
This paper proves that a smooth Reinhardt domain in complex space with a complete…
A smooth Reinhardt domain with a well-defined complete Bergman-Einstein metric is biholomorphic to the complex unit ball; for n at least 2…
This paper proves that two specific ways of deforming a curved annulus in the hyperbolic…
Hyperbolic spiral and stretch maps are extremal quasiconformal maps on hyperbolic annuli, with constant distortion given by simple…
This paper proves that Newton's method
Newton iteration for the inverse Grötzsch modulus converges globally for every y > pi/2 and is strictly increasing for y > pi, resolving…
· “Global convergence and monotonicity of Newton iteration for the inverse Gr\"otzsch modulus”
This paper proves an effective Second Main Theorem for entire curves in the complement of…
Every algebraically nondegenerate entire curve in the complement of a general ordered pair of smooth transverse plane curves of admissible…
· “Invariant two-jets and effective hyperbolicity for complements of two plane curves”
This mathematics paper proves a sharp unique continuation result at infinity for the…
Any weak solution to ∂̄u=Vu on an exterior domain with bounded V that decays like e^{-C|z|} for C>2||V||∞ must vanish, and this threshold…
The paper proves that the volume of the largest centered ellipsoid fitting inside a…
For any pseudoconvex family of domains, the negative log of the maximal inscribed ellipsoid volume is plurisubharmonic, and no equivariant…
This math paper proves that if a complex algebraic variety carries a variation of Hodge…
For varieties supporting a variation of Hodge structure with an immersive period map, all exterior powers of the logarithmic cotangent…
· “Intermediate hyperbolicity of varieties supporting a variation of Hodge structure”
This paper proves that for a three-dimensional Kähler manifold with a J-semistable pair…
On any compact Kähler threefold, a J-semistable pair has a J-null locus that is a finite union of curves and divisors, and the J-flow…
Compression isotopy yields Runge embeddings and Loewner PDE answers
A stagewise compression condition yields global Runge embeddings and C^n-valued Loewner PDE solutions.
· “Runge embeddings, approximation of biholomorphisms on Stein manifolds, and the Loewner PDE”
Polyhedral surfaces get a discrete uniformization theorem
Every hyperbolic polyhedral surface maps to a standard model; Euclidean and spherical cases need one bound.
A third-order ODE now governs the extremal function in every dimension
Interpolation over the extremizer's zeros yields the equation and closes the even-dimension gap.
· “The H\"ormander--Bernhardsson function in higher dimensions”
All Stirling-Ramanujan constants are exponential periods
Explicit integrals now cover zeta derivatives, L-function values, and sphere determinants.
· “General Stirling-Ramanujan Constants are exponential periods and applications”
One elliptic quotient map integrates the exceptional foliations
Leaves are parametrized; symmetry groups, first integrals, and the positive-characteristic dichotomy are made explicit.
First nonhomogeneous Kähler–Einstein metrics embed into CH^∞
First complete nonhomogeneous Kähler–Einstein examples induced by CH^∞, with flat and Sasakian consequences.
· “Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains”
Elliptic points yield holomorphic discs with half-order smoothness
New blow-up method lifts the regularity of disc fillings and handles higher-index complex points.
· “The Bishop family of holomorphic discs: regularity and higher index”
DPS conjecture proven for compact Kähler spaces
Nef anti-canonical class on total space forces pseudo-effective anti-canonical class on base, with no projectivity.
· “On a conjecture of Demailly-Peternell-Schneider: the Kahler case”
Full n→∞ asymptotics found for Jacobi-Piñeiro polynomials
A 3×3 Riemann–Hilbert analysis gives explicit prefactors and O(1/n) uniform error everywhere, including near endpoints.
· “Strong asymptotics for Jacobi-Pi\~neiro orthogonal polynomials”
Defect cube roots converge for every finite-lower-order holomorphic curve
The exponent 1/3 is the sharp endpoint, and the bound holds uniformly for all hyperplanes in a countable general-position family.
· “Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves”
Boundary mass governs when disk iterates hit the sharp growth rate
A logarithmic integral of the Herglotz measure decides whether hyperbolic distance grows with bounded error.
· “Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization”
Explicit maps embed every exceptional model in C^3
One construction realizes all dimension-three models, completing the classification.
· “Classification of compact homogeneous strongly pseudoconvex hypersurfaces in mathbb C^n”
Rank-two symmetric quotients: π1 isomorphism forces biholomorphism
Holomorphic maps between finite-volume Kähler-Einstein quotients are biholomorphisms when the domain is symmetric of rank at least 2.
Conical Kähler–Einstein limits have unique tangent cones
A log version of Donaldson–Sun theory recovers the boundary divisor from the metric and pins down the tangent cone at every point.
For n≥4, positive Hessian need not yield open or discrete maps
A Pogorelov-type function collapses a line segment to a point with Hessian determinant staying positive.
· “A Pogorelov-type counterexample to the discreteness and openness of gradient mappings”
Commutator estimates hold exactly for p ≤ 2
For 1≤p≤2, φ(A,B) is controlled by input commutators; for p>2 even finite-rank counterexamples block any such bound.
· “Commutator estimates for functions of noncommuting self-adjoint operators”
One formula fixes Christoffel decay for deep-zero measures
When a measure dies off as e^{-h(|x|)}, the decay of its Christoffel function is now explicit.
· “Christoffel functions of measures on the real line with divergent logarithmic integral”
A Laguerre trick rescues deconvolution where ETFE divides by zero
A change of basis turns ill-conditioned frequency division into a well-conditioned circular convolution solved with one FFT.
· “Impulse Response Estimation via Laguerre-Fourier Expansion”
A single kernel sum decides which point sets are Dirichlet zero sets
The test replaces arc measure with extremal-function measures, making the old sufficient condition a special case.
· “Extremal functions and zero sets for the Dirichlet space”
Rigid escaping sets: Hénon automorphisms are iterates
Every global automorphism preserving the escaping set is, up to a fixed linear map, an iterate of the Hénon map.
· “On automorphisms of super-level sets of Green's functions of H\'{e}non maps”
A stretched-domain exhaustion ends in the domain or a disc
Bounded distortion forces a dichotomy: the limit is the original domain, or a disc (ball).
Exact formula found for angle between S^2's Hardy quantizations
The overlap is a Legendre ratio that yields the unitary intertwiner and exact trace series.
· “Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere”
Sharp cutoffs found for two regularity problems on tent spaces
Fractional integration and randomized Taylor coefficients: complete characterizations on analytic tent spaces.
Ball masses decide Schatten class on pluriharmonic Fock space
A measure's fixed-radius ball masses in L^p characterize S_p membership for all 0<p<∞, with sharp norm bounds.
Theta_n varieties are determinantal complete intersections
For pure Theta_n-contractions the same matrices give a dilation, model, and boundary inequality.
Multivalued Hardy-Rogers maps gain fixed points in b-metric spaces
Distance-dependent coefficients and one of three mild conditions guarantee a fixed point, even for higher iterates.
· “Fixed Point Theorems for Hardy--Rogers Type Multivalued Mappings in b-Metric Spaces”
Bidisk submodules z^k−w^k repeat their invariants in k-blocks
Each invariant of [z-w] is repeated k times in a row, making k readable from the sequence.
· “Block Repetition of Numerical Invariants for the Submodules [z^k-w^k] in H²(mathbb D²)”
Exactly 18 two-word patterns are the minimal convergence criteria
The same framework yields two Gaussian-integer rules and an infinite set that beats every finite one.
· “Optimal local convergence criteria for integer and Gaussian integer continued fractions”
A diagonal kernel restores the ξ-Bergman kernel's missing structure
Built from Riesz representatives, the new kernel is positive definite, real analytic, and equals K_ξ on the diagonal.
· “A Canonical Positive Definite Kernel Associated with the xi-Bergman Kernel”
Hayman–Wu constant is π²
The bound is sharp: no conformal map can push a line's preimage longer than π², and examples approach it.
A partial isometry can have an off-center circular numerical range
A rank-four counterexample in six dimensions breaks the conjecture that such discs must be centered at zero.
· “A Counterexample to the Gau--Wang--Wu Conjecture on Partial Isometries”
Complex automorphisms approximate every R^n diffeomorphism
Complex automorphisms can approximate every real diffeomorphism of R^n; volume-preserving versions follow too.
· “Holomorphic Approximation for Real Diffeomorphism Groups”
Slice monogenic Corona solved in every Clifford dimension
Matrix Corona plus a symmetry average yields bounded slice solutions for any n≥2.
· “The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem”
Positive powers keep operator-orbit frames intact
If a positive operator's iterates frame the space, every p-th iterate still does—so these frames have infinite redundancy.
· “Power Stability of Finitely Generated Dynamical Frames for Positive Operators”
No Fermat triples in capacity-zero model spaces
When the boundary spectrum has capacity zero, every n≥4 Fermat triple is one function scaled.
· “Fermat's Last Theorem in star-invariant subspaces: capacity-zero boundary spectrum”
One identity settles Sommen's bivector bills
The gap between cell counts and homology is boundary data; tracking it explains both the n=4 and n=5 overcounts.
Singular fibers can hide from the topology of a smooth variety
A 5-dimensional construction answers four open questions about when singular fibers must be visible.
Derivative lower bound survives in quaternionic cubics
For roots inside radius $k \ge 2^{1/3}$, the derivative norm is at least $3/(1+k^3)$ times the polynomial norm.
· “On Turan Type inequality for Quaternionic Canonical Generalized Polynomials”
Every strip summation formula comes from a Nevanlinna-type function
The classification now covers Guinand–Weil formulas for the Selberg class, via log-derivative generating functions.
· “Classification of Fourier summation formulas on a horizontal strip”
Irregular points decide when Chebyshev norms stay bounded
For semi-regular sets, boundedness is equivalent to a Green-function sum over irregular points being finite.
· “Widom factors for Chebyshev and residual polynomials on semi-regular subsets of mathbb{R}”
A parameter λ redraws sharp Bohr radii
New best-possible radii follow from inserting λ into Bohr inequalities, from 1/3 down to 1/√(2λ+1).
· “Bohr type inequalities with one parameter for the class of self analytic maps on unit disc”
Vanishing Berezin transform misses noncompact L1 Toeplitz operators
For L1 symbols, boundary decay of the Berezin transform no longer certifies compactness; the disk endpoint p=3 is sharp.
On Hopf manifolds, nef bundles escape semi-positive metrics
Nonvanishing Ueda classes block smooth metrics; all positivity cones are now classified.
· “Nef but Non-Semi-positive Line Bundles on Hopf Manifolds”
Vector-valued resurgence governs false theta functions
Eichler integrals of weight 1/2 and 3/2 forms are modular resurgent; Stokes data follow the S-matrix.
Square-zero q-Dirac pairs now cover all Clifford-valued coordinates
The stable range delivers full-coordinate square-zero q-Hermitian families with exact radial restriction and boundary trace $[2m-2n]_q$.
Endpoint counterexamples show Beurling–Carleson support is not enough
At the sharp exponents, one measure on a single Beurling–Carleson set defeats both regularity and deficiency classes.
· “Beurling--Carleson Endpoint Counterexamples: Singular Inner Functions and a Semilinear Equation”
Finite Gram reduction makes multiplier submodule sheaves coherent
Higher-rank L2 integrability reduces to scalar multiplier ideals, giving coherence and jumping discreteness.
· “Finite Gram Scalarization and Further Properties of Multiplier Submodule Sheaves”
No regularity needed: all diamond isometries are holomorphic
Preserving the Kobayashi distance already forces C^1 smoothness, so the old regularity assumption was redundant.
Generalized Toeplitz operators reduce to block Toeplitz form
Equivalence after extension transfers kernels, invertibility, and Fredholm data from classical operators to generalized ones.
· “Equivalence after extension for general Toeplitz operators”
One condition keeps zeta singularities on the negative axis
Under one stated condition, the continuation stays holomorphic except on the negative real axis.
· “Analytic continuation of local zeta functions to a slit complex plane: the one-dimensional case”
Bloch radius 3(1-k-M)^2/128 proven for harmonic maps
A new term in the derivative equation shrinks the guaranteed schlicht disk, but does not destroy it.
· “Bloch and Landau Type Theorems for Harmonic Mappings with Inhomogeneous Analytic Dilatation”
Octonionic Hessian complex proved elliptic
Moufang identities tame non-associativity, giving the octonionic ∂∂̄-lemma and Hartogs–Bochner extension.
Exact higher-Schwarzian bounds for lune and bean map classes
Lune-starlike, lune-convex, and bean maps get sharp bounds, each attained by an explicit extremal function.
· “Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains”
Closed leaves force a first integral on tree-shaped singularities
A smooth-space integrability criterion now works on singular surfaces, and formal integrals become holomorphic.
Exact formulas solve the damped wave equation on a quarter-plane
Fokas transform yields closed-form solutions, with corner conditions that make them smooth to the boundary.
16/27 and 71/324: sharp Hankel bounds for cardioid starlike maps
Paper also pins the third Hermitian–Toeplitz determinant to the interval [-841/2295, 1].
Solution map for degenerate Beltrami equations is locally C^{1,1}
Even at the edge of ellipticity, normalized Beltrami solutions vary differentiably with the coefficient.
· “An Orlicz variational formula for David-type Beltrami equations”
Two fields, not two numbers, classify vortex-free equations
Vortex-free cases need a density plus a current modulo Möbius maps; equal mass and charge can still differ.
· “Moduli of the Sourceless Framed Beltrami-Vekua Normal Form”