Papers

Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth TQFTs can distinguish (stable) diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth 6-manifolds with finite $\pi_2$ are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite $\pi_2$ that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a $\mathbb{C}P^2$ connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all $n\geq2$ and $m\geq 2n$, there is a pair of $(2n-1)$-connected closed $(4n+m)$-manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.

Winkelmann conjectures that every smooth flexible complex affine variety of dimension at least two is a Rosay–Rudin space. We verify this for every positive-dimensional quotient $G/H$ with $G$ a connected complex semisimple algebraic group and $H$ a closed connected reductive subgroup. Weak and strong tameness coincide, every injection between tame discrete sets extends to a holomorphic automorphism, and complements of tame, finite, or empty sets are Oka. Every sufficiently sparse enumerated sequence can be sent to a fixed sequence by a composition of $n$ complete holomorphic flow maps, with $n\le6$; injective self-maps of the fixed sequence admit such a realization with $n\le2$. These flows preserve an invariant algebraic volume form. The proof uses entire interpolation and regular functions invariant under two solvable subgroups to construct the automorphisms. If $G$ is simple of rank at least two and $G/H$ is spherical and admits a closed equivariant embedding into an irreducible module, the normalization bound improves to $n\le4$. We also prove that $\SL_2/N(T)$, where $T$ is a maximal torus, is an RR-space. This quotient is the complement of a smooth conic in $\mathbf P^2$.

Dose optimisation of two-agent combinations in early-phase oncology trials must jointly balance toxicity, efficacy, and pharmacologically meaningful drug exposure, yet existing model-assisted combination designs use only binary toxicity–efficacy outcomes and leave pharmacokinetic (PK) data unused for dosing decisions. We propose PKComb-BOIN12, a utility-based Bayesian optimal interval design embedding a continuous PK exposure metric into two-agent dose optimisation via a PK-guided escalation rule and a PK-sufficiency constraint on the final admissible set. Two contributions distinguish this design from a direct combination extension of existing single-agent PK-informed designs. First, PK-based elimination uses a coordinatewise order-ideal (down-set) removal rule, so that eliminating a combination for insufficient exposure is licensed exactly by the assumed monotonicity of exposure in each agent's dose, rather than an unjustified scalar summary of combined dose intensity. Second, the pre-specified target exposure is treated as transportable-but-verifiable: a hierarchical calibration mechanism updates the assumed monotherapy-derived target using accruing combination-arm PK data, with a formal bound on the resulting selection bias under transportability violations. A time-to-event extension, TITE-PKComb-BOIN12, accommodates late-onset outcomes. A factorial simulation study spanning realistic scenarios, including one calibrated to a published combination trial, decomposes the source of PKComb-BOIN12's gains over its non-PK predecessor and evaluates robustness to non-monotone exposure, PK-variance misspecification, and target-transportability violations. An R Shiny application supports design evaluation and real-time conduct.

Together with Bourgain's slicing problem and the variance conjecture, the Kannan-Lovász-Simonovits (KLS) conjecture has been a driving force for mathematical developments in high dimensional geometry in recent decades. We present here a proof of this conjecture. The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages.

We explicitly compute and parametrize the centralizers of elements, encompassing dege\-ne\-rate and unipotent ones, in the orthogonal, symplectic, and unitary gro\-ups over the real and complex fields, including the indefinite cases. The same approach simultaneously yields the isotropy groups of $H$-skew-symme\-tric, Hamiltonian, and $H$-Hermitian matrices under the adjoint representations of the corresponding classical groups. The resulting groups are conjugate to subgroups of nonsingular block matrices whose blocks exhibit a rectangular block Toeplitz structure.

We give a simpler proof of the theorem of J.~Tao and R.~Travkin saying that (the neutral block of) the affine Hecke category is the monoidal colimit of finite Hecke categories. The argument is axiomatized to apply to various sheaf-theoretic and group-theoretic contexts. We also prove an extension of the Tao-Travkin equivalence for the whole affine Hecke category as a Morita equivalence.

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

We study the non-parametric estimation of the roughness of a path from discretely sampled observations. Our approach is based on the concept of normalized power variation of a path. The estimator identifies a variation index by comparing coarse increments with local sums of fine increments. We prove pathwise consistency under Hölder regularity, nondegenerate coarse critical power sums, uniform local bounds on fine critical power sums, and separation of the two sampling scales and obtain a convergence rate for the estimator. Neither convergence of critical variation nor monotonicity of the finite-sample statistic is required. The result applies to fractional Brownian motion under an explicit dyadic sampling rule and to Schauder series with bounded coefficients bounded away from zero. For signed Takagi–Landsberg functions we obtain a uniform asymptotic formula and a quantitative bound for every root of the estimating equation in a fixed parameter interval. We illustrate the results with numerical experiments for Takagi– Landsberg functions and fractional Brownian motion.

In Networked Markov Decision Processes, transition dynamics are often unknown and the state–action space grows rapidly with the number of agents. In this setting, Taylor representations naturally approximate $Q$-functions, but a naive order-$n$ expansion over $N$ agents requires $\Theta(N^n)$ coefficients. We justify these expansions under smooth expected future local rewards with controlled derivatives. Under this condition, finite-speed information propagation and discounting imply that local-critic Taylor coefficients decay exponentially with the graph distance to the farthest agent involved. Discarding distant-agent coefficients and marginalizing then yield scalable local Taylor representations with a bound controlled by graph locality. Building on these representations, we propose a scalable model-free actor–critic algorithm, establishing finite-sample critic and near-stationarity guarantees for a linear LSTD critic. We then introduce a more expressive neural TD parameterization. Unlike prior constructive spectral methods, our approach covers settings without access to a known local dynamics map, such as hidden switched linear–quadratic regulation. Across three control benchmarks, our method matches or outperforms spectral baselines while scaling efficiently to large graphs.

It was recently shown by Paolini and Shelah, with another proof by Laskowski and Ulrich, that torsion-free abelian groups are Borel complete. This means that isomorphism for torsion-free abelian groups is as complicated as possible, so that torsion-free abelian groups do not admit any meaningful complete isomorphism invariants. We give a proof following the ideas of the Paolini-Shelah argument but with several expository advantages. In particular, the combinatorial part of the argument is much simplified. We make use of the ideas of this proof to show that there is a torsion-free abelian group with no computable copy but with a copy computable from every non-computable set. This answers a long-standing question of Downey and Goncharov.

We revisit an old non-diagonal modular invariant of $F_4$ at level $6$, discovered in 1989. At the time of its discovery, it was noticed that it could not be obtained from a conformal embedding of affine Lie algebras. Later, it was shown that it can be obtained from an extension of a vertex operator algebra which is not of Lie type. To this modular invariant, one can attach a full CFT defined (or formulated) as a module category $\mathcal M$ for the fusion category $\mathcal A = \mathcal C(F_4,6)$. More recently, it was recognized that $\mathcal M$ is associated with an étale algebra of $\mathcal A$. In other words, the resulting quantum module is a quantum subgroup (it is also said to have self-fusion, to be of type I, or to be flat). We study the dual $\mathcal O$ of $\mathcal A$ with respect to $\mathcal M$. In particular, we describe the algebra of quantum symmetries (we obtain the chiral generators and display the Ocneanu graphs for the fundamental irreps of $\mathcal A$). The chiral graphs are not connected, each one containing two pairs of isomorphic components. The vertex sets of these components are identified with the simple objects, up to equivalence, of two module categories $\mathcal M$ and $\mathcal M^\prime$ sharing the same modular invariant. $\mathcal M$ enjoys self-fusion, whereas $\mathcal M^\prime$ does not. The restriction of the action of the Verlinde algebra $A$ to the associated abelian groups $M$ and $M^\prime$ defines two families of matrices with non-negative integer entries (nimreps) and their adjacency graphs. As a by-product, we obtain the matrices describing the defect lines of the theory, the integer and rational fusion polynomials attached to $A$ and its modules, and the relevant numerical invariants, quantum dimensions, quantum masses, and induction-restriction rules.

The recursive-line and signed Zarankiewicz numbers maximize the number of squares in augmentations of a maximum $C_4$-free base, subject to two sufficient irreducibility criteria. The count includes one square per base cell and one per selected pair of unused cells. We compare these parameters with the second-order number, which uses irreducibility itself. Every maximum $m\times n$ base admits a recursive-line augmentation with at least $mn/2-C\max(m,n)$ squares, for an absolute constant $C$. Combining this bound with a two-column extension of known fixed-width families, we show that all three parameters are asymptotically equivalent, uniformly as the larger dimension tends to infinity. For individual displays, a transfer graph shows that once the signed closure identifies every selected pair, the signed criterion is equivalent to irreducibility. We determine the second-order number for every six-column rectangle and give eventual exact formulas for all three numbers at widths seven, nine and eleven. We also prove signed and recursive-line equality at $8\times7$ and, together with earlier values, whenever the shorter side is at most six, except possibly at $14\times4$. The exact-value results are computer-assisted, using exhaustive enumeration, checked propositional refutations and symbolic certificates with a proved lifting argument for arbitrary lengths.

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

We establish a maximum-variance inequality for discrete $\alpha$-concave random variables in the regime $-1/3<\alpha<0$, extending corresponding results beyond the discrete log-concave setting. As applications, we obtain a reverse entropy power inequality and bounds for the Lévy concentration function. We further establish polynomial-type concentration inequalities, reflecting the heavy-tailed nature of the class. A main tool in our approach is a convex majorization principle that reduces the relevant extremal problems to suitable $\alpha$-affine distributions.

For each fixed $n$, we prove a positive lower bound for the diameter of a compact simple geodesic $n$-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of $n$ for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer $q\ge2$, a geodesic construction with $2q+3$ sides and diameter less than $3/q$, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.

We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by $g(t)$ satisfies in dimension $n\geq 3$ $$g(t)=c_n \sqrt{t} |\log t|^{\frac{-1}{n - 2}} (1 + o(1)), t\to 0,$$ and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.

We discuss surface-pinned, topologically charged, nonsingular defects (SD) in light scattering by a plasmonic object. They follow from exact Maxwell boundary conditions and are purely topological—unaffected by surface forces. At optical frequencies, in (quasi) 2D, they appear in the electric and Poynting vector fields for $H^{(z)}$ polarization, not for $E^{(z)}$. Exact solutions for a circular cylinder reveal topological charge-conserving bifurcations involving SDs with bulk singularities or SDs alone, and contraction of defect regions under loss.

We study learning in continuous games when players receive only pairwise preference feedback, revealing which of two actions is preferred but neither payoff values nor preference magnitudes. We first show that standard external regret and coarse correlated equilibria (CCE) are not identifiable from this ordinal information: the same sequence of play can incur zero and linear regret in two ordinally equivalent games, while the distributions that remain CCE across all cardinal representations consistent with the same preferences are exactly those supported on pure Nash equilibria. Motivated by this gap, we develop a first-order ordinal theory based on normalized unilateral preference directions, introducing an ordinal directional regret benchmark and corresponding equilibrium notions. We show that block-normalized pseudogradient dynamics achieve sublinear ordinal regret and, under additional structure, Nash-convergence guarantees. We then use a single-comparison estimator to implement these dynamics from finite pairwise comparisons. With one comparison per player and round, the resulting algorithm achieves sublinear finite-resolution ordinal regret against arbitrary opponent behavior and, in ordinal potential games, almost-sure last-iterate convergence to the Nash set. Our results provide regret, dynamics, and equilibrium guarantees directly from preference feedback without reconstructing cardinal utilities.

Let $X$ be a smooth proper integral variety of dimension at most four over a field of characteristic zero. We construct a finite sequence of blowups along smooth centers with projective endpoint, preserving the intersection of all maximal quasi-projective opens of $X$. The construction depends only on a fixed functorial principalization procedure and is natural under isomorphisms, including those over different ground fields. The proof has two parts. A colength argument on regular local surfaces gives, in every dimension, a finite preparation after which some maximal quasi-projective open has complement mapping into codimension at least three in $X$. In dimensions at most four these images are points or curves. Ample numerical classes on the normalized components of that complement give finite bounds for summing base ideals of linear systems. The resulting ideal descends to the original field; geometric integrality is not required.

This work studies the asymptotic behavior of a class of viscous Hamilton–Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a $\Gamma$-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.

This manuscript aims to advance existing approaches to the abelianization of symplectic quotients. Two such approaches include the imploded cross-sections of Guillemin-Jeffrey-Sjamaar, and the Gelfand-Cetlin abelianizations of the first author and Weitsman. Using our notion of generalized Gelfand-Cetlin data, we generalize the second approach to account for irregular values of the moment map. Connections to symplectic implosion are emphasized throughout this manuscript.

For $d\ge3$, let $\lambda_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, while it is not sharp for $d\ge9$. We prove $\lambda_d=C_d$ for the remaining cases $5\le d\le8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.

In this paper we give necessary conditions for 4-class association schemes that are generated by double covers of strongly regular graphs. These conditions are applied to open cases for diameter 4 antipodal distance-regular graphs. Using these conditions we are able to show the nonexistence of four cases in the table of Brouwer, Cohen and Neumeier: $\{20,18,3,1;1,3,18,20\}$, $\{22,21,3,1;1,3,21,22\}$, $\{ 54,50,5,1;,1,5,50,54 \}$, $\{ 170,162,9,1;,1,9,162,170 \}$. More generally, we show there is no distance-regular graph with intersection array $\{k,b_1,b_2,1;1,b_2,b_1,k \}$ where $b_2 \neq 1$ and $\frac{ k b_1 }{4}(1+k+\frac{ k b_1 }{2b_2})$ is odd. Tables are also given for more general 4-class association schemes generated by double covers of strongly regular graphs.

A group is an {\it $\SJ$-group} if it admits a general abelian series all of whose terms are subnormal of finite defect in the whole group. We construct a group $G$ which is not an~\hbox{$\SJ$-group}, although every countable subgroup of $G$ is an $\SJ$-group.

V. Manuilov introduced the universal Roe algebra, namely the uniform Roe algebra associated with the largest uniformly locally finite coarse structure $\mathrm{Univ}$ on $\mathbb{N}$, together with several variants. We study these algebras through embeddability properties of $\mathrm{C}^*$-algebras. Among other results, we show that the universal Roe algebra admits an embedding from $\prod_n\mathbb{M}_n$, answering a question by Farah. Thus, Ozawa's non-embeddability theorem for $\prod_n\mathbb{M}_n$ in uniform Roe algebras of uniformly locally finite metric spaces does not extend to arbitrary uniformly locally finite coarse spaces. Nevertheless, embeddability properties distinguish the variants introduced by Manuilov and the quasi-local algebra of $\mathrm{Univ}$, apart from the multiplier algebra $M(\mathbb{I})$. As a by-product, we also obtain an operator-norm analogue of Birkhoff's Problem 111: the norm-closed convex hull of permutation operators is exactly the set of doubly substochastic operators. Moreover, we show that every separable $\mathrm{C}^*$-algebra embeds into the quasi-local algebra of $\mathrm{Univ}$. Although we do not know whether the analogous statement holds for the universal Roe algebra, we prove that every separable quasidiagonal $\mathrm{C}^*$-algebra embeds into it, and hence into the reduced $\mathrm{C}^*$-algebra of some étale locally compact second countable groupoid.

Recently, it was shown in arXiv:2607.27629 that under a very general condition, a generic family of curved Kakeya sets in $\mathbb{R}^3$ not trivially contained in a hypersurface has Hausdorff dimension strictly greater than 2. In this paper we leverage large language models to give an explicit construction of a family of curves failing the worst compression phenomenon in arXiv:2607.27629, but whose union has Hausdorff dimension and upper box dimension $5/2$. The example shows that there are Kakeya compression phenomena besides the trivial example of concentrating near a hypersurface. This answers a question raised in section 1.9 of arXiv:2607.27629. The models used are ChatGPT 5.6 Sol and ChatGPT 6 Astra.

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré–Sobolev constants $\Lambda^{\mathcal{A}}_{p,q}(\Omega)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $\Lambda^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann–Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $\Lambda^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

In this paper, we compute the orders of upper triangular symplectic Renner monoids using admissible sets. We obtain both recursive and closed-form formulas for these orders. We then derive their exponential generating functions and establish connections between these functions and the Laguerre polynomials.

We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.

We prove the finiteness of the genus of quaternion division algebras over some formally real fields. In particular, if $F$ is a formally real field and the second power of the fundamental ideal $I^2(F)$ is torsion-free, then the genus of any quaternion division $F$-algebra is trivial. For instance, this immediately implies that the genus vanishes for any quaternion division algebra over a formally real pythagorean field. The proofs are based on methods from the theory of quadratic forms.