We discuss birational properties of actions of finite groups on algebraic varieties, linearizability, torsors, and versality.

We consider a problem to develop a framework for model identification adhering to the tenets of neuromorphic computation, without resorting to neural networks as the mathematical substrate. In particular, all computations take place in continuous time. We are naturally led to adaptive observers, where the main technical obstacle is the possible mismatch between the unknown plant order and the observer order. The key concept that informs the proposed framework is an overparameterized model, an input-output equivalent model that provides a suitable parameterization in the overmodeled case, with theoretical extensions also addressing the undermodeled case. A discrete algorithm orchestrates successive experiments to incrementally learn the model order, while a standard parameter adaptation law learns the parameters.

We present more than seventy optimal degree-diameter Cayley graphs that are non-isomorphic to previously known optimal Cayley graphs with the same degree and diameter. For each graph, we provide comprehensive data on its parameters and structural invariants. Comparison with known constructions reported by Marston Conder on the Combinatorics Wiki and in related works reveals significant differences in girth, algebraic connectivity, domination number, automorphism group structure, cycle distributions, distance-related properties, and other graph invariants. These findings show that optimal degree-diameter Cayley graphs are frequently non-unique. The new examples enlarge the catalogue of known optimal graphs and provide benchmarks for investigations in degree-diameter problems, algebraic graph theory, and extremal graph theory. Since some of the graphs were identified through randomized search techniques, the collection is not exhaustive.

We study topologies on Ran spaces. In the literature, two distinct topologies frequently appear: the Hausdorff topology, and a finer one constructed as a colimit, that we call the final topology. In this work, given a metric space $M$, we construct new metric topologies on $\mathrm{Ran}(M)$, called weighted topologies. They interpolate between the Hausdorff and final topologies, the later being recovered as a limit in the category of spaces. This structure equips the final topology with a uniformity, which we show to be complete. Finally we study the Ran spaces as stratified spaces. We show that, whenever $M$ is a Riemannian manifold, the weighted topologies are conically stratified, while the final topology is only so in a weak sense.

We study efficient online methods for dynamic inverse problems with infinite time horizon. We concentrate, in particular, on problems whose forward model arises from a PDE. Our motivating application is flow monitoring with Electrical Impedance Tomography (EIT). The idea of such online methods is to take single steps of of standard optimisation algorithms, on each time index; each data frame. A predictor, based on problem dynamics, is used to transfer iterates one from time index to the next one. If we monitor a fast flow with a correspondingly fast measurement modality, such as EIT, basic methods are unable to solve the PDE before new data arrives. Our idea, then, is to not solve it, and instead, on each iteration, each time index, take single or few steps of standard iterative solvers towards the solution of both the PDE and an adjoint PDE. This is what ``single loop'' refers to. To the overall problem, we apply standard online optimisation methods, at the outside developed for exact gradients $\nabla E_k(x^k)$ of the iteration-dependent data fidelity $E_k$ that incorporates the PDE. We replace the gradient by a single-loop estimate $\tilde\grad E_k(x^k)$ that satisfies standard smoothness properties with summable errors. This allows standard regret proofs to go through. Our numerical experiments on dynamic EIT validate the theoretical predictions and highlight the potential of the proposed approach for the real-time solution of PDE-constrained dynamic inverse problems.

The concept of $ρ$-frequent hypercyclicity is introduced in order to provide a refined form of frequent hypercyclicity. This is achieved by replacing the denominator in the definition of frequent hypercyclicity by an appropriately chosen calibration function $ρ$. A $ρ$-Frequent Hypercyclicity Criterion is determined and the $ρ$-frequent hypercyclicity of weighted backward shifts is investigated.

In this work, we propose a novel approach, based on the path integral formulation of Koopman spectrum, to discover the phase-space geometry of the planar Cislunar Restricted 3 Body Problem (CR3BP). In contrast to existing techniques, which use trajectory-based (usually) local analysis, we leverage the Koopman operator framework, which generates a global linear \emph{representation} of the system, to reconstruct the global phase space geometry of the CR3BP. In particular, we compute the principal eigenfunctions of the Koopman operator via the path integral approach and show how the zero level curves of these eigenfunctions encode the phase space characteristics of the planar CR3BP.

We study representations of \(1/n\) as a sum of three unit fractions whose denominators are all divisible by a prescribed integer \(m\). After scaling, this is equivalent to representing \(m/n\) as a sum of three unit fractions. Our main focus is the Erdős--Straus case \(m=4\). We introduce a divisor-based function \(\mathrm{fab}(n,a,b)\), prove that its admissible parameters recover exactly the decompositions of \(1/n\) with all three denominators divisible by \(4\), and compare this parametrization with well-known Type I/II descriptions.

For a finite dimensional algebra $A$ of finite global dimension we study the Auslander--Yoneda algebra defined as the Yoneda algebra of the direct sum of all indecomposable $A$-modules. We show that the Auslander--Yoneda algebra is an Auslander--Gorenstein algebra if and only if every indecomposable left and right $A$-module is spherical in the sense of Auslander and Bridger. This motivates the study of spherical algebras defined by the condition that every indecomposable module is spherical. We characterize spherical algebras by a certain natural pair of subcategories being a split torsion pair. Moreover, we prove that representation-finite algebras which are spherical are directed and give a full classification of spherical Nakayama algebras. Furthermore, we show that replicated algebras of hereditary algebras are spherical. As a final application of the new notion of spherical algebras, we give a negative answer to a question of Venjakob on Auslander regular algebras in general, but show that there is a positive answer when assuming that every indecomposable left $A$-module is spherical.

We consider maps between two Riemannian manifolds and study a functional given in terms of the $L^\infty$-norm of the derivative. This functional is not differentiable, but we can define critical points with the help of a subdifferential. The resulting notion includes, for example, minimisers in a given homotopy class. We derive a geometric condition equivalent to criticality in this sense. The condition is formulated in terms of a vector-valued $1$-current on the domain manifold, which encapsulates some of the key properties of the critical point. Moreover, this $1$-current is itself a critical point of a generalised mass functional.

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperkähler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperkähler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

This paper shows that the Alexander-Briggs code of a knot gives rise to a new invariant that distinguishes prime alternating knots. The restriction to prime alternating knots precisely follows from the fact that our approach relies on Tait s flyping theorem. We also provide examples where the new invariant succeeds in separating knots that the well known invariants, such as some knot polynomials, fail.

We give a structural account of the finite free convolutions of Marcus, Spielman, and Srivastava in terms of reproducing kernel inner products on polynomial spaces and a multilinear model over the squarefree algebra. In this model, additive convolution becomes algebra multiplication, and the nilpotent logarithm linearizes it, recovering the finite free cumulants of Arizmendi and Perales. This perspective leads to a class $\mathcal{LC}_n$ of multilinear polynomials characterized by nonpositivity of higher-order cumulants, closed under additive convolution and satisfying several key permanence properties associated with negatively dependent measures. We show that every graph Laplacian pencil belongs to this class, with higher-order cumulants given by Hamiltonian cycle counts in induced subgraphs.

This article studies the high-frequency behavior of eigenstates of perturbations of the Laplace-Beltrami operator on the two-sphere $\mathbb{S}^{2}$ by a measure supported on an equator. We are interested in understanding to what extent this behavior can be described in terms of the geodesic flow of the sphere. This is done by analyzing quantum limits and semiclassical measures of sequences of high-frequency eigenfunctions, which describe how their $L^2$-masses concentrate in phase space. When the Laplacian on $\mathbb{S}^{2}$ is perturbed by a bounded potential, it is known that the family of all possible semiclassical measures is contained in the set of positive measures on the unit cosphere bundle $S^*\mathbb{S}^{2}$ that are invariant under geodesic flow (with equality in the unperturbed case). In this article, we show that the presence of a singular delta potential on a closed geodesic results in the existence of sequences of eigenfunctions whose semiclassical measure is not invariant under geodesic flow. In particular, one can find a sequence of eigenfunctions whose energy asymptotically concentrates on the hemisphere bounded by the equator on which the potential is concentrated.

These are lecture notes for the PCMI 2024 Graduate Summer School for the mini-workshop on motivic explorations in enumerative geometry. Motivic homotopy theory allows to do enumerative geometry over an arbitrary field, which leads to additional arithmetic and geometric information. The goal of the mini-workshop is to explain why and how this works. We will also provide a toolbox for solving enumerative geometry problems in this setting, including the use of tropical geometry. We start with two classical examples in enumerative geometry, namely Bezout's theorem and the count of lines on a smooth cubic surface. We then explain how to solve these problems, first over the complex and real numbers, and then over an arbitrary field, using the A1-degree from motivic homotopy theory. Then we introduce tropical geometry, more precisely we focus on tropical plane curves and show how they can be used to prove Bezout's theorem for curves over an arbitrary field. Finally, we discuss tropical correspondence theorems.

We study Euler characteristics of tropical subvarieties of tropical abelian varieties. We prove that every H-regular subvariety, locally modeled on tropicalizations of sufficiently well-behaved very affine varieties, has nonnegative signed Euler characteristic. This gives a tropical analogue of a theorem of Green-Lazarsfeld for subvarieties of complex abelian varieties. The main input is a local vanishing theorem for H-regular tropical fans, which also yields a Lefschetz-type theorem for affine H-regular tropical varieties. We further show that the signed Euler characteristic inequality fails for general tropical subvarieties of tropical abelian varieties, and we construct a 3-dimensional tropical fan whose link is not homotopy equivalent to a bouquet of 2-spheres.

The Dyn-Farkhi conjecture asserts that the square of the Hausdorff distance from a compact set to its convex hull is subadditive with respect to Minkowski addition. The conjecture is elementary in dimension 1, was recently proved by Meyer in dimension 2, and was disproved in dimensions $n\geq3$ by Fradelizi, Madiman, Marsiglietti, and Zvavitch. The symmetric case $A=B$, however, remained open. We show that the conjecture already fails in this restricted setting. More precisely, for every $n\geq3$, we construct a compact set $A\subset\mathbb{R}^n$ such that $$d(A(k))=d(A)>0$$ for every $1\leq k\leq n-1$, where $d(X)$ is the Hausdorff distance from $X$ to its convex hull and $A(k):=\frac1k (A+\dots+A)$ is the $k$-fold iterated Minkowski average of $A$. We also prove that the threshold $k=n$ is sharp: for every nonempty compact $A\subset\mathbb{R}^n$ with $n\geq 2$, we have $$d(A(n))\leq \left(1-\frac{n-1}{n(2n-1)}\right)d(A).$$

In this paper, we combine a data-driven system representation with a framework to systematically construct (approximate) solutions to nonlinear optimal control problems. By immersing the unknown dynamics into an extended state space, solutions are characterised via purely data-dependent algebraic conditions. This allows us to design dynamic state-feedback controllers with local stability and performance guarantees for unknown nonlinear, input-affine systems directly using data, without explicitly identifying the dynamics.

In this paper, we study some refined properties of uniformly quasi-greedy bases. In particular, we characterize the stability of uniformly quasi-greediness under scalings with bounded quotient and show that unconditionality is sufficient to guarantee this property. For the "isometric" case, i.e., when the uniformly quasi-greedy constant is $1$, we prove that 1-uniformly quasi-greediness implies disjointness if the underlying norm is strictly monotone. We also introduce a stronger notion of quasi-greediness by requiring uniform order boundedness of the greedy sums over greedy orderings. The characterization and properties of such bases are derived along the lines of uniformly quasi-greedy bases.

We study a one-sided degenerate Cahn-Hilliard equation with anisotropic traction flux, arising as a reduced continuum description of mechanically biased cell interactions in digit-forming organoids. The equation combines a one-sided degenerate mobility with a density-weighted anisotropic higher-order transport term. This traction term is not generated by the variational derivative of the Cahn-Hilliard energy and therefore produces sign-indefinite contributions in the energy balance. For nonnegative initial data, we prove the global-in-time existence of nonnegative weak solutions. The proof combines an energy estimate for the diffusive flux with a mobility-matched entropy method adapted to the vacuum degeneracy. A key point is that the entropy variable cancels the mobility, turning the anisotropic traction contribution into a coercive first-order term in the entropy inequality, while the energy estimate supplies a weighted control of the diffusive flux.

Let $Γ= (G, σ)$ be a signed graph and $v$ a non-isolated vertex of $Γ$. Let $Γ-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $Γ$, and $d_Γ(v)$ the degree of $v$ in $Γ$. In this paper, we prove that the largest eigenvalue $λ_1(Γ)$ of $Γ$ satisfies \[ λ_1(Γ) \le \sqrt{λ_1^2(Γ- v) + 2d_Γ(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $Γ$ is connected and $d_Γ(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.

We study Stein fillings of planar contact 3-manifolds under certain constraints on their relative trisections. We partially classify the diffeomorphism types of such fillings admitting relative trisections with genus at most 2.

Positive structures in Lie groups with respect to a subset $Θ$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.

Two (pseudo-)Riemannian metrics are said to be geodesically equivalent if they share the same geodesics considered as unparametrized curves. In 1865, E. Beltrami posed the problem of describing all geodesically equivalent metric pairs locally. At generic points, this problem was solved by Dini in the Riemannian setting and later extended to the pseudo-Riemannian case by Bolsinov, Matveev, and Pucacco. In the present paper, we solve the Beltrami problem in dimension two by providing a complete local classification of geodesically equivalent pseudo-Riemannian metrics in a neighbourhood of a singular point. This yields a full solution of the problem in dimension two.

In many dynamical systems applications, one seeks representative trajectories summarizing families of periodic orbits arising from parameter variation or uncertainty. While the Fréchet mean provides a natural notion of averaging in nonlinear metric spaces, its application to periodic trajectories is complicated by the interplay between geometric shape and temporal dynamics. Inspired by ideas from shape analysis, we develop a framework for computing Fréchet means of periodic orbits by representing trajectories as closed curves and introducing a metric structure on a quotient space that accounts for circular phase shifts. Within this framework, we establish the existence of empirical Fréchet means in the resulting infinite-dimensional quotient space. A central finding is that the resulting Fréchet mean depends strongly on the chosen parametrization: time parametrization preserves dynamical information, whereas arc length parametrization emphasizes geometric structure. To reconcile these viewpoints, we propose a decoupled approach that computes a geometric Fréchet mean using arc length parametrization and subsequently reconstructs representative dynamics through harmonic averaging of aligned speed profiles. We further introduce curvature- and medoid-based diagnostic measures that quantify the representativeness of the resulting mean and identify situations in which averaging produces geometric artifacts or fails to capture heterogeneous ensembles. Numerical experiments for the Van der Pol oscillator, the Rosenzweig-MacArthur predator-prey model, and the Morris-Lecar neuronal model demonstrate that the proposed methodology yields robust geometric summaries together with consistent averaged dynamics. The framework provides a principled approach for constructing representative periodic trajectories and assessing their validity in uncertainty quantification and dynamical systems.

We prove the uniformization theorem for domains in the Riemann sphere via the Kobayashi metric. Our proof does not rely on the existence of the elliptic modular function and instead uses the characterization of complete Kobayashi hyperbolicity for domains in the sphere.

We consider the problem of \emph{disjointness of rescalings} $(φ_{κt})_{t\in \mathbb{R}}$, $κ\in\mathbb{R}$ of a flow $(φ_{t})_{t\in \mathbb{R}}$ in the context of smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces. We show that, when the genus of the surface is $g\ge 2$, almost every locally Hamiltonian flow with 2g-2 non-degenerate simple saddles is such that any distinct two rational rescalings $(φ_{κt})_{t\in \mathbb{R}}$ and $(φ_{κ' t})_{t\in \mathbb{R}}$ with $κ=p/q$ and $κ'=p'/q'$ of different absolute values, are disjoint. Previous results on disjointness of rescalings were available only for rescalings for locally Hamiltonian flows and their special flow representations in genus one. The result is proved using a criterion for disjointness based on the study of the distribution of Birkhoff sums of a special representation and in particular estimates on their exponential tails decay. A key novel geometric ingredient in the proof is the existence of a sequence of rigidity times which display what we call bounded-type rigidity, so that a large set of points comes back in time $q$ with distance $O(1/q)$. To produce such bounded-type rigidity times we exploit a particular way of degeneration of a translation surface to a flat torus for which the vertical flow has bounded-type rotation number.

We prove a phase space criterion for dynamical Amrein-Berthier uncertainty principles. The abstract result says that, for a Fourier integral operator $A\in FIO(χ)$ associated with a tame canonical transformation $χ$, the localized operator $\mathbf{1}_E A\mathbf{1}_F$ is compact on $L^2(\mathbb {R}^d)$ whenever $χ$ satisfies a vertical non refocusing condition: high frequency covectors issued from a spatially localized region cannot return to a vertical direction over the observation region. In the linear symplectic case this condition is equivalent to the familiar nondegeneracy $\det B\neq0$ of the upper right block of the symplectic matrix. We apply this compactness theorem to Schrödinger propagators for Yajima--type Hamiltonians, including quadratic electric and linear magnetic growth, and obtain two--time Amrein--Berthier inequalities for compact localization sets at all nonrefocusing times. The result extends the compactness mechanism behind the dynamical Amrein-Berthier principle to a genuinely microlocal setting.

We characterize all algebraic numbers $α$ of degree $8$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the linear relation $α_{1}=α_{2}+α_{3}+α_{4}$. Analogous characterization is obtained for the linear relation $α_{1}+α_{2}+α_{3}+α_{4}=0$. In particular, when an algebraic number $α$ of degree $8$ has a non-even minimal polynomial and possesses exactly six distinct linear relations of the form $α_{i_1}+α_{i_2}+α_{i_3}+α_{i_4}=0$, we prove that $α$ is a sum of a quadratic and a quartic algebraic number.

Stability of robust heteroclinic cycles and networks is typically studied by constructing return maps and analysing their associated transition matrices. This analysis can be simplified with the network's projected map, derived by projecting the linear action of the transition matrix onto a simplex. This projection produces a piecewise-smooth map, which is one-dimensional for heteroclinic networks in $\mathbb{R}^{4}$. We consider two such networks, the Kirk--Silber network and the $Δ$-clique network. For the Kirk--Silber network, the projected map is discontinuous on its switching manifold, while it is continuous for the $Δ$-clique network. In this paper, we address the dynamical phenomena that produce a discontinuity in the case of the Kirk--Silber network, and explain the value of the projected map at the switching manifold. We construct a completed return map near both networks, which captures the behaviour of all trajectories that begin near the network but may move away from it temporarily. We show that the discontinuity in the projected map of the Kirk--Silber network emerges due to two phenomena: first, there exists a discontinuity in a component of the completed return map in the limit as trajectories approach the Kirk--Silber network, as a result of the presence of a separatrix near the network, and, second, the procedure that defines the projected map. For the $Δ$-clique network, there is no such separatrix, and so no such discontinuity emerges. This analysis is a necessary step towards understanding the more complicated dynamics observed near heteroclinic networks of five or more equilibria.