We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^d\rightarrow\mathbb R^m$, where $M^d$ is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of $M$ is greater than $50\log((n+1)!)$, then for each dimension $0<k<d-m$ there is a point $z\in\mathbb R^m$ such that any cell structure on the fiber $p^{-1}(z)$ has more than $n$ cells of dimension $k$. The proof is based on a freedom theorem for ideals in group rings of hyperbolic groups proved in arXiv:2309.16791.

We obtain an optimal proximity bound for integer linear programs in standard form max{cx: Ax=b, x nonnegative integer}, where A is an integer mxn matrix of rank m<n and b is an integer vector. Specifically, we show that the Euclidean distance from any optimal vertex solution of the LP relaxation to a nearest optimal integer solution is bounded by $\sqrt{\det(AA^t)}-1$ and that this estimate is asymptotically tight. We also derive bounds for the optimal integer solutions involving the product function $\prod_{i=1}^{n}(x_i+1)$ and discuss their applications in the knapsack setting.

Koopman linearization opens many possibilities for control synthesis and analysis of nonlinear systems. Whether or not any given nonlinear control system admits a finite-dimensional Koopman representation remains a crucial question to address. A related problem is to categorize the class of all Koopman linearizable nonlinear control systems. In this work, we present differential geometric conditions on the drift and control vector fields of a control-affine nonlinear system, that must be necessarily satisfied for Koopman linear transformation to exist. The same conditions are also shown to be sufficient for (a slightly weaker notion of) Koopman linearizability on control-invariant manifolds. Further, these conditions, together with an additional condition, become necessary and sufficient for Koopman linearizability to a controllable linear system. Our examples illustrate the ease of checking these conditions, and also shed light on how Koopman linearizing transformation may not exist for a control-affine system even though one can linearize the autonomous part of the system via Koopman lifting.

We study dimension-free Markov--Bernstein inequalities for polynomials with respect to product probability measures. In the Gaussian case, for $p\ge4$, we prove that \[ \|\nabla f\|_{L^p(γ^n)} \le C(p)d^{\frac12+θ_p} \|f\|_{L^p(γ^n)} \] for every polynomial $f$ of degree at most $d$, where $θ_p\le \frac{2}{3p}$ and $θ_p=0$ whenever $p$ is an even integer. Thus, for even integer exponents, we establish the sharp dependence on the degree conjectured by Eskenazis--Ivanisvili. For general $p\ge4$, the estimate improves upon their dimension-free inequality. We also obtain dimension-free Markov--Bernstein inequalities with sharp dependence on the degree for even integer exponents beyond the Gaussian setting. We first prove such estimates for the uniform distribution on the unit cube and then extend them to products of absolutely continuous measures with unimodal densities. Finally, we treat products of one-dimensional Freud measures with densities proportional to $e^{-|t|^{2m}}$.

This paper investigates the Kähler hyperbolicity modulus on complete Kähler manifolds, with a particular focus on hyperconvex domains and bounded strongly pseudoconvex domains. Our main result establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function. As applications, we compute the Kähler hyperbolicity modulus for bounded symmetric domains. Furthermore, we obtain lower bounds for the Kähler hyperbolicity modulus on bounded strongly pseudoconvex domains equipped with Kähler-Einstein metrics or Bergman metrics.

We present a proof of Nash's smooth embedding theorem with an emphasis on accessibility and rigor.

We study the large sample asymptotic behaviour of the Maximum Likelihood Estimator of the discount and strength parameters $(α,θ)$ in the Ewens--Pitman model for random partitions, under mild assumptions on the data-generating mechanism. We show that four distinct regimes arise, depending on the limiting behaviour of the frequency spectrum. In particular, in contrast with previous work, we find that $θ$ may play a crucial role asymptotically. We further show that the existing literature implicitly focuses on only two of these regimes, and we relate this restriction to the constraints imposed by infinite exchangeability. Under the latter, indeed, the number of distinct blocks and the frequency spectrum are necessarily tied by a rigid structural relation. We prove that this lack of flexibility can be overcome through what we call the scaled Ewens--Pitman model, in which $θ$ is allowed to grow with the sample size $n$. Finally, we provide empirical evidence from real-world data showing that such extensions are needed to capture frequency spectra that fall outside the classical Ewens--Pitman framework.

We prove that a large class of $2$-step solvable Lie algebras equipped with a complex structure $J$ admits a Levi-Malcev type decomposition, adapted to $J$. As an application, we prove that the Fino--Vezzoni conjecture holds true for $2$-step solvable unimodular Lie algebras. Finally, we give a structural characterisation of $2$-step, unimodular, completely solvable Lie algebras admitting an SKT metric.

Singular volume integral operators associated with constant-coefficient partial differential operators extend the applicability of potential theory to inhomogeneous problems, for example arising from nonlinearities or variable coefficients. Typically the PDE kernels in these operators give rise to singularities at all $\mathcal{O}(1/h^3)$ volume discretization/evaluation points in a mesh of characteristic size $h$, while the slowly-decaying nature of such kernels give rise to long-range interactions that require coupling to fast summation algorithms. The presented method uses Green's identities to regularize a wide variety of both scalar-valued and vector-valued volume integral operators by use of a certain regularizing volume density interpolant. The analysis shows how the regularizing effect of the interpolant is global in the sense that the interpolation quality increases in an exactly compensatory fashion as the distance to the Green's function singularity decreases. High-order convergence estimates with tabulated simplex quadratures are established, including with exact representation of curved domains.

This is essentially the content of a talk at the workshop on Character Sheaves on Loop Groups at the University of Minnesota (June 8,2026). It describes the early days of the theory and also some recent developments.

We prove Hadamard ill-posedness for the non-autonomous linearised Prandtl equations around time-dependent shear-flow equilibria in function spaces up to Gevrey class 4. More precisely, we construct compactly supported smooth initial data, Gevrey class 4 in the tangential variable, for which the system admits no weak solution for any positive lifespan. In this regard, we improve previous results by showing that classical semigroup-type instabilities do not, by themselves, imply Hadamard ill-posedness in the non-autonomous case when the initial time is not a variable of the system. Our argument is based on a family of exact unstable modes whose $L^2$-norms grow, at tangential frequency $k$, like $\exp(c \sqrt k t)$ up to times of order $t \sim k^{-1/4}$. Their construction relies on an inner-outer gluing scheme, in the spirit of matched asymptotic expansions, which combines unstable inner solutions near a non-degenerate critical point with an exact outer solution and yields exponentially small matching errors for short times.

A skew-Hadamard difference set (SHDS) in a finite group $G$ is a classical combinatorial object with deep connections to design theory, coding theory, group theory, and the construction of Hadamard matrices. Even though the abelian case has been extensively studied -- with strong structural constraints known, such as the necessity of $G$ being a $p$-group for some prime $p \equiv 3 \pmod{4}$ -- there are still some open questions regarding existence of SHDSs for the abelian case. The non-abelian case remains largely unexplored, despite the known existence of non-abelian SHDSs. In this paper, we establish new necessary conditions on the order and structure of a finite group $G$ that admits an SHDS. These results provide the first general structural restrictions for SHDSs in non-abelian groups. In particular, we prove that if a group $G$ is nilpotent and admits an SHDS, then $G$ is a $p$-group. Our method makes use of the structure of the rational group algebra, and completely avoids the use of the group characters.

In this paper, we study the structure of Kubo-Ando means on the cone of positive Hermitian matrices over the real numbers, complex numbers, and quaternions. Given a Kubo-Ando mean $σ$ with representing function $f$, we obtain an explicit decomposition of $\text{A} σ\text{B}$ in terms of the spectrum of $\text{A}^{-1}\text{B}$. More precisely, we show that $\text{A} σ\text{B}$ can be expressed as a finite linear combination of matrices of the form $\text{A}\left(\text{A}^{-1}\text{B}\right)^{k}$, with coefficients depending only on $f$ and the eigenvalues of $\text{A}^{-1}\text{B}$. We first investigate the linear case and characterize the pairs of matrices for which every Kubo-Ando mean admits an affine representation. We then focus on the cone $\mathscr{P}_{3}(\mathbb{D})$, where we derive explicit formulas for the decomposition coefficients in terms of spectral invariants. Finally, we show that the same techniques extend to a broad class of alternative means, yielding explicit decompositions in the commutative setting and extending recent results of Choi, Kim, and Lim.

Urban waste management faces increasing operational and environmental challenges driven by population growth, heterogeneous waste streams, traffic congestion, and the need for sustainable collection infrastructures. We present an integrated optimization framework for the design of multi-type urban waste collection and recycling systems. In the proposed network, waste generated at distributed collection points is gathered by waste-specific small vehicles and transported to intermediate consolidation facilities, where it is aggregated before being transferred to treatment plants using larger vehicles. The problem consists of jointly determining the location of consolidation facilities and treatment plants, together with the associated two-echelon routing operations, under a limited infrastructure budget. We develop an integrated location-routing model that simultaneously captures facility-location decisions, multi-product routing structures, vehicle-capacity restrictions, and repeated unloading operations. The objective is to minimize total operational costs, including transportation, routing, and handling activities, while satisfying investment-budget constraints. The framework promotes compact collection regions, improves vehicle utilization, and reduces unnecessary transportation effort in dense urban environments. Computational experiments based on realistic urban scenarios derived from the city of Medellín (Colombia) demonstrate the effectiveness of the proposed approach. The results show that integrated infrastructure and routing decisions can substantially improve operational efficiency while supporting more sustainable recycling operations. The framework provides a practical decision-support tool for urban planners seeking to design cost-efficient and environmentally responsible waste-management systems.

In 2018, Hamaker and Reiner introduced weak order for monotone triangles, which extended the usual notion of weak order on the symmetric group. Monotone triangles on $\{1, \ldots, n\}$ are well-known to be in bijection with the set ASM$(n)$ of $n \times n$ alternating sign matrices. Hamaker and Reiner defined weak order on ASM$(n)$ to be induced from weak order on monotone triangles via the standard bijection. Recently, the present authors used an a priori different definition of weak order on ASM$(n)$ to give a combinatorial characterization of the codimension of ASM varieties and to show that the natural K-theoretic representatives of these varieties satisfy a divided difference recurrence. In the present work, we establish compatibility of these definitions of weak order on ASM$(n)$. Additionally, we give three different explicit means of computing weak order covering relations on ASM$(n)$: on ASMs themselves, on monotone triangles in a manner different from that given by Hamaker and Reiner, and on bumpless pipe dreams, which are a newer family of combinatorial objects also in correspondence with ASMs. Finally, using the language of bumpless pipe dreams, we characterize the fibers of the weak order operators, each of which forms a sublattice of the strong Bruhat order on ASM$(n)$.

We solve a long-standing problem of whether the homology groups of the Torelli subgroups $\mathcal{I}_g\le\mathrm{Mod}_g$ are finitely generated in stable range. Namely, we prove that the group $H_k(\mathcal{I}_g;\mathbb{Z})$ is finitely generated, provided that $k\le g-2$. Two main ingredients of our approach are as follows. First, we show that the action of any symplectic transvection $t_x\in\mathrm{Sp}_{2g}(\mathbb{Z})$ on the homology of $\mathcal{I}_g$ satisfies the following unipotency condition: $(t_x-1)^{k+1}H_k( \mathcal{I}_g;\mathbb{Z})=0$. The proof of this fact relies on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of homologous curves on $Σ_g$. The second key ingredient is Tavgen's theorem asserting that the group $\mathrm{Sp}_{2g}(\mathbb{Z})$ is boundedly elementarily generated. For homology with coefficients in $\mathbb{Q}$, we further prove that $H_k(\mathcal{I}_g;\mathbb{Q})$ is an algebraic $\mathrm{Sp}_{2g}(\mathbb{Z})$-representation in the same stable range $k\le g-2$. Kupers and Randal-Williams have obtained a conditional result: they computed the algebraic part of the rational cohomology of Torelli groups in stable range under the assumpition that the rational cohomology groups are finite-dimensional in this stable range. Our results turn this conditional computation into a precise theorem that describes the whole rational cohomology ring of Torelli groups in stable range. As further applications, we, firstly, prove Morita's conjecture asserting that the $\mathrm{Sp}_{2g}(\mathbb{Z})$-invariant part of the rational cohomology of $\mathcal{I}_g$ stabilizes to the polynomial ring $\mathbb{Q}[e_2,e_4,\ldots]$ in the even Miller-Morita-Mumford classes; secondly, we prove the uniform representation stability for the series of groups $\left\{ H_k\bigl(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}$.

Cospectral graphs (graphs that share the same eigenvalues) expose the limitations of using the graph spectrum to uniquely identify graphs, and they also help to understand what structural properties a graph spectrum cannot capture. Switching methods, which are standard tools for constructing cospectral graphs, require specific structural and algebraic conditions to hold for the operation to preserve the graph's spectrum. However, there is no guarantee that the obtained cospectral switched graph is non-isomorphic. In this paper we study this isomorphism problem for a recent and prolific switching method to produce cospectral graphs with respect to the adjacency spectra: Wang-Qiu-Hu (WQH) switching. We do so by using common-neighbour multisets associated with a WQH partition, which allows us to derive an external common-neighbour criterion for certifying non-isomorphism after WQH-switching. Then, we apply the new criterion to clique extensions and to weak tensor products, with coclique extensions as a special case. As an application we obtain infinite families of cospectral non-isomorphic graphs, including some known constructions. Finally we extend the conditions of WQH-switching to generalized adjacency matrices and, under an additional degree condition, to Laplacian and signless Laplacian matrices.

Building on work of Ransford, we prove that whenever $E$ is a closed subset of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ such that if $T$ is a contraction on a Hilbert space with $σ(T)\subset E$ and $\|T^{-n}\|=O(u_n)$, then $T$ is unitary. This confirms conjectures of Esterle and Ransford. Our main new idea is a spikes-in-collars principle for positive subharmonic functions.

Hypergraph edge partitioning is a central problem in theoretical and applied computer science, with broad impact on distributed computation, communications, optimization, and machine learning. In this setting, one is given a collection of hyperedges -- each consisting of up to $d$ vertices from a ground set of size $n$ -- and seeks to assign these hyperedges across $N$ partitions so as to minimize, for example, the vertex footprint, i.e., the maximum number of vertices that appear in any partition. We here identify the fundamental limits of hypergraph edge partitioning -- optimized over all conceivable algorithms -- for a broad class of probabilistic hypergraph models where each hyperedge may appear independently with \emph{its own} probability; a model sufficiently general to encompass well-known models such as the Degree-Corrected or Mixed-Membership models, the Hypergraph Stochastic Block model, the Latent-Space/Geometric or Kernel Models, and others. By pairing our deterministic partitioner with a new converse, we first show that, for any $n,d$, and under the very mild condition of $N \leq \binom{\lfloor\sqrt{\frac{nd}{2}}\rfloor}{d}$, as long as the hyperedge set $\mathbf{X}$ satisfies $|\mathbf{X}| \gtrsim n N \log N$, then with probability at least $1-2/3n^z$, no algorithm can provide a footprint $π_{\mathbf{X}}$ less than $$π^{\bigstar}_{\mathbf{X}} = \frac{1}{2\sqrt{2}}\frac{n}{N^{1/d}}. $$ We then show that our hypergraph partitioner comes to within a small constant factor from $π^{\bigstar}_{\mathbf{X}}$, for each $\mathbf{X}$. This optimality captures dense and sparse hypergraphs alike (with sizes down to linear in $n$), and it additionally entails a near-optimally balanced allocation of hyperedges across partitions.

A generalised inverse first passage problem asks whether, given a probability measure $p$ on $[0,\infty]$, one can find a boundary $b:[0,\infty]\to \mathbb{R}$ such that the stopping time:\[τ:=\inf\left\{t:Λ\int_0^t ω(W_s-b(s))ds \geq U\right\}\] has distribution $p$, where $U\sim Exp(1)$, $Λ\in(0,\infty)$ and $ω$ is a monotonic decreasing function. We construct a branching-selection particle system whose hydrodynamic limit is governed by a free boundary problem and connect this to the generalised inverse first passage problem. In the $N$-particle system, particles move as independent Brownian motions, branch at a prescribed rate, and are removed at a rate proportional to their location relative to a position $b^N(t)$ which is a function of the empirical distribution. We identify the limit of $b^N$ as the solution of the inverse first passage problem.

We present a multilevel B-spline-based fast integral method for the solution of the electric field integral equation (EFIE), combining fast Fourier transformation (FFT)-compatible kernel interpolation with robust high-order interpolation. Existing FFT-accelerated global Lagrange-based approaches rely on equidistant interpolation points and can, therefore, suffer from Runge-type instabilities at high interpolation orders, limiting robust high-accuracy compression. In contrast, B-splines on equidistant knot vectors overcome these instabilities and enable robust high-order interpolation for accurate matrix compression. Replacing Lagrange interpolation by B-spline interpolation is, however, non-trivial: B-spline coefficients do not coincide with function values at the interpolation points, and the associated sampling matrices can become ill-conditioned. To address these challenges, we introduce a knot-removal stabilization strategy, combined with exact interlevel transfers based on knot insertion, yielding accurate, well-conditioned multilevel interpolation. Moreover, we propose a factorization strategy that preserves the null space of the scalar potential operator up to machine precision and is compatible with low-frequency preconditioning techniques. Numerical results for both canonical and realistic geometries demonstrate robust high-order interpolation without the breakdown observed for Lagrange-based approaches and confirm $\mathcal{O}(N)$ complexity.

In this paper we deal with integral Hardy type inequalities on finite segments. The interval is assumed to be finite and avoiding the origin. We prove new sharp $L_2$-inequalities and their $L_p$-analogues. Constants in the proven inequalities depend on the first root of the corresponding Lamb type equation for the Bessel function. In the $L_2$-case the extremal function is found. We consider Hardy-type inequalities in differential form. Using the one-dimensional Hardy inequalities, we establish an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli.

A finite subset $X$ of $\mathbb R^d$ is called Ramsey if for every $k$ there exists an $n$ such that whenever $\mathbb R^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. K\v rí\v z showed that if there is a soluble group of symmetries of $X$ that acts transitively on $X$, then $X$ is Ramsey. Determining which sets are Ramsey is a major unsolved problem. In this paper we show that if there is a finite group of isometries of $\mathbb R^d$ that acts transitively on a set $X$, and also on a set $Y$, then the `prism' formed by $X$ and $Y$ in $\mathbb R^{d+1}$ (meaning the set $X$ together with a translate of $Y$ in the direction perpendicular to $\mathbb R^d$) is itself contained in a finite set on which a group of isometries acts transitively. Moreover, if the initial group of isometries is soluble then so is the final group. This provides a new tool for generating Ramsey sets.

This paper introduces pro-tangles, a natural generalization of classical tangles, which are functors from the Boolean cube to Bar-Natan's cobordism category. By employing the simplicial Yoneda embedding, we construct the Khovanov simplicial presheaf of a pro-tangle as a homotopy colimit and prove that this simplicial presheaf is representable, with representing object the classical Khovanov simplicial object. We establish a fully faithful embedding showing that the weak equivalence class of this simplicial presheaf is determined by the chain homotopy type of the Khovanov complex. Furthermore, we utilize Boolean cube decompositions to construct an algebraic spectral sequence for pro-tangles. This spectral sequence converges to the total Khovanov homology, and its $E_1$ page is explicitly expressed in terms of the Khovanov homology of reduced tangles. This categorical setup yields a functorial interpretation of Reidemeister invariance in terms of morphisms of spectral sequences. By applying the tangle TQFT construction, we study this spectral sequence for Hopf clasps, the fundamental structural building blocks in tangle and link theory. We show that the spectral sequence collapses at the $E_3$ page, which further specializes to an $E_2$-collapse under the restriction to Hopf sums. Finally, we investigate connected sums of pro-tangles and pro-links. To address the module-action dependencies arising from tensor products in multi-connected sums, we introduce a state-dependent modified tensor operator and prove a structural decomposition theorem that generalizes the classical result at the chain complex level.

We introduce a new product on the ambient space $F_q^n$ as a generalization of Euclidean, Hermitian and $δ$ products. We give some general properties of the dual codes, relation with Euclidean duals, definition and characterization of self orthogonal, self dual, dual containing and LCD codes along with certain existence conditions. Also, we calculate the dual codes of some classes of codes like repetition, binary and $λ$-constacyclic codes with respect to this product. Further, we extend and analyse this notion of the product for codes over semisimple rings.

We show that $χ(X,T_X)<0$ for the currently known families of smooth Fano fourfolds of Picard rank $1$ and index $1$. Combining this with the known Picard rank $1$ index $> 1$ cases, we show that among all currently known smooth Fano fourfolds of Picard rank $1$, the only variety satisfying Bott vanishing is the projective space. By a result of Kawakami--Totaro, the existence of an endomorphism of degree greater than 1 implies Bott vanishing. Therefore, among the currently known smooth Fano fourfolds of Picard rank $1$, any variety admitting an endomorphism of degree greater than 1 must be $\mathbb P^4$. Together with Burt Totaro, we develop new Schubert2 functions for symmetric and skew-symmetric degeneracy loci, and weighted projective spaces.

We prove a distributional convergence result for a multidimensional version of symmetric cooperative motion which was introduced and studied in one dimension in \cite{HRW, SCM1}. Our approach relies on framing the associated recursive distributional equation as a discretization of the porous medium equation. A major challenge is to analyze the behaviour of finite difference schemes which approximate weak solutions of the porous medium equation with unbounded initial data. In overcoming this difficulty, we perform a detailed analysis of the probability mass function of symmetric cooperative motion, in which we introduce several new comparison arguments for the discrete process. Consequently, along the way, we establish a novel multidimensional convergence result for a finite difference scheme approximating the ZKB/Barenblatt solution of the porous medium equation, which is of independent interest.

The classification of curvature homogeneous hypersurfaces in real space forms was established by Tsukada in 1988, with the remaining rank-two cases in $\mathbb{S}^4$ and $\mathbb{H}^4$ settled by Bryant-Florit-Ziller in 2025. It is obvious that curvature homogeneity implies constant Ricci eigenvalues. In this paper, we prove that for hypersurfaces in real space forms, the converse also holds: a connected hypersurface immersed in real space forms has constant Ricci eigenvalues if and only if it is curvature homogeneous. Hence, hypersurfaces with constant Ricci eigenvalues in real space forms are also classified, which, in particular, generalizes the classification of Einstein hypersurfaces obtained by Lawson for minimal hypersurfaces and by Ryan for general cases in 1969. Moreover, as a byproduct, curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues can not be isometrically immersed in any real space form of codimension one. Finally, we show that a hypersurface with constant Ricci eigenvalues is isoparametric if it is a complete hypersurface either in $\mathbb{S}^{n+1}$ or nonflat in $\mathbb{R}^{n+1}$ for $n \geq 3$; or if it is not of constant sectional curvature $-1$ in $\mathbb{H}^{n+1}$ for $n \geq 5$.

Motivated by the polygonal Pólya--Szegő conjecture for torsional rigidity, we study two monotonicity problems for torsional rigidity. The first concerns a mixed torsion problem on fixed-area right triangles, with a Dirichlet condition on one leg and Neumann conditions on the other leg and on the hypotenuse. We prove that the mixed torsional rigidity strictly increases as the ratio of the Neumann leg to the Dirichlet leg increases. The proof uses a Hadamard shape derivative, a Pohozaev-type identity, and a monotonicity result for the mixed torsion function. We also prove a similar result for the mixed ground state of Laplacian. The second concerns regular polygons. If \(P_N\) denotes the regular \(N\)-gon of area \(π\), we prove, by a purely analytic Schwarz--Christoffel/Bergman analytic-content argument, that \[ T^D(P_{N+1})>T^D(P_N),\qquad N\ge3, \] where \(T^D\) is the Dirichlet torsional rigidity. We also obtain the asymptotic expansion \[ T^D(P_N)=\fracπ{8}-\frac{πζ(3)}{N^3} +\frac{π^5}{45N^4}+O(N^{-5}). \]

We introduce a new class of quadratic in the momenta Hamiltonians with potential on Banach spaces of infinite dimension which possess an infinite family of conserved quantities in involution with respect to a naturally defined Poisson structure. We show that the Hamiltonian flows of the constructed integrals are locally well defined and that they are related to a hierarchy of evolution PDEs via nonlinear analytic transformations of the phase space, called $Q$-transforms. Our framework offers a new geometric interpretation of hierarchies of integrable PDEs, linking infinite-dimensional phase spaces directly to the classical geometric phenomenon of geodesic equivalence.