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1. 287
   Isomorphism of the free group factors

   Operator algebras 1 manuscript, 23 pages Lean Reasoning summary

   Resolves the free group factor isomorphism problem: $L(\mathbb F_2)\cong L(\mathbb F_3)$, and hence all interpolated free group factors, including $L(\mathbb F_\infty)$, are isomorphic. Their common factor has fundamental group $\mathbb R_{\gt 0}$.

2. 231
   The free uniform spanning forest is a factor of IID

   Probability and statistical mechanics 1 manuscript, 19 pages Lean

   On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.

3. 259
   A group without fixed price

   Group theory 1 manuscript, 27 pages

   Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.

4. 250
   Boone–Higman embeddings with higher finiteness

   Group theory 3 manuscripts, 83 pages Lean

   A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F_∞: a classifying space with finitely many cells in each dimension. A single group of type F_∞ can also contain every finitely presented group.

   Closest manuscript: A universal group of type F∞

5. 196
   A counterexample to Kaplansky’s zero-divisor conjecture

   Algebra 1 manuscript, 26 pages Lean

   Constructs a finitely presented torsion-free group G whose group algebra $\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

6. 247
   An infinite finitely presented residually finite 2-group and a finitely presented nil algebra

   Group theory 2 manuscripts, 73 pages Lean

   Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽₂-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.

   Closest manuscript: An infinite finitely presented residually finite 2-group

7. 248
   Thompson's group F is nonamenable

   Group theory 1 manuscript, 13 pages Lean

   Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.

8. 197
   A torsion-free group algebra that is not directly finite

   Algebra 4 manuscripts, 93 pages Lean Reasoning summary

   Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

   Closest manuscript: A Counterexample to the Group-Ring Determinant Conjecture

9. 296
   The generator problem for finite factors

   Operator algebras 1 manuscript, 18 pages Lean

   Proves that every type II₁ factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $P\subset M$ of such factors, the unitaries u with $M=W^*(P,u)$ form a dense G_δ subset in the trace 2-norm topology.

10. 256
    Nonsingular systems of equations over arbitrary groups

    Group theory 2 manuscripts, 29 pages Lean

    Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

    Closest manuscript: The Kervaire theorem for groups

11. 249
    A finitely generated Eilenberg–Ganea counterexample

    Group theory 1 manuscript, 25 pages Lean

    Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.

12. 252
    A torsion-free hyperbolic group that is neither residually finite nor linear over any field

    Group theory 1 manuscript, 21 pages Lean

    Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.

13. 286
    Rigidity and arithmetic of lattice von Neumann algebras

    Operator algebras 2 manuscripts, 190 pages

    Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-$(T)$ lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.

    Closest manuscript: The arithmetic category and stable recovery of lattice factors

14. 142
    Deterministic polynomial factorization over prime fields

    Theoretical computer science 1 manuscript, 48 pages

    Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽_p, including multiplicities, in bit complexity polynomial in $(n+1)\log p$. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

15. 011
    Prime-factor statistics of $p-1$

    Number theory 3 manuscripts, 187 pages

    Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

    Closest manuscript: Prime Predecessors with an Even Number of Prime Factors

16. 253
    An infinite finitely presented simple amenable group

    Group theory 1 manuscript, 58 pages Lean

    Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.

17. 207
    The ℓ¹-Bass conjecture for all discrete groups

    Algebra 2 manuscripts, 66 pages Lean

    Proves the ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.

    Closest manuscript: The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

18. 236
    The exact factor-of-IID threshold for free Ising spins on trees

    Probability and statistical mechanics 1 manuscript, 50 pages Lean

    Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $\tanh\beta\le(d-1)^{-1/2}$, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

19. 285
    Counterexamples to Baum–Connes and Kadison–Kaplansky

    Operator algebras 3 manuscripts, 184 pages

    Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C^∗-algebra, disproving the Kadison–Kaplansky conjecture.

20. 014
    Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

    Number theory 8 manuscripts, 348 pages

    Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

21. 305
    Four-dimensional disk embedding and Wall's conjecture

    Topology 3 manuscripts, 133 pages

    The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F₂ is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

    Closest manuscript: A boundary-only obstruction to four-dimensional disk embedding

22. 204
    Tensor saturation for even spin groups

    Algebra 1 manuscript, 46 pages

    Proves saturation factor one for $\mathop{\mathrm{Spin}}\nolimits (2n)$, n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.

23. 057
    Fundamental groups of special complex varieties and root orbifolds

    Algebraic and complex geometry 3 manuscripts, 83 pages

    Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.

24. 314
    Cyclic length and chromatic fixed-point loss

    Topology 1 manuscript, 30 pages

    Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

25. 018
    The Margulis–Platonov conjecture over global fields

    Number theory 2 manuscripts, 159 pages

    Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of $G(k)$ is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.

    Closest manuscript: The Margulis–Platonov conjecture over global function fields

26. 291
    Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

    Operator algebras 4 manuscripts, 180 pages Lean

    Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C^∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.

27. 020
    Squarefree quartics and power-free polynomial values

    Number theory 1 manuscript, 55 pages Lean

    Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d-2)$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.

28. 310
    Quillen's conjecture in rational homology

    Topology 1 manuscript, 63 pages

    Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

29. 116
    Uniform black-box noncommutative identity testing across characteristics

    Theoretical computer science 3 manuscripts, 52 pages Lean

    For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.

    Closest manuscript: Uniform Matrix Hitting Points in Every Positive Characteristic

30. 258
    Gersten’s conjecture and virtual compact specialness of one-relator groups

    Group theory 2 manuscripts, 198 pages

    Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup $\mathrm{BS}(m,n)$, with $m,n\ne0$, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

    Closest manuscript: Baumslag-Solitar-free one-relator groups are hyperbolic