Papers

Borcherds-Kac-Moody (BKM) Lie algebras are constructed by combining a number of $\mathfrak{sl}_2$-s and $3$-dimensional Heisenberg algebras in a certain way, generalizing the celebrated Kac-Moody algebras. Such algebras have a wide range of applications, including the examples of the monster and fake monster Lie algebras being BKM algebras. Among their main features are existence of an invariant bilinear form, denominator identity and Weyl-Kac character formulas for irreducible representations. We introduce a construction of closely related algebras that we call Virasoro-Borcherds-Kac-Moody (VBKM) algebras: if it happens that certain $3$-dimensional Heisenberg algebras inside a BKM algebra form the infinite-dimensional Heisenberg algebra, we replace it by the Virasoro algebra in the construction. We construct a family of Lie algebras over $\mathbb{P}^1$, where the generic fiber is a VBKM algebra and a special fiber is a BKM algebra. This allows us to visualize VBKM algebras as deformations of BKM algebras and prove structural properties about VBKM algebras similar to those of BKM algebras, despite the absence of an invariant bilinear form. Those include denominator identity and character formulas of irreducible representations. We give examples of realizations of such $\mathbb{P}^1$-families of VBKM algebras in lattice vertex operator algebras. In particular, we include the fake monster Lie algebra into such a family.

We extend the Malliavin theory for $L^2$-functionals on product probability spaces that has recently been developed independently by Decreusefond and Halconruy (2019) and by Duerinckx (2021), by characterizing the domains and investigating the actions of the three Malliavin operators in terms of the infinite Hoeffding decomposition in $L^2$, which we identify as the natural analogue of the famous Wiener-Itô chaos decomposition on Gaussian and Poisson spaces. We further explore the corresponding Ornstein-Uhlenbeck semigroup in terms of its Mehler representation and prove new moment bounds for iterated gradients. As an illustration of the abstract framework, we prove an infinite version of the quantitative de Jong CLT that has recently been proved by G. Peccati and the author (2017) and by the author (2024).

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

We give a Green-function proof of the stable Bernstein theorem in $\mathbb R^7$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.

Bilu and Linial conjectured that every finite $d$-regular graph with $d \geq 2$ admits an edge signing $\sigma$ whose signed adjacency matrix $A_\sigma$ has spectral radius at most $2\sqrt{d-1}$. We prove that every signing meeting the mixed-root condition $r_{A_\sigma}\leq\sqrt{2(d-1)}$ satisfies \[ \rho(A_\sigma) < \frac{3+\sqrt5}{2}\sqrt{d-1}, \] where $r_{A_\sigma}$ is the largest root of the mixed determinantal polynomial $\chi[A_\sigma,-A_\sigma]$. The interlacing theorem of Ravichandran and Srivastava guarantees a signing satisfying the mixed-root condition, so our result improves the coefficient $2\sqrt2$ in their two-sided spectral bound. In the proof, we construct a positive matrix-valued probability measure supported on the roots of $\chi[A_{\sigma},-A_{\sigma}]$. The second moment gives a simple matrix inequality $A_{\sigma}^2 + dI \preceq 4r_{A_\sigma}^2I$, which yields a preliminary coefficient $\sqrt{7}$. Estimates for the fourth moment use information about short walks to obtain the coefficient $(3+\sqrt{5})/2$. For triangle-free graphs, the coefficient improves to $\sqrt6$, and for graphs with girth at least five, it improves to $\sqrt{35/6}$. As a result of independent interest, we extend the construction to $\chi[A_1,\ldots,A_k]$ for Hermitian matrices $A_1,\ldots,A_k$ with zero diagonal, and compute the first two moments explicitly. Finally, an explicit signing of $K_8$ shows that the mixed-root condition alone cannot guarantee a coefficient below $(4+\sqrt5)/\sqrt6$.

The capacity of the relay channel is characterized by three schemes: decode-forward, coordinated compress-forward, and uncoordinated compress-forward. The proof relies entirely on properties of typical sequences.

We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length $n$, every first nontrivial deduction between them requires an identity with at least $n+1$ variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.

To quantitatively characterize the long-time dynamics of the periodic damped stochastic Korteweg–de Vries (sKdV) equation driven by additive noise, we investigate the uniform-in-time error estimates for a Lie–Trotter operator splitting approximation. This splitting combines the exact deterministic KdV flow with the exact Ornstein–Uhlenbeck solution map for linear damping and additive forcing. Long-time error analysis for the sKdV equation is highly nontrivial because of the interplay among the lack of dissipative smoothing, the loss of one spatial derivative arising from the Burgers-type nonlinearity, and the fluctuations of stochastic forcing. To overcome these difficulties, we develop new strategies based on exponential Lyapunov estimates, the continuous dependence estimate, and the local error decomposition. We establish strong and weak order \(1\) convergence uniformly in time under sufficiently large damping, as well as strong order \(1\) convergence on finite time intervals under a weaker damping condition. Under stronger assumptions, the splitting approximation admits a unique invariant probability measure on \(H_0^2\), which approximates the invariant measure of the exact dynamics with order \(1\). To the best of our knowledge, these constitute the first uniform-in-time strong error and quantitative invariant-measure approximation results for the periodic damped sKdV equation with additive noise. Our results provide an avenue for long-time simulation of the damped sKdV equation.

We prove that every rational proper holomorphic map from $\mathbb{B}^2$ to $\mathbb{B}^4$ has degree at most $5$, confirming D'Angelo's conjecture in this case. This sharp bound follows from Meylan's degree estimates and an intersection-theoretic argument ruling out degree $6$.

We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBenedetto, Gianazza and Vespri. In contrast to the exponential change of variables and expansion-of-positivity used in the local case, our technique is nonlocal and relies on comparison arguments and barriers in a variational framework. Indeed, we obtain the stronger conclusion where the forward comparison starts immediately after the reference time, without an additional time gap, which is false in the local case. The latter leads to elliptic-type Harnack estimates for parabolic equations. Our result is new even in the linear case.

We prove that a Noetherian integral domain is regular if, and only if, the derived pushforward of the structure sheaf along every blowup is a perfect complex. For suitable local rings, we give a construction which realizes a shift of the residue field as a direct summand of the derived pushforward of the structure sheaf along an explicit blowup.

We describe the ring of Laurent symmetric polynomials in terms of generators and relations, giving an analogue of the Fundamental Theorem of Symmetric Polynomials. We extend the Hall inner product, and give an algebraic proof that the Laurent Schur polynomials form an orthonormal basis of this ring. In the case of infinitely many variables, we realize the ring of Laurent symmetric functions as a direct limit of inverse limits, and relate them to characters of rational and algebraic representations of general linear groups.

We study bounded entire solutions of the Allen-Cahn equation in dimension three with quadratic energy growth. We prove that such a solution of finite Morse index I has at most 2I ends. Furthermore, we show that Morse index one solutions have two ends and are axially symmetric, which generalizes an earlier result by Florit-Simon.

Quadrotor navigation in cluttered 3-D environments requires global guidance while local motion remains subject to collision and motion limits. Harmonic potentials provide dense guidance from a global boundary value problem, but coupling a volumetric harmonic field to constrained physical quadrotor motion remains an open experimental problem. We couple a precomputed volumetric harmonic field with a constrained predictive planner that queries the field at predicted positions instead of extracting a global reference path. In Structured 3-D tests, harmonic guidance yields larger minimum clearance and lower RMS jerk than matched Dijkstra guidance, at the cost of longer paths; the same pattern remains when both methods use the same passage. Long maze tests span routes far beyond one prediction horizon, and Crazyflie trials validate physical execution. To the best of our knowledge, this is the first physical quadrotor demonstration of volumetric harmonic field navigation. The results show that globally constructed harmonic guidance can directly support local constrained motion generation on a physical quadrotor.

This paper studies projection-free algorithms for stochastic constrained multi-level compositional optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Since projection onto the constraint set can be computationally expensive, we develop projection-free methods that rely on linear minimization oracles. For non-convex objectives, we propose variance-reduced projection-free algorithms and establish complexity guarantees under both the Frank-Wolfe gap and the gradient mapping criteria. We also develop momentum-based methods that achieve convergence guarantees under weaker smoothness assumptions. Additionally, by using a stage-wise design, we derive a parameter-free variant that preserves the same complexities for the Frank-Wolfe gap. Such a design can be further used to develop algorithms for convex and strongly convex functions whose rates match those of single-level projection-free counterparts. Finally, we consider finite-sum problems and derive complexities for non-convex, convex, and strongly convex objectives. Numerical experiments across multiple tasks demonstrate the effectiveness of the proposed methods.

Let $\sigma_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. Previous work determined $C(\sigma_{n,d})$ when $d$ is odd or a power of two, but the general even non-power-of-two case remained open. We determine the certificate complexity for every degree $1\le d\le n$, thereby completing the classification for elementary symmetric Boolean functions. Writing $d=2^t m$ with $m$ odd, we obtain an explicit formula in which the possible deficit from the maximal value $n$ is controlled by $2^t$, while the exact value is determined by a binary containment condition involving $m$. In particular, \[ n-2^{\nu_2(d)}+1\le C(\sigma_{n,d})\le n, \] and we characterize when the upper bound is attained. Moreover, we determine the least positive period of the certificate-complexity deficit $\Delta_d(n)=n-C(\sigma_{n,d})$: it is $1$ for odd $d$, equals $d$ when $d$ is a power of two, and equals $2^{\lfloor\log_2 d\rfloor+1}$ for even non-power-of-two $d$. This least-period problem is distinct from the classical periodicity of the underlying value sequence $\binom{j}{d}\bmod 2$. The known odd-degree and power-of-two formulas are recovered as special cases.

Let $(M,\nabla)$ be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood $N$ of the zero section in $TM$ is a complex structure on $N$ such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization $P$ on $N$ is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to $P$. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply $\mathcal P$-complete) if the ac-structure (resp. the ac-polarization) exists on $TM$. Although on a small enough $N$ an ac-structure always exists, entire connections are rear and if a maximal domain of definition $N_{max}$ of an ac-structure exists (different from $TM$), $N_{max}$ is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole $TM$. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are $\mathcal P$-complete.

We study the Cuntz semigroup of arbitrary rings with stable rank one. We prove that Cuntz subequivalence between countably generated projective right modules $P$, $Q$ is equivalent to $P$ being isomorphic to a pure submodule of $Q$, whilst Cuntz equivalence amounts to isomorphism.~We also prove order-cancellation of finitely generated projective modules (and, more generally, pairs of modules admitting certain complements) from direct sums with countably generated projective modules.

Motivated by Robert Schneider's trailblazing work toward developing a unifying algebraic theory of integer partitions, we explore various binary operations on partitions to identify algebraic structures on sets of partitions. In particular, we discover that several sets of restricted partitions form abelian groups under reduced versions of concatenation, component-wise addition, and component-wise multiplication. One type of restricted partition from a group structure also enjoys a bijection with ordinary partitions of any given size. We extend two partition groups to vector spaces over the finite field $\mathbb{Z}_p$, where $p$ is a prime. We further discover that partitions are equipped with a commutative ring structure. Finally, we consider subgroups, subspaces, and ideals of our algebraic partition structures to investigate properties of related types of restricted partitions. The new examples of algebraic structures described in this paper open the door to partition analysis via algebraic tools, decompositions, extensions, and geometry.

After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra $\hat{\mathfrak{g}}$. The auxiliary algebras $\hat{\mathfrak{g}}$ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras $\hat{\mathfrak{g}}$ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).

Let $(\mathscr{C}, \mathbb{E}, \mathfrak{s})$ be a weakly idempotent complete extriangulated category, and let $\xi_1\subseteq\xi_2$ be proper classes of $\mathbb{E}$-triangles. We prove an analogue of Cole's mixing theorem in this setting: two compatible admissible model structures, relative to $\xi_1$ and $\xi_2$ respectively, give rise to a mixed admissible model structure $\mathcal{M}_m$ relative to $\xi_1$, which has the trivial objects of the second model structure and the fibrant (or cofibrant) objects of the first. We also describe the cofibrant objects of $\mathcal{M}_m$ explicitly in terms of the two original model structures. For exact categories this recovers the mixing theorem for exact model structures, and for triangulated categories it applies to proper classes of triangles; we give a non-degenerate example on the homotopy category of a quasi-Frobenius ring.

We prove multivariate stability for a class of affine triangular recurrences whose two coefficients are raised to an arbitrary positive integer power. The variables keep track of cross-step heights in a weighted lattice-path model. The applications include stability and real rootedness of parametrised Eulerian, Stirling, and Lah families. For the parametrised Eulerian polynomials, we obtain multivariate refinements of known real-rootedness results and prove simplicity and strict interlacing of consecutive rows. A kernel-preserving bijection with increasing binary trees yields a synchronised Foata–Strehl action on pairs and a combinatorial interpretation of their gamma coefficients along with a recurrence. The same bijection identifies refinement posets that admit symmetric Boolean and chain decompositions for parameter one and cover-supported $\mathfrak{sl}_2$ operators prove their PECKness. We extend the stability and strict interlacing results to synchronised Stirling permutations and prove PECKness for ordinary Stirling permutations with a fixed number of plateaux.

We give an algorithm counting the permutations that avoid a fixed pattern of the following form: the direct sum of an increasing pattern and 231. The first members of the family are 1342 and 12453. For each member the algorithm uses polynomially many operations and stored integers, with degrees that grow linearly in the length of the pattern. It comes from a recurrence that reads a permutation from left to right and records the constraints that the letters read so far impose on those still unread. This recurrence has exponentially many states, but part of each state is protected: later steps carry it along unchanged and do not depend on it, and factoring the protected part out leaves a dynamic program of polynomial size. For 12453 a translation symmetry sharpens the bounds to degree seven for the operations and degree four for the storage. We also compute the number of 12453-avoiding permutations of every length up to 150. The previously published series, due to Biers-Ariel (2019), reached length 38. We also give a sampler of uniformly random avoiders. A floating-point implementation of it, proved to be within total variation distance $3.5\cdot10^{-5}$ of uniform for ideal random bits, draws the one million 12453-avoiding permutations of length 300 shown in a heatmap. The literal and kernel recurrences for 1342 and 12453 are verified in the Lean 4 proof assistant.

For every $\alpha\ge3$ and all sufficiently large $n$, we identify a single graph that simultaneously maximizes the number of independent sets of every size among all $n$-vertex $2$-connected graphs with independence number $\alpha$. This graph is unique up to isomorphism and is a balanced cycle-linking of the disjoint union of $\alpha$ cliques whose orders differ by at most one. More precisely, for each $3\le\beta\le\alpha$, the graphs maximizing the number of independent $\beta$-sets are exactly the cycle-linkings whose clique-size cyclic words are $\lfloor\beta/2\rfloor$-balanced. These results extend the corresponding extremal result for connected graphs to the $2$-connected setting. The proof combines generalized Turán-type clique counting and the edge-extremal theory of $2$-connected graphs with a coefficientwise balancing-switch argument. The switch also shows that a shortest imbalance of length $r$ first affects the independent-set count in degree $2r$.

In this paper, we construct two extremal positive definite even unimodular lattices of rank $88$ using a Hermitian tensor product over $\mathbb{Q}(\sqrt{-23})$, thereby extending the known existence range for such lattices from rank $80$ to rank $88$.

In this work, we propose an unfitted finite element scheme to approximate the solution of the heat equation on moving domains. We use the {\phi}-FEM paradigm in which the computational domain is described implicitly by a level-set function {\phi}. This function is incorporated into the variational formulation in order to enforce the boundary conditions, which allows the use of unfitted meshes in space. Such a strategy avoids the need for remeshing at each time step and makes it possible to handle complex geometrical evolutions. Moreover, the {\phi}-FEM approach has the advantage of being simple to implement within standard finite element libraries. We introduce a fully discrete scheme that combines the {\phi}-FEM spatial discretization with the lowest-order discontinuous Galerkin method in time. Under regularity assumptions on the level-set function and a mild restriction linking the time step to the mesh size, we establish an optimal a priori error estimate in the L2(0,T;H1) norm. Finally, we present several numerical experiments that confirm this convergence rate, exhibit a second-order convergence in the L{\infty}(0,T;L2) norm, and illustrate the robustness and accuracy of the proposed method.

We prove that a very general sextic threefold in $\mathbb{P}^4$ is algebraically hyperbolic, settling the last open case and completing the classification of algebraic hyperbolicity for very general hypersurfaces in projective space. The proof follows the Coskun–Riedl scroll construction, with the moduli of stable maps and the geometric canonical height as new ingredients.

The problem of determining whether a category is balanced (i.e., every morphism that is both an epimorphism and a monomorphism is an isomorphism) has been widely investigated, especially in algebra. For complete categories with an extremal projective generator, we characterise the property of being balanced, and the stronger property that every monomorphism is regular, in terms of implicit operations. These characterisations apply, for example, to any quasiequational class of algebras, possibly with a proper class of function symbols and unbounded arities, that admits free algebras.

Mather's $\beta$-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of $d \ge 1$ directions of perturbation satisfying a natural symmetry condition, the jet of the $\beta$-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the $\beta$-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint $C^\infty$–holomorphic dependence of the invariant curves, and hence of the $\beta$-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.

In this paper, building on our previous Sylow criteria, we establish small-subgroup criteria of liftability and $F$-liftability for the linear automorphism group $G$ of smooth hypersurfaces $X$ over algebraically closed field of characteristic zero. When $\mathrm{dim} X = p-2$ for some odd prime $p$, we prove that ($F$-)liftability of any finite subgroup of $G$ can be tested on its $p$-subgroups of order at most $p^2$. When $X$ is a degree $p$ hypersurface of dimension $2p-2$, we prove that ($F$-)liftability of $G$ can be tested on all its $p$-subgroups of order at most $p^3$, which is sharp for $p\geq5$. When $p=3$, this bound improves to $9$, giving the corresponding criteria for smooth cubic fourfolds.