Papers

We prove Chvátal's conjecture, posed in 1972: every hereditary family of subsets of a finite set has a largest intersecting subfamily that is a star. More generally, we prove a sharp correlation inequality for increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. Writing $g^*(x)=1-g(1-x)$, we show that $$ \sum_{\varnothing\ne S\subseteq[n]}\hat{g}(S)^2\max_{i\in S}\mathrm{Inf}_i[f]\le\frac{2\mathrm{Cov}(f,g)\mathrm{Cov}(f,g^*)}{\mathrm{Cov}(f,g)+\mathrm{Cov}(f,g^*)}. $$ When $g$ is antipodal, that is, $g=g^*$, this yields $\mathrm{Cov}(f,g)\ge\frac{1}{4}\min_{i\in[n]}\mathrm{Inf}_i[f]$, the correlation formulation of Chvátal's conjecture due to Friedgut, Kahn, Kalai and Keller.

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $\pi$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $\pi$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

We study complete two-sided constant-mean-curvature hypersurfaces of dimensions $2\le n\le6$ and finite Morse index in simply connected space forms. In the round sphere the immersed domain is compact; in Euclidean space every noncompact example is minimal; in hyperbolic space of curvature $-1$ we obtain compactness under explicit dimension-dependent mean-curvature thresholds, from $H^2>1$ for surfaces to $H^2\ge5/3$ in dimension six. The conclusions also hold for the volume-constrained index, without properness, volume-growth, or curvature-bound assumptions. A common Green-function argument uses explicit rational combinations of eight identities and the stable Bernstein theorem of Hong-Li-Wang. Stable hyperbolic tubes show that no mean-curvature threshold independent of dimension can give compactness in all dimensions.

In this paper we affirmatively resolve the singularity conjecture for finitely supported non-degenerate random walks on cocompact Fuchsian groups. The method we use is based on constructing a pair of geodesic currents, one for the Lebesgue measure and one for the random walk measure, and checking that they are different by using Fourier analysis on the boundary.

A prime circle of order $2n$ is a circular ordering of $1,\ldots,2n$ such that the sum of every two adjacent terms is prime. We prove that a prime circle exists for every sufficiently large $n$. The proof is based on a perfect matching and robust expansion. We also study Hamilton cycles in graphs and digraphs defined by primitive sums and differences over finite fields. In particular, the primitive-sum graph on $\F_q$ is Hamiltonian for every prime power $q>18\,888\,871$, and for the graph on a full prime field $\F_p$, the bound improves to $p>61$.

We establish an upper bound for the symmetry rank of closed, simply connected, almost non-negatively curved manifolds of dimension 4 through 9 and in the case of maximal symmetry rank provide an equivariant diffeomorphism classification of this class of manifolds in dimensions 4, 5, and 6.

The generalized packing–covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. We also prove the bound on generalized Hamming weights $d_t(C)\le2R_t(C)+1$ whenever the alphabet size $q$ satisfies $q\ge R_t(C)$. Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.

We consider solving (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) – with heuristic enhancements and GPU implementation – has been very successful in solving huge-scale linear optimization problems (LPs). However, its application to more general conic convex optimization problems is not so well-studied. We analyze the theoretical performance of rPDHG for general (convex) conic linear optimization, and LP as a special case thereof. We show a relationship between the geometry of the primal-dual $\delta$-(sub-)level sets ${W}_\delta$ and the convergence rate of rPDHG. Specifically, we prove a bound on the convergence rate of rPDHG that improves when there is a primal-dual (sub-)level set ${W}_\delta$ for which (i) ${W}_\delta$ is close to the optimal solution set in Hausdorff distance, and (ii) the ratio of the diameter to the “conic radius” of ${W}_\delta$ is small. And in the special case of LP, the performance of rPDHG is bounded only by this ratio applied to the (sub-)level set corresponding to the best non-optimal extreme point. Depending on the problem instance, this ratio can take on extreme values and can result in excellent or poor performance of rPDHG both in theory and in practice.

Let $q$ be a prime power and $\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\infty)$ be the decomposition group of the infinity place $P_\infty$ of $H$ which is the common pole of $x$ and $y$. For every subgroup $G\le A(P_\infty)$, we construct explicit generators of Galois subfield $H^G$ of $H$ with respect to $G$ and determine an absolutely irreducible equation defining the smooth affine plane model for such a Galois subfield. For $p$-subgroups, the generators can be chosen so that the defining equation has an additive polynomial on the left-hand side and an $\mathbb{F}_p$-quadratic polynomial on the right-hand side. For $q=27$, we can construct a genus-two subfield $D\subset H$ that is not isomorphic to $H^J$ for any subgroup $J\le \text{Aut}(H)$ from the explicit equations of Galois subfields of the Hermitian function field. To the best of our knowledge, this is the first example of a maximal function field covered but not Galois-covered by the same Hermitian function field.

We study robust bond valuation with endogenous short-rate feedback under volatility uncertainty. Within the $G$-expectation framework, the dependence of the short rate on the bond price yields a nonlinear fixed-point problem, represented by a quadratic $G$-BSDE for the logarithmic price. Under suitable assumptions, we establish existence, uniqueness, comparison, and stability for bounded finite-horizon solutions. An additional strict monotonicity condition yields a unique bounded infinite-horizon solution and exponential convergence of finite-horizon approximations on compact time intervals. We apply these results to inverse short-rate design, constructing discount-rate coefficients that reproduce admissible smooth bond-price targets at a fixed maturity. For long maturities, we construct feedback rules under which the compensated logarithmic price converges exponentially to a prescribed bounded state-dependent profile, while the asymptotic yield equals a specified target.

Given a finite dimensional Lie algebra $L$ on a finite field $\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\mathcal{L}(L)$ and introduce the notion of “subalgebra commutativity degree” of $L$. This represents the probability to find that two randomly chosen subalgebras $A$ and $B$ of $L$ are permutable. We investigate the subalgebra commutativity degree of $L$ in connection with recent techniques of algebraic combinatorics and number theory, providing upper and lower bounds which may influence the structure of $L$. A specific study for the subalgebra commutativity degree of Heisenberg algebras is executed.

We study the $\theta$-congruent number problem for $\cos\theta=\pm3/5$ and $\pm4/5$ using the generalized theta series construction of Sirolli–Tornaría. We describe its specialization to newforms of weight $2$ over $\mathbb Q$ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every $\theta$-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight $3/2$ whose Fourier coefficients determine the central $L$-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-$\theta$-congruence unconditionally, and the converse holds assuming the Birch–Swinnerton-Dyer conjecture. We also prove unconditional non-$\theta$-congruence for primes in explicit arithmetic progressions.

We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada–Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.

Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure—one whose square is conditionally negative definite (CND)—to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and show that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification using geodesic and Wasserstein distances.

For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the Discrete Series found by Harish-Chandra. In this paper, we study the structure of symmetry breaking operators for Discrete Series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of T. Kobayashi, Nakahama and Pevzner. This we do by using reproducing kernels for the representations and our previous duality principle in order to find some explicit details on the nature of the differential operators representing symmetry breaking operators, in particular to what extent they are given by differentiations in normal directions to the $H$-orbit in $G/K$.

As robot capabilities increase, quadratic programming (QP)-based controllers must account for a similarly increasing number of constraints to ensure safe, reliable operation. Yet, with each added constraint, this introduces more chances of momentary conflict: in which case, a QP solver that returns an "infeasible" status leaves the controller with nothing to execute. To address this, we introduce ElastiQP, a modified dual active-set QP solver that relaxes every inequality constraint with an exact, per-constraint l1 penalty while keeping equality constraints (dynamics) hard. Notably, ElastiQP does so by folding the slack variables into the solver analytically, maintaining a constant size of the condensed linear system. On a suite of robot control benchmarks, ElastiQP achieves microsecond-level performance, matching or outperforming leading modern solvers on feasible problems. On infeasible problems, ElastiQP handles these gracefully, confining violations to strictly the conflicting inequality terms, returning a usable solution up to 40x faster than the best alternative solvers. ElastiQP is available as an open-source C++ header-only library, with Python and JAX interfaces, at https://github.com/StanfordASL/elastiqp.

This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system $x_{k+1}=f(x_k,u_k)$ with measured state and a $C^r$ transition map, $r\geq 1$, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon $N$. We prove that this behavior is a $C^r$ embedded submanifold of the ambient trajectory space with intrinsic dimension $n+Nm$, where $n$ and $m$ are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates $(x_0,\mathbf u)$ is a $C^r$ diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder–decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least $n+Nm$. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.

We propose a notion of simplicity for Cayley permutations that is compatible with inflation and agrees, under a natural relational encoding, with indecomposability in the sense of Schmerl and Trotter. We establish a substitution decomposition analogous to that of permutations, in which the components are modules of the associated relational structure. We use it to enumerate simple Cayley permutations as well as simple restricted growth functions.

The test error of a model plotted against its number of parameters $d$ falls, peaks when the model can just fit the training data, and falls again, exhibiting the double descent phenomenon. We explain the phenomenon with statistical mechanics. The training trajectory of a stochastic gradient-based method is a particle wandering over the energy landscape of the training loss at an induced temperature $T$, and a run that has equilibrated visits every parameter vector of a given training loss equally often, the fundamental postulate of statistical mechanics, with probability given by the Boltzmann distribution. Because training starts at an initial point and has only finite time to diffuse, it carries an effective weight decay, which makes every parameter a quadratic degree of freedom. The equipartition theorem then distributes the energy among the $d$ degrees of freedom in shares of $T/2$, so at a fixed training loss adding parameters lowers the temperature and drives the Boltzmann distribution toward the stationary path. Finally, adding parameters can only lower the $L^2$ norm of the stationary path, so a solution sampled at fixed loss is less likely to be large with increasing $d$, effectively increasing weight regularization.

Let $X$ be a normal variety in arbitrary characteristic, and let $\tilde X$ be the successive blowup along subvarieties $Z_1,\dotsc,Z_n\subseteq X$ of codimension at least two that have simple normal crossings and satisfy $Z_h\not\supseteq Z_i$ whenever $h<i$. We prove that the relative cone of curves $\overline{\operatorname{NE}}(\tilde X/X)$ is generated by the classes of finitely many elementary curves, and that every face admits a contraction over $X$. Moreover, we prove that every small face contraction admits a $D$-flip for every $\mathbb R$-Cartier divisor $D$ that is relatively anti-ample. We also compute the relative Cox sheaf and prove its finite generation.

We prove the André–Oort conjecture for hypersurfaces $V \subset Y(1)^n \cong \mathbb{A}^n_\mathbb{C}$ defined by an equation $a_1 x_1^m + \ldots + a_n x_n^m = b$, where $a_1, \ldots, a_n, b \in \overline{\mathbb{Q}}$ and $m \in \mathbb{Z}_{>0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1, \ldots, a_n, b$. This is the first effective proof of a uniform André–Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces $V \subset Y(1)^n \times \mathbb{G}_m^l$.

In 1854 Riemann presented his Habilitationsvortrag Über die Hypothesen welche der Geometrie zu Grunde liegen to the Faculty of Philosophy of the University of Göttingen. The lecture does not contain analytical developments and Riemann never explained the reason for this choice. According to Dedekind this choice was done in order to make the lecture understandable to the non-mathematician professors of the Faculty. This is the suppression thesis. Nevertheless, in 1851, three years before, Riemann has presented his Inauguraldissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse to the same professors of the Faculty of Philosophy. The Inauguraldissertation contains many analytical developments but, on that occasion, they were not suppressed. Therefore, the presence of non-mathematician professors is unrelated with the presence or absence of analytical developments in the lectures. The ideas contained in Riemann's Habilitationsvortrag –manifold, line element, curvature– were so new at the time that there was not even a proper language to express them. Therefore, Riemann was forced to adopt a non-technical expository style. This explanation for the absence of analytical developments in Riemann's Habilitationsvortrag is more credible than the suppression thesis.

Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$. We also construct a nine-dimensional projective $F$-rational variety $\mathfrak{X}$ and a closed embedding $\mathfrak{X}\hookrightarrow\mathbb{P}^{29}_k$ under which the same conclusion holds.

Let $G$ be a compact connected semisimple Lie group, and let $\pi:G\to U(V_\lambda)$ be a finite-dimensional irreducible unitary representation associated with the highest weight $\lambda$. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_\lambda$. Similar extremal properties hold for power functions in place of the entropy function. The proof uses a second variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the uncertainty principle characterizing coherent states. This solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups.

Taming the exponential increase of the Hilbert space dimension with system size in the simulation of gapped quantum lattice models is of the highest relevance for understanding and designing exotic quantum materials, where nonlocal interactions are of particular interest. High-order linked-cluster expansions provide access the solution of the eigenvalue problem for the infinite system, yet rely on the computation of high-dimensional oscillatory lattice sums with a graph structure, only approachable with Monte Carlo methods so far. This work resolves this issue, rendering all required graph lattice sums, referred to as graph zeta functions for kernels involving power-laws, computable. The resulting method reduces the evaluation time for state-of-the art series expansions from tenthousands of core-hours to minutes. After factorizing the lattice sum over blocks, each block is evaluated by the cheapest available strategy depending on its treewidth $\mathrm{tw}$. Basic blocks admit analytic forms in terms of generalized zeta functions. Series-parallel blocks with $\mathrm{tw}\le 2$ can be computed at linear cost in the number of graph nodes and in the size of the momentum grid using a semi-analytical algebra based on Epstein zeta functions and rapidly decaying Fourier series. Finally, for $\mathrm{tw}>2$, the method is combined with tensor-network bucket elimination yielding polynomial scaling of numerical work and memory in momentum grid size with exponents only growing with $\mathrm{tw}$ rather than with the number of vertices. Through use of FFT, the full momentum grid is recovered at the cost of a single momentum evaluation. We provide a detailed analysis of the precision and runtime of our method against analytic and numerical benchmarks. We further reproduce published Monte Carlo data for the transverse-field Ising model on different 1D, 2D, and 3D lattices, obtaining full agreement.

In this paper, we develop methods for constructing $\mathbb{Z}/2$-harmonic 1-forms in dimension three by varying the background metric. On $\mathbb{R}^3$, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at each vertex as the monodromy locus of a $\mathbb{Z}/2$-harmonic 1-form on $B^3$. We also construct desingularization models for every critical $\mathbb{Z}/2$-eigensection, agreeing exactly with the associated homogeneous model outside a compact set. Finally, we construct a nondegenerate $\mathbb{Z}/2$-harmonic 1-form on every closed connected oriented three-manifold for a suitable smooth metric.

\cite{mss2015} proved Weaver's discrepancy result existentially, resolving the Kadison–Singer conjecture . Finding such signs efficiently for general inputs remained an open algorithmic question. In the real-arithmetic model, we give a deterministic algorithm running in polynomial time with discrepancy at most $35\sqrt\varepsilon$. The algorithm walks from the origin of the hypercube to a vertex, fixing coordinates as they hit a face. Its potential measures a soft spectral edge of the discrepancy matrix perturbed by an operator-valued free semicircular element. The perturbation's covariance vanishes as the coefficients reach their endpoints. Inspired by the free interpolation approach of Bandeira, Boedihardjo, and van Handel \cite{bbvh2023}, we combine Lehner's variational formula \cite{lehner1999} with spectral Tsallis–$1/2$ regularization used in \cite{allenZhuLiaoOrecchia2015} and \cite{pesentivladu2026}. The resulting potential has a finite-dimensional SDP formulation, allowing the discrepancy and remaining covariance to be analyzed together. We analyze the optimizer's stability through the linearized Karush–Kuhn–Tucker (KKT) system of a regularized min–max problem, whose stationarity equations are related to the matrix Dyson equation \cite{erdos2019}. This gives the movement rule: either a coordinate can move toward its nearer endpoint at small spectral cost, or a low-curvature direction orthogonal to the current coefficient vector allows further progress. Choosing the better sign of this direction controls discrepancy while increasing the squared distance from the origin. Upcoming work \cite{kathuria2026higherRank} will address higher-rank Kadison-Singer and spectrally thin trees. Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.

We develop a geometric framework for Hamiltonian engineering in finite-dimensional quantum systems using ideal control pulses. Starting from a bilinear Schrödinger equation with unbounded control amplitudes, we construct the closed pulse group and use extensions of control systems and Filippov's relaxation theorem to obtain a family of effective Hamiltonians given by the convex hull of the drift's adjoint orbit plus the Lie algebra of the pulse group. The geometry of this orbitope describes possibilities beyond group averaging. Using the isotypic decomposition of the adjoint representation, we characterize its affine hull and show that the group average lies in its interior. This yields locally accessible families of effective Hamiltonians around the invariant part of the drift. We apply the framework to recover the necessary and sufficient condition for dynamical decoupling from arbitrary interactions with a finite-dimensional bath. For connected abelian pulse groups, we describe the relevant representation decomposition through restricted roots. Finally, for qubit networks with identical pairwise couplings, we give a qualitative characterization and quantitative estimation of effective Hamiltonians that can be generated using our framework.

A compactification of the discrete space $\omega$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $\omega$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $\omega$, and the Arkhangel'skiĭ–Bereznitskiĭ–Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $\omega$. We investigate symmetric compactifications of $\omega$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson–Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $\omega$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.

We construct skein theory for 3-manifolds with embedded line defects, starting from the data of a ribbon tensor category and its balanced braided module category. We focus on line defects arising from quantum symmetric pairs, and prove finiteness properties for their defect skein modules. As an application, we consider $\mathbf{Z}_2$-equivariant skein theory and establish an equivalence with defect skein theory in certain settings, leading to a skein theoretical construction of a family of $\mathrm{C}^\vee\mathrm{C}$ DAHA-modules in the Type AIII case.