数学前沿

The quantum mechanics of one $N\times N$ Hermitian matrix in the infinite $N$ limit is well known to define a two-dimensional string theory when the quartic coupling constant $g$ is tuned to a special negative value $g_*$. We study the thermodynamics of this Matrix Quantum Mechanics (MQM), treating the SU$(N)$ symmetry as a global symmetry. Crucially, the non-singlets, i.e. states in non-trivial representations of SU$(N)$, implement the effects of vortices on the discretized worldsheets. This random lattice model has the low-temperature $c=1$ and high-temperature $c=0$ phases, with the Berezinskii-Kosterlitz-Thouless (BKT) transition separating them. We map out the phase diagram in the coupling--temperature plane, which is separated by the critical stability curve $T_c(g)$ extending from $g_*< g <0$. We study this curve using the non-perturbative matrix bootstrap and low- and high-temperature perturbation theory, which explain the thermal instability. At low temperatures, thermally excited non-singlets backreact on the Fermi sea of matrix eigenvalues, causing it to spill over. At high temperatures, the theory is described by an effective matrix integral obtained by dimensional reduction on the thermal circle; the eigenvalue distribution of the zero mode becomes unstable at the critical coupling. We conjecture that the BKT transition occurs at a special point on the critical curve and estimate its position. Finally, we switch to the regular square lattice and study the MQM-inspired Villain model, where the link factor is the one-dimensional propagator on a circle. Using Monte Carlo simulations, we map out the BKT transition in this model.

In this paper, we theoretically and numerically study the minimizers for the Cahn-Hilliard energy with the Flory-Huggins potential under the strong anchoring condition, i.e., the Dirichlet boundary condition. We reveal bifurcation phenomena mediated by the boundary condition, the transition layer thickness, and the temperature of the system. Numerical simulations are also presented to approximate the minimizers of this energy by solving a gradient-flow equation, namely the Allen-Cahn equation constrained with strong anchoring conditions and random initial data. The effects of varying the transition layer thickness and temperature are presented to confirm the theoretical analysis.

We prove global uniqueness and Lipschitz stability for a scalar diffusivity in the heat equation from a single final state at any prescribed positive time. On each prescribed uniformly elliptic $C^{1,\alpha}$-bounded class with known coefficient boundary values, we choose a constant initial datum and a spatially constant exponential boundary input that work for the entire class. The result holds on smooth bounded connected domains in every dimension $n\ge2$ and gives an affirmative answer to \cite[Problem~8.2]{Alessandrini} within these classes. The stability estimate controls the $L^2$ diffusivity difference by the $H^2$ final-state difference, and this data norm is optimal within the Sobolev scale for the chosen experiment.

In this paper, we introduce an adaptive regularization strategy for stochastic mirror descent (SMD) to solve a class of risk functional minimization problems in infinite-dimensional Banach spaces. Using a Schauder basis, we construct nested finite-dimensional subspaces whose dimensions adapt to the sample size $n$ and balance approximation and optimization errors. We restrict each SMD subproblem to the corresponding subspace and project the stochastic gradient onto its dual space. This yields closed form solutions and coordinate-wise coefficient updates with low computational and storage complexity. For risk minimization in $\mathcal{L}^p$ spaces with $1<p<\infty$, we construct Bregman distances adapted to the underlying Banach space geometry using $\max\{2,p\}$-convex functionals induced by uniform convexity. At the $\mathcal{L}^1$ endpoint, we instead construct a locally strongly convex functional based on the entropy function. By developing a new analytical framework, we establish a convergence rate of $\mathcal O\left(n^{-\min\{\frac12,\frac1p\}}\right)$, up to logarithmic factors. In the misspecified setting, we prove convergence of the risk to its minimum under weaker regularity assumptions and derive explicit polynomial rates when the regularity exceeds a specified threshold. Finally, we apply the method to statistical inverse problems and illustrate its empirical performance through numerical experiments in both settings.

Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology ($THH$) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: $THH$ of such an algebra is equivalent, as an underlying spectrum, to $THH$ of its center, normalized on $\pi_0$ by the reduced trace instead of the unit map. This comparison $\Phi_A$ is not shown to be \'etale-locally cyclotomic. But what is cyclotomic \'etale-locally is a separate Morita trace, agreeing with $\Phi_A$ only on $\pi_0$. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of $\Phi_A$. It is insensitive to the Brauer class, reflected instead by $K$-theoretic and module-categorical pairings $THH$ does not see. We establish the analogous statement for derived Azumaya algebras over connective $\mathbb{E}_\infty$-rings, under a connectivity hypothesis.

In Ong et al.~(2025), a new finite automaton model with probabilistic and nondeterministic transitions was introduced, along with a corresponding regular expression language. The key difference from previous approaches is that nondeterminism is formalized in terms of multisets instead of powersets. Automata and expressions are interpreted over the space of probability distributions over multisets of strings with finite multiplicities. The main result of that paper was a full Kleene theorem showing that the two formalisms are equally expressive. In this paper we give a decision procedure for the semantic equivalence of two automata or expressions. The algorithm is based on polyomial algebra and makes use of Gr\"obner basis techniques for ideal membership testing.

We consider the defocusing nonlinear wave equations $\partial_t^2u-\Delta u+u^p=0$, with $p$ odd, for real-valued functions on $\mathbb R^{1+d}$, in the energy-supercritical case. For $(d,p)=(9,3)$, $(7,5)$ and $(6,7)$, we construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-\frac{2}{p-1}}W^*\big(\omega^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2\pi$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb R^{d}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first blow-up solutions of defocusing energy-supercritical NLW with smooth real-valued data. For comparison, we also give in an appendix, for every energy-supercritical exponent, a construction of a bounded, finite-energy, radial solution which is continuously self-similar in a backward light cone and whose supremum blows up; it is not smooth across the light cone, and the difficulty of the problem lies entirely in this regularity. Existence of the real profile is proved by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients, with rigorous error control. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.

We present a general construction of eventually periodic projective resolutions for modules over the quotient of a ring by a regular central element. Our approach combines the iterated mapping cone technique with the higher homotopies of Shamash. The periodic part of each resolution comes from a matrix factorization of the central element. The construction is applied to the integral group rings of quotients of groups of finite cohomological dimension by infinite cyclic central subgroups. We give explicit calculations for several families of groups, including hyperbolic triangle groups and the mapping class group of the plane with four punctures.

Three-dimensional (3D) printing makes customized parts in small batches, but 3D printers are expensive and slow, so the grouping, orientation, and sequencing of parts decide both delivery times and printer use. We study the allocation of parts with due dates to jobs on identical powder bed 3D printers, where the orientation of a part sets its height and the plate area it occupies. We formulate a bi-objective mixed integer linear programming model that minimizes the weighted earliness and tardiness of the parts and the unused plate area of the jobs, and we show that the optimal timing of a fixed job sequence is an isotonic regression. The Pareto front is computed with the epsilon-constraint method, and a non-dominated sorting genetic algorithm built on this timing result handles larger instances. On random instances, choosing the orientation lowers the earliness and tardiness by 63\% on average compared with the orientation given in the data. For 15 to 40 parts, the genetic algorithm keeps 98\% to 99\% of the hypervolume of the best known fronts in less than a minute per run on average, whereas a time-limited epsilon-constraint method keeps about half of it with 30 and 40 parts.

A celebrated conjecture of Gromov states that for closed aspherical manifolds, the vanishing of simplicial volume implies the vanishing of the Euler characteristic. We prove Gromov's conjecture for closed convex projective manifolds and, more generally, for closed orientable smooth manifolds admitting an immersed straightening. Generalising the straightening of simplices in properly convex domains in real projective space, we introduce the notion of immersed straightening and prove that, for every integer $d\geq 1$, there exists a constant $C_d>0$ such that every closed orientable smooth $d$-manifold $M$ admitting an immersed straightening satisfies $|\chi(M)|\le C_d||M||$.

Let $E$ be the Cantor set, $G=\operatorname{Homeo}(E)$, let $P\le G$ be the group of order preserving homeomorphisms and $L\le G$ the group of homeomorphisms preserving the Cantor--Lebesgue measure $\lambda$. We study how $G$ is built from these two subgroups. In 2003, Glasner and Weiss stated that $G=LP=PL$. We show that this is not the case, in fact $PL$ and $LP$ are two different proper subsets of $G$. On the other hand, $PL$ is a dense $G_\delta$ subset of $G$, the multiplication map $P\times L\to PL$ is a homeomorphism, and $G=PLP=LPL$. In particular, every homeomorphism of $E$ is conjugate to an element of $PL$. We classify the double cosets $PgL$ using special clopen values set. There are continuum many of them, and $PL$ is the only one which is not meager. We prove that $G$ is topologically generated by one element of $P$ and one element of $L$ and these two generators produce the Thompson's group $V$. We also show that the generic homeomorphism of $E$ is the product of a generic element of $P$ and a generic element of $L$. The equality $G=PL$ fails also if $\lambda$ is replaced by any other full nonatomic measure.

A system $(X,\mu, T)$ is partially rigid with constant $\delta>0$ if there exists an incresing sequence $(n_k)$ such that $\displaystyle \liminf_{k \to \infty } \mu(A \cap T^{-n_k}A) \geq \delta \mu(A)$ for every measurable $A \subset X$. The partial rigidity rate is the largest constant of partial rigidity. We characterize the partial rigidity rate of subshifts arising from $q$-additive functions $\pmod M$ in terms of quantities arising from B\'esineau's theorem. With this, we build a family of strongly $q$-additive functions to deduce that the collection of partial rigidity rates for substitution subshifts is dense in $[0,1]$ answering a question of Donoso, Maass and the author.

The alternating-current optimal power flow (ACOPF) problem is fundamental to power system operation and planning, but its nonconvexity makes globally optimal solutions of large-scale instances computationally challenging. Convex relaxations provide tractable lower bounds for assessing solution quality and supporting global optimization, although their effectiveness depends strongly on relaxation strength. This paper proposes an offline convexification and bound tightening framework for constructing strong linear programming (LP) relaxations for families of related ACOPF instances. Rather than repeating the tightening procedure for each instance, we compute variable bounds and second-order cone outer-approximation (SOC-OA) cuts offline using a demand-slack model and reuse them for subsequent instances with the same topology and admissible demand variations. Computational results on PGLib instances show that the fully offline approach achieves an average gap of $4.32\%$, compared with $3.80\%$ for the fully online approach, while achieving an average speedup of approximately $272\times$. The results demonstrate that offline convexification and bound tightening can recover most of the relaxation improvement of online tightening while significantly reducing the computational burden for repeated ACOPF evaluations.

Stochastic Push-Pull achieves linear speedup with respect to the network size $n$ on static strongly connected digraphs, but whether this speedup persists on time-varying networks remains unclear. In this work, we construct a quadratic decentralized problem on period-two networks showing that stochastic Push-Pull may fail to attain linear speedup over time-varying digraphs. For $p$-periodic networks, we identify an effective node count $n_{\mathrm{eff}}\le n$ that governs the leading stochastic term. Specifically, for smooth nonconvex objectives, we establish convergence upper bounds and matching lower bounds for stochastic Push-Pull with constant stepsizes, both exhibiting the leading-order dependence $\Theta(1/\sqrt{n_{\mathrm{eff}} T})$. When $p=1$, the identity $n_{\mathrm{eff}}=n$ recovers the usual linear speedup for static digraphs. However, periodic networks can have $n_{\mathrm{eff}}=o(n)$, demonstrating the loss of linear speedup under temporal variation. Numerical experiments illustrate effective-count scaling and the separation between static and periodic networks, consistent with our theoretical findings.

We consider the relationship between the Stanley--Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen--Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley--Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.

Let $M$ be a closed, connected, non-orientable hyperbolic manifold, and let $\mu$ be a good measure on $M$. Denote by $Homeo_0(M,\mu)$ the identity component of the group of $\mu$-preserving homeomorphisms. For every $d\geq2$, we prove that $\dim\overline{EH}_b^d\bigl(Homeo_0(M,\mu)\bigr)=\infty$, where $\overline{EH}_b^d$ denotes reduced exact bounded cohomology in degree $d$.

We introduce families of Lagrange functions associated with height and distance-squared functions on implicit plane curves. We show that degeneracies of these families characterize geometric properties of the curves, such as inflections and vertices, and give geometric interpretations of the corresponding Lagrange multipliers in terms of contact with lines and circles.

Let $K$ be a complete nonarchimedean valued field with $v(K^\times)=\mathbf R$, and let $V=\mathcal O_K$. We prove that $K$ is spherically complete if and only if $V[[T]]$ is coherent, and that this is also equivalent to $V[[T]]$ being a GCD domain. If $K$ is perfect of characteristic $p$, the same characterization holds for the Witt vector ring $W(V)$. Thus, this settles the previously unresolved full-real-value-group case in the coherence problems for both formal power series and Witt vector rings. In particular, this result also gives affirmative answers to Questions~9 and~10 of Anderson--Kang--Park. The proof combines a coherence criterion for complete rings with a spherically complete valuation quotient and a uniform construction of non-finitely generated intersections of two principal ideals from an empty ball chain.

We construct counterexamples, along dyadic lacunary sequences, to two problems of Erd\H{o}s on the almost-everywhere behavior of sums $\sum_{j\le N}f(n_jx)$ on the circle. The main device is a dyadic spike block: rare positive spikes, each counted many times by a run of consecutive dilates, make the lacunary averages large, while a deterministic lower bound on every block prevents the rest of the series from cancelling them. We obtain a mean-zero $f\in\bigcap_{1\le q<\infty}L^q(\mathbb T)$ and a sequence $n_j=2^{m_j}$ with $n_{j+1}/n_j\ge2$ such that $\|f-S_Nf\|_2\ll(\log\log N)^{-1/2}$ and $\limsup_{N\to\infty}\frac1N\sum_{j\le N}f(n_jx)=+\infty$ for almost every $x$. This answers negatively a question of Erd\H{o}s on whether the condition $\|f-S_Nf\|_2\ll(\log\log\log N)^{-C}$ suffices for almost-everywhere convergence, and shows that Matsuyama's condition $\|f-S_Nf\|_2\ll(\log\log N)^{-c}$, $c>1/2$, cannot be relaxed to $c=1/2$. A second choice of parameters gives, for every $2\le p<\infty$, functions $f\in L^p(\mathbb T)$ with $\limsup_{N\to\infty}\sum_{j\le N}f(n_jx)/(N(\log N)^{1/p-\varepsilon})=+\infty$ for every $\varepsilon>0$, almost everywhere. This is sharp up to $\varepsilon$, and the case $p=2$ answers negatively a second question of Erd\H{o}s. Finally, a bounded variant in the spirit of Marstrand's disproof of Khintchine's conjecture gives sets of arbitrarily small measure whose lacunary visiting frequencies have limsup $1$ almost everywhere.

We study tropical Tevelev degrees arising from maps between certain tropical moduli spaces of curves. Building on work of Dawson and Cavalieri, who defined and computed tropical Tevelev degrees in the case of degree $d = g+1$ and $n = g+3$ marked points, we extend the theory by introducing an additional integer parameter $\ell$. In our framework the curve degree and number of marked points vary as $d = g + 1 + \ell$ and $n = g + 3 + 2\ell$, and we analyze the resulting tropical Tevelev degrees for both positive and negative values of $\ell$. This tropicalizes results of Cela, Pandharipande, and Schmitt on algebraic Tevelev degrees. We then further broaden the framework by introducing generalized tropical Tevelev degrees, providing the tropical counterpart to the generalized Tevelev degrees studied by Cela and Lian. These results establish a wider set of computational and structural patterns for intersection calculations on tropical moduli spaces and reveal new behavior beyond the classical setting.