Papers

Let $H$ and $K$ be subgroups of a virtually polycyclic group $G$. We prove the Hirsch length inequality $$h(H)+h(K) \leq h(H\cap K)+h(G).$$ We show that equality holds when the number of $(H,K)$-double cosets is finite, and that the converse holds when $G$ is nilpotent. For homomorphisms $\varphi,\psi \colon G \to H$ between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is $$h(\mathrm{Coin}(\varphi,\psi)) \geq h(G)-h(H).$$ In this formulation, equality holds when the Reidemeister number $R(\varphi,\psi)$ is finite, and the converse holds when $H$ is nilpotent.

This paper studies generalized Hamiltonian gradient flows for contact-type Hamilton–Jacobi equations, adopting the variational framework of Herglotz's principle and its fundamental solution \(h_L(t,x,y,u)\). Two main results are presented. First, precise first-order sensitivity relations are derived, linking derivatives of \(h_L\) to dual arcs of minimizing trajectories. Second, quantitative second-order estimates show that \(h_L\) is locally semiconcave and, over short time, semiconvex, indeed uniformly convex in certain variables. These regularity properties follow from a detailed variational analysis of minimizing trajectories. The work establishes a foundation for intrinsic methods in analyzing singularity propagation and generalized gradient flows in contact context, with implications for weak KAM theory, optimal transport, and regularity of viscosity solutions.

This paper provides a complete characterization of the sign of the propagation speed in the symmetric two-species Lotka-Volterra competition-diffusion model under strong competition. The system admits a unique bistable travelling front and the sign of its speed determines which of the two species invades the other. We prove that for every strong competition intensity greater than $1$ and every diffusion ratio $d\neq1$, the travelling front propagates so as to expand the territory occupied by the faster-diffusing species. The front has a zero speed exactly when $d=1$. We also establish the smooth dependence of the wave speed and the travelling front on the model parameters. The main step in the sign characterization is to prove that a monotone standing front cannot exist when the diffusion rates are different. Combined with continuity of the wave speed with respect to the parameters, the species-exchange symmetry of the system, and an explicit travelling front at a particular parameter value, we obtain the sign of the propagation speed throughout the entire parameter region. This establishes the “Unity is not strength” theorem, which was previously known only in restricted parameter regimes.

Peer prediction seeks to incentivize agents to truthfully report an observed signal by rewarding joint sets of reports without observing a ground truth. Following the generalization of information-theoretic mutual information introduced in Kong and Schoenebeck (2019), we call a function of a joint distribution over signals a mutual information when it is non-negative and disincentivizes garbling reports for all information structures. An unbiased estimator for a mutual information takes some number of samples from the distribution and returns rewards for both agents, such that the expected reward is equal to the mutual information. We seek to characterize the set of mutual informations with unbiased estimators for a given number of samples. We show that for three or fewer sampled report pairs, the only mutual information with an unbiased estimator is trivially zero, and for four or five samples with a binary report space, the Determinant Mutual Information (DMI) of Kong (2024) is the unique mutual information (up to a scalar multiple). We further show that DMI ceases to be unique at six samples. We provide an improved estimator of DMI for any given number of samples and characterize its convergence rate. We also examine mutual information estimators that accept a randomized number of samples. First, we show that mutual information estimators on an ex-ante bounded number of samples (termed "stop-short estimators") can achieve a lower variance than an equivalent fixed-sample estimator (for DMI). Second, we introduce the class of scoring-rule-based mutual informations and identify in this family a mutual information that can be estimated with under three samples in expectation.

We study quadratic polynomials that are nonnegative on the non-compact set \[ F:=\{(x_1,x_2)\in\mathbb R^2:\ x_1\ge 0,\ x_2\ge 0,\ x_1x_2\le 1\}. \] All extreme rays of the cone of nonnegative quadratic polynomials are characterized on this set. The characterization allows us to study parameterized valid inequalities for quadratic convexifications involving $F$, which yields a semidefinite representation of its lifted convex hull in the quadratic space. Our analysis on extreme ray characterization separates the positive-semidefinite (PSD) and non-PSD branches, reduces the latter to boundary nonnegativity, and classifies the boundary contacts of the relevant extreme rays. By reparameterization, we also find a non-trivial and non-permutation-symmetric six-dimensional linear section of the cone of nonnegative homogeneous ternary octics that are sum-of-squares. We also show that our lifted convex hull result yields a degree-bounded preordering certificate of nonnegative quadratics on $F$ and a degree-bounded certificate for a family of nonnegative quartics on the half-strip.

We obtain arithmetic conditions which are satisfied by solvable groups admitting an abelian $\pi$-subgroup. First, we extend a previous characterization by removing the square-free hypothesis on some fixed irreducible character degree. We then show the same arithmetic conditions follow from a faithful action of an abelian $\pi$-subgroup on the $\pi'$-part of the Fitting subgroup, where $\pi$ is the set of prime divisors of the aforementioned character degree.

The aim of this manuscript is to show that the group of autoequivalences of an exact Courant algebroid over a compact base manifold is a tame Fréchet Lie group. Moreover, we compute its Lie algebra. Furthermore, we show that the space of generalized almost complex structures is a tame Fréchet manifold and that the canonical action of the group of autoequivalences on the space of generalized almost complex structures is smooth tame.

Battery-swapping stations (BSSs) can shorten electric-vehicle energy replenishment while using centrally managed battery inventories as flexible grid-connected storage. Realizing both benefits requires the station to schedule charging, grid discharge, and swapping service before future customer demand and electricity prices are known. This paper develops a forecast-aware rolling-horizon operating framework for this problem. A lightweight DLinear model predicts 24-hour price and demand trajectories, Stein variational gradient descent quantifies their uncertainty through representative scenarios, and a two-stage stochastic model predictive controller converts those scenarios into station decisions. The controller accounts for service shortfall, terminal readiness, a protected service buffer, and electrochemical degradation without assuming perfect future information. The application contribution is an implementable controller that coordinates the station's mobility-service and energy-storage roles. The methodological contribution is a modular forecast-to-control interface that separates the operational value of mean-forecast accuracy from that of uncertainty representation. In a 120-day closed-loop evaluation, DLinear-SVGD SMPC achieves the lowest cost among the implementable controllers. Relative to deterministic DLinear MPC, it reduces final cost by 1.2% and service-shortfall hours by 80.7%, with 99.10% of the evaluated hours free of shortfall.

We establish the Moran–Hutchinson formula for attractors of finite systems of surjective similitudes on semimetric spaces. More precisely, for a complete, normal semimetric space satisfying strong regularity and geometric doubling, we prove that the open set condition implies that the Hausdorff measure of the attractor at the similarity dimension is positive and finite. We also prove the converse, specifically, positivity of the Hausdorff measure at the similarity dimension implies the open set condition. This provides a partial answer to a question posed by Bessenyei and Pénzes in 2022.

We study the relation between linear stability of generated linear series on smooth curves and slope stability of their associated syzygy bundles. Motivated by conjectures of Mistretta and Stoppino, we establish new cases in which linear stability implies slope stability, focusing first on generated linear series over general curves and then on curves lying on polarized K3 surfaces. In the case of general curves, we use Brill-Noether-theoretic arguments to relate the numerical conditions on the linear series to the semi-stability of the syzygy bundle. For curves on K3 surfaces, we combine Lazarsfeld-Mukai bundles with Bridgeland stability conditions and restriction techniques to obtain slope-stability results under explicit degree bounds. These results provide further evidence for the expected equivalence between linear stability of linear series and slope stability of syzygy bundles.

Though the sharp $L^{2}$-Caffarelli–Kohn–Nirenberg (CKN) inequalities have been extensively studied in the entire Euclidean spaces, the corresponding problem on domains whose boundary contains the origin remains largely unexplored. We investigate the sharp $L^{2}$-CKN inequalities on half-spaces and orthants $\mathbb R^{n}_{k,+}$ by computing explicitly the optimal constants, determining all possible extremal functions, and establishing exact identities for the deficits. Since the singular weights $|x|^{-2b}$ rule out the lifting argument that is available for the simpler Heisenberg Uncertainty Principle, we develop an approach based on the transformations $u(x)=|x|^{m}v(x)$ for an appropriately chosen $m$ combined with spherical harmonic decompositions and weighted identities. Moreover, we establish weighted Poincaré inequalities associated with measures of the form \[ e^{-\delta|x|^{\tau}}|x|^{\beta}\bigl(\prod_{i=n-k+1}^{n}x_i^{2}\bigr)\,dx, \] together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincaré inequality. On the full orthant, the linear modes cease to be admissible competitors, since all odd spherical harmonics are annihilated by the lifting; the first non-radial mode is then of degree two, and both the sharp constant and the manifold of optimizers change accordingly. Finally, we establish several stability estimates, and second-order stability estimates, of the CKN inequalities on the half-spaces and orthants throughout the full parameter range.

We have previously, for a given positive integer $n$, explicitly characterized by certain linear inequalities the $\mathfrak{k}$-highest weight tableaux of shape length $2n$ or $2n-1$ produced by $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson (LR) rule. We now extend that characterization for lower shape lengths. Then using the composition of promotion operators defining the Naito-Suzuki-Watanabe bijection between $\mathfrak{k}$-highest weight tableaux and $\widehat{\mathfrak{g}}$-dominant tableaux, we also explicitly characterize by certain linear inequalities the corresponding $\widehat{\mathfrak{g}}$-dominant tableaux when the shape length is $2n$ or $2n-1$. In addition, when the given $n$ is $3$ or $ 4$ the $\mathfrak{k}$-highest weight and the $\widehat{\mathfrak{g}}$-dominant tableaux via $1$-$0$-slack recording tableaux are characterized by linear inequalities for lower shape lengths. Since recording tableaux in the quantum Littlewood-Richardson rule are in natural bijection with Littlewood-Richardson-Sundaram (LRS) tableaux, we relate our results on the inverse quantum LR rule with other two bijections for the Naito-Sagaki conjecture and establish bijections between $\mathfrak{k}$-highest weight tableaux, $\widehat{\mathfrak{g}}$-dominant tableaux and the lattice points in a (disjoint) union of flagged hive polytopes.

We consider the Euclidean distance geometry problem (EDG): given a subset of the pairwise distances of an unknown cloud of $n$ points in $\mathbb{R}^\ell$, recover the point cloud up to rigid motions. When $n$ is large, a popular practical approach is to minimize a nonconvex quartic, known as the squared-stress or s-stress, over point clouds in $\mathbb{R}^k$, with $k$ potentially larger than $\ell$. It is a long-standing open problem to understand the optimization landscape of the s-stress when all pairwise distances are known (Malone and Trosset, 2000; Parhizkar, 2013). It was recently shown that the landscape is not benign when $k=\ell$, and it was conjectured that the landscape becomes benign as soon as $k\ge \ell+1$ (Song et al., 2025; Criscitiello et al., 2026). Here, we show that the complete-graph s-stress has a benign landscape whenever $k\ge 2(\ell+1)$, establishing the conjecture up to a factor of two. A key idea is to view second-order criticality as a containment of two ellipsoids; finding a descent direction then corresponds to finding a separating hyperplane that violates this containment. This dual perspective yields the stated landscape result, and also applies to any measurement operator whose inverse satisfies a simple frame condition.

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

We study the quantum drift-diffusion, or Derrida-Lebowitz-Speer-Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt\varrho$. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^2(\Omega)$, explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint. The generated semigroup, which contracts the Hellinger distance between the densities, thus yields a canonical solution - existing, unique, and stable for every nonnegative $L^2$ initial datum and in every space dimension - independent of any approximation scheme: it is in fact the unique contraction semigroup extending the classical evolutions that emanate from smooth, uniformly positive data. The implicit Euler scheme converges to it, and $\sqrt u\in L^2_{\rm loc}(H^2)$ along the flow. When the datum belongs to the domain of the operator, the solution is strong and satisfies the equation pointwise, with no reaction term created on the vacuum $\{u=0\}$. We characterize the trajectories in several equivalent ways - as Bénilan integral solutions and through one-sided weak formulations - prove the maximality of the operator also in the $H^2$-$H^{-2}$ duality and, in dimension $d\le3$, identify the flow with the weak solutions in the uniqueness class of Fischer. A second-order estimate of independent interest underlies the construction: on a convex domain with Neumann conditions the dissipation $\int_\Omega(\Delta u)^2/u\,\mathrm{d} x$ is finite exactly when $\sqrt u\in H^2(\Omega)$, and it then controls the full Hessian of $\sqrt u$, in every dimension.

The aim of the present paper is to develop a spectral perturbation theory from a higher perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, our first main result establishes a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which only depends on the “data”, i.e. the limit eigenspace and the magnitude of the perturbation, through the resolution of a minimization problem. Remarkably, we make no assumptions on the perturbed eigenelements (besides, naturally, the spectral stability). Moreover, we cover both the cases of simple and multiple limit eigenvalues in full generality. In the second part, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights (also motivated by optimization in spectral geometry); secondly, we investigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Riemannian manifold changes when a second small manifold is glued on a small portion of it.

The first singular homology of a Peano continuum $X$ with torsion-free first Cech homology, $\check H_1(x)$, splits as $H_1(X) = \check H_1(X) \oplus K$ where $K$ is the homology shape kernel of $X$. Consequently if a Peano continuum $X$ is a subspace of $\mathbb R^3$, then $H_1(X) = \mathbb Z^\lambda \oplus K$ where $K$ is the homology shape kernel of $X$ and $\lambda$ is a countable cardinal. In the process we construct cotorsion quotients of subgroups of the first homology which correspond to path-connected fibrations of $X$.

Arrow allegories provide a convenient abstract framework to work with lattice-valued relations, or more precisely, relations that use the elements of a given Heyting algebra as truth values. One characteristic of arrow allegories is that all relations of the given arrow allegory use the same Heyting algebra ${\mathcal H}$. In this paper we want to extend this approach to allegories where relations between different objects may use different lattices of truth values and even further to relations that use a different lattice of truth values for every pair in the relation. Therefore, we define three concrete allegories, $\mathrm{Rel}({\mathcal H})$, $\mathrm{Rel}^u({\mathcal H})$ and ${\mathcal H}{\rm-Rel}$, where the allegory listed later is a full suballegory of the previous ones. These three allegories capture the three different situations mentioned above. In particular, ${\mathcal H}{\rm-Rel}$ is the standard example of an arrow category. We investigate these allegories and provide suitable categorical definitions for these structures.

We study empirical Bayes estimation of the prior in high-dimensional linear regression $\mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{\varepsilon}$, where the regression coefficients are drawn independently from an unknown sub-Gaussian prior. In contrast to the sequence model, the design matrix couples the latent coefficients, so that recovering the prior requires deconvolving it from both the noise and copies of itself. We introduce the Empirical Bayes Method of Moments (EBMoM), a computationally efficient procedure for general designs that recursively estimates the prior moments through a lower-triangular system of estimating equations and runs in time $O(np^2)$. Under mild design conditions, satisfied in particular by a broad class of correlated random designs, we show that EBMoM consistently estimates a growing number of moments and hence the prior itself, provided that $n\geq p^{1-o(1)}$. A matching information-theoretic lower bound, valid for a broad class of designs, shows that this sub-linear sample complexity is optimal for nonparametric prior estimation. This improves on existing results for likelihood-based methods whose consistency requires a linear sample size $n=\Omega(p)$.

This paper develops fluid antenna array (FAA)-inspired subarray sizing and transmit-power design for a two-hop amplify-and-forward (AF) unmanned aerial vehicle (UAV) relay using progressively contracting user-location posteriors. A contiguous reconfigurable subarray is shared by first-hop reception and second-hop forwarding, such that its active size jointly determines the receive gain, forwarding gain, and beamwidth. By adaptively controlling the effective aperture, the proposed design exploits geometric reconfigurability to balance array gain against pointing robustness under location uncertainty. Projecting the position covariance onto the array direction yields a closed-form direction-limited size inversely proportional to directional uncertainty. Posterior samples are propagated through the two-hop rate model, and the subarray size and transmit power are then selected to minimize UAV power subject to a worst-user lower-tail rate requirement and hardware power limits. The planned configuration is further audited over instantaneous two-hop Rician channels at the true user positions. At t = 8 s, the proposed design saves 3.17 dB over full-array narrow-beam transmission on paired feasible geometries and achieves 60.0% service success at a 0.15-W budget, compared with 43.2% for a fixed eight-element subarray.

We investigate the subspaces obtained by images of multilinear graded polynomials evaluated on the matrix algebra endowed with the canonical $C_n$-grading, where $C_n$ denotes the cyclic group of order $n$. Moreover, this graded algebra admits a natural action of $C_n$ arising from a fine refinement of the grading. We prove that every $C_n$-submodule of the trivial component of the grading is the image of some multilinear polynomial in graded variables. In particular, we answer in the negative a recent conjecture posed by T.~de Castilho and L.~Centrone (2023).

We extend a construction of Marker and Steinhorn for producing countable non-archimedean real closed fields which have no non-trivial automorphisms. The original construction produces such a field of transcendence degree two over the field of real algebraic numbers. We show how to use the same method to produce examples of all finite transcendence degrees and of countably infinite transcendence degree over the real algebraic numbers, answering a question of Marker and Steinhorn.

Fix $\varepsilon>0$, let $x>1$ be a large real number and let $\text{rad}(n)=\prod_{p\mid n}p$ be the radical of an integer $n\geq1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd(a,b,c)=1$, such that $c>(\text{rad}(abc))^{1+\varepsilon}$, is called exceptional triple. Recent works have proved that the cardinality $\#\mathscr{E}(x)$ of set $\mathscr{E}$ of exceptional triples satisfies $\#\mathscr{E}(x)=O(x^{33/50})$. This note proves that the cardinality of the exceptional set $\mathscr{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally for any $\varepsilon<2/35$.

We construct two families of non-binary entanglement assisted (EA) quasi-cyclic (QC) quantum low-density parity-check (QLDPC) codes over arbitrary finite fields, each possessing a precisely determined code rate. The first family is derived from a pair of non-binary classical QC-LDPC codes, designed such that the unassisted portion of the overall Tanner graph of the resulting EA-QC-QLDPC code is free of 4-cycles. The second family, on the other hand, is constructed from a single non-binary classical QC-LDPC code whose Tanner graph itself is 4-cycle-free. In developing the codes belonging to the first family, we employ a single Bell pair to establish entanglement between the transmitter and the receiver, thereby minimizing the required entanglement resources. Furthermore, these constructions demonstrate that careful graph-based design can effectively balance error-correction performance with entanglement consumption, providing a practical approach for realizing efficient non-binary EA-QC-QLDPC codes.

Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence–convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-$\beta_2$ regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.

Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k<n/2$, there are convex potentials with a $k$-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent \[ p_{n,k}=\frac{n(n-k)}{2k}. \] Their Hessians belong to $L^p_{\loc}$ precisely for $p<p_{n,k}$, lie in the weak endpoint space $L^{p_{n,k},\infty}_{\loc}$, and fail to belong to any finite-index Lorentz endpoint. This matches the critical minimum-set theorem of Collins and Mooney. At the regularity required in the question, the construction produces branch sets of every integer dimension $k<n/3$. We also give a topological reformulation of the problem. In the convex branch, gradient fibers are exactly contact sets with supporting affine functions, and openness and discreteness are equivalent to strict convexity. For a general gradient under the hypotheses above, critical Sobolev mapping theory already supplies continuity and sense preservation. The question asks whether the monotone factor in the Eilenberg–Whyburn monotone–light factorization is trivial.

We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) 58 (2025), no.~4, 943–970], which treats $m$-fold symmetry solutions with $m\geq4$. We prove that every nonconstant $W^{1,p}$ solution satisfies $\|g_\theta(t)\|_{L^p}\to\infty$ as $t\to\pm\infty$ for $1<p\leq\infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D_\theta g_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{\alpha,r}$ for $\alpha r<1$. For such data, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit. This structural assumption on $D_\theta g_0$ is automatic for $C^1$ data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. 10 (2023), no.~1, 1–100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for $m\geq4$, they cover the full well-posed scale-invariant range $m\geq3$.

An essay, based on a public lecture delivered at the 2026 International Congress of Mathematicians, on how the mathematical community might respond to the arrival of artificial intelligence tools that are capable of performing research-level mathematical tasks. Rather than debating the capabilities of such tools, we condition on the hypothesis that these capabilities will arrive, and examine instead a question that is orthogonal to it: what the goals and values of mathematical research actually are. The problem-solving component of mathematics is used as a case study.

We consider the probability that the convex hull of the first $n$ partial sums of a $d$-dimensional random walk contains the origin. Under symmetric exchangeability of the increments and a general-position assumption, this absorption probability is distribution-free and admits an explicit formula, previously obtained by Kabluchko, Vysotsky and Zaporozhets [Geom. Funct. Anal. 27 (2017)] using characteristic polynomials of hyperplane arrangements. We give a different proof, based on a wall-crossing method which we develop here. Starting from a deterministic configuration of increments, we count the signed permutations for which the convex hull of the corresponding partial sums contains the origin and show that this count remains unchanged under generic deformations of the increments, and hence is the same for all configurations outside a natural exceptional set of measure zero. Evaluating the invariant at a single well-chosen configuration reduces the remaining calculation to the enumeration of permutation records combined with Wendel's theorem. Our method also reproves Wendel's theorem on convex hulls of random points with a sign-flip-invariant joint distribution and, in dimension one, Sparre Andersen's theorem. Finally, we derive new probabilistic representations and recurrence relations for the absorption probabilities of random-walk convex hulls and their random-bridge analogues.

We propose a refinement of the quantum Hikita conjecture of Kamnitzer, McBreen, and Proudfoot that bridges the representation theory of Coulomb branches with the enumerative geometry of Higgs branches. We also introduce a general framework for proving it, which we carry out for ADE quiver gauge theories with minuscule framings and for the gauge theory corresponding to the Jordan quiver. As an application, we use the resulting quantum Hikita isomorphisms to give a geometric description of graded traces on quantized Coulomb branches.