I’m collecting my mathematical research here, with an explanation of each result, its current status, and links to the paper and supporting materials.
I use AI extensively in this work. Unless otherwise stated, these are preprints that have not undergone independent human peer review.
Papers
-
Free spanning-forest degree can decrease under squaring
Adding every two-step connection takes an infinite graph from 8 neighbors per vertex to 40. Yet the expected number of edges touching a vertex in its free random spanning forest decreases. The paper gives an exact calculation and a checkable certificate.
- A continuous free product system not generated by its tensor core
A continuous free product system can contain sectors that its maximal tensor subsystem cannot generate.
- Logarithmic Bray sampling in symplectic and orthogonal groups
Logarithmic sample bounds for Bray’s construction in large symplectic and orthogonal groups over odd finite fields.
- Missing faces and topological embeddings of sphere skeleta
Adding a missing middle-dimensional face to the skeleton of a triangulated sphere prevents embedding it back into that sphere.
- Positive tangent-bundle Seshadri constants characterize projective space
A positive tangent-bundle Seshadri constant at one fixed point forces a smooth projective variety to be projective space, in every characteristic.
- An explicit odd-order p-group with as many automorphisms as elements
An explicit odd-order p-group with exactly as many automorphisms as elements: a computer-assisted proposed solution to Kourovka Notebook problem 16.63.
- Direct Sequence Laws under the Network Multispecies Coalescent: A genealogy equivalence, canonical coalescent-response behavior, and finite determination with an epoch-complexity obstruction
What multilocus DNA sequence laws reveal about gene genealogies, and why demographic complexity limits finite-locus identification under the NMSC.
- Minimum Bell-Setting Complexity for Qubit POVM–PVM Separation
Why qubit POVMs need at least three Bell settings on one side and two on the other to beat projective measurements, when shared randomness is allowed.
- Positive Recurrence for Single-Linkage Bimolecular Weakly Reversible Stochastic Reaction Networks
Positive recurrence for finite weakly reversible bimolecular stochastic reaction networks with one linkage class, without the pure-species condition.
- Raw Strongly Tree-Child Level-2 Species Networks from Metric Genealogy Laws under the NMSC: Signed pair co-occupancy, rooted graph reconstruction, and demographic fibers
Which species-network histories can exact genealogy laws distinguish? A level-2 NMSC classification identifies exposed parameters and unavoidable ambiguities.
- A polynomial versus superpolynomial gap in vertex-distribution-free graph testing
A fixed graph property has polynomial testing cost under uniform vertex sampling but superpolynomial cost with arbitrary vertex weights.
- A smooth Reshetnyak-harmonic sphere with unbounded distortion
A smooth sphere homeomorphism locally minimizes Reshetnyak energy while its infinitesimal distortion has no finite bound.
- Coarse endpoint concentration at every diverging scale for one-dimensional space-time random walks
One-dimensional random walks in iid space-time environments have sharply concentrated coarse endpoint laws at every deterministic diverging interval scale.
- Finite-energy lattice percolation with internal critical probability one
An invariant finite-energy percolation on every lattice Z^d, d ≥ 2, has a unique infinite cluster destroyed by every independent thinning below full retention.
- Polynomial-time optimization and exact separation for rational mixed-integer bipartite covers
Exact rational optimization and convex-hull membership for mixed-integer bipartite covers, with polynomial binary-input complexity and unrestricted signs.
- Post-Lie nonexistence with a semisimple target
A unimodular-source obstruction for finite-dimensional complex post-Lie algebras, proving nonexistence for perfect nonsemisimple sources and semisimple targets.
- Property (T) above one-third density in random free-product quotients
Kazhdan property (T) above density one-third for fixed random free-product syllable models with at least three factors, through every integer relator length.
- Pure point Dirichlet spectrum on quasi-conical domains
A connected quasi-conical domain with a complete Dirichlet eigenbasis and dense positive eigenvalues, addressing the Krejčířík–Lotoreichik spectral question.
- Three anti-invariant classes on a nonintegrable four-manifold
A compact four-manifold with nonintegrable almost complex structures and three anti-invariant de Rham classes, addressing the Draghici–Li–Zhang conjecture.
- Signed impulse estimates and global smooth solutions for finite multispecies relativistic Vlasov-Maxwell
A finite-species extension of the OpenAI signed-impulse argument for relativistic Vlasov–Maxwell, with distinct masses and charges and no neutrality assumption.
- A counterexample to Kourovka Notebook Problem 16.45
An explicit group of order 100920 separates minimal permutation bases from independent sets, answering Cameron’s Kourovka Notebook Problem 16.45.
- A two-dimensional complex with embedding-dimension gap two
A finite two-dimensional simplicial complex that embeds piecewise linearly in R³ but needs R⁵ when every original triangle must remain flat.
- An application of separable-state nonrepresentability to Grassmann orbitopes and calibrations
A reduction from separable quantum states rules out finite semidefinite lifts of an oriented Grassmann mass ball and yields a calibration obstruction.
- Asymptotic LOCC dense coding with the symmetric four-qubit W state
A many-copy protocol lets two independent senders use the symmetric four-qubit W state to exceed two bits per copy with LOCC decoding.
- Consistent spectral recovery of smooth reversible diffusion tensors at a fixed lag
A consistency theorem recovers a smooth anisotropic diffusion tensor from stationary observations at one fixed lag using all positive empirical spectral modes.
- Exact Diffusion Design for Maximally Collective Stable Turing Patterns in Binary-Complex Mass-Action Networks
Mass-action networks with an unstable subsystem of size n−1, exact diffusion thresholds, and locally stable patterns for every species count n ≥ 4.
- Exact Tree–Theta-Trinet Collisions under the Kimura 2- and 3-Parameter Models
Exact Kimura-model examples show a phylogenetic tree and a level-two network can share every DNA site-pattern probability, even with strictly positive parameters.
- Measure-only preserving Markov transports characterize mass-stationarity
Measure-only randomized transports characterize mass-stationarity, including periodic random measures and information beyond the mass itself.
- Non-symmetry of biconstrained two-step reachability
Swapping two connection thresholds changes the guaranteed two-step reachability in finite graphs, with a proved gap of at least 1/108.
- Nonconvexity of fixed odd-power sum-of-squares cones
A finite sextic example shows that requiring a fixed odd power to be a sum of squares need not define a convex set.
- The spatial locus of common tangents to three disjoint convex sets is null
Common supporting tangent lines to three disjoint convex sets in three-dimensional space occupy zero volume, even for nonsmooth or nonclosed sets.
- A flow reduction for equilibrium uniqueness in rank-one bimatrix games
Deciding whether a rational rank-one bimatrix game has exactly one Nash equilibrium is coNP-complete, even when a strict pure equilibrium is supplied.
- A Minimal-Surface Counterexample to a Seshadri Lower Bound
A family of minimal complex algebraic surfaces violates a proposed Seshadri lower bound, using a classical plane pencil and a double base change.
- A modular-storage proof of NP-completeness for chip-firing halting on simple digraphs
An attributed proof that conservative chip-firing halting is NP-complete on simple strongly connected digraphs, including graphs without opposite arcs.
I’m gradually adding papers to this collection. Additional work is available in the earlier research archive.
Collaboration
I welcome researchers interested in checking, extending, or developing these results. Each paper page describes specific directions where collaboration would help. You can reach me at me@aleckriebel.com.