Papers

We show that for a generic compact initial surface in $\mathbb{R}^3$, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

Let $\mathcal D$ be a product of Drinfeld upper half spaces and let $J$ be the corresponding product of projective linear groups acting on $\mathcal D$. We prove that if $Y$ is a reduced, quasi-compact separated strict $\mathbb C_p$-analytic space and $f: Y\to\mathcal D$ is analytic with $\dim Y<\dim\mathcal D$, then $J f(Y(\mathbb C_p))$ is closed and nowhere dense in $\mathcal D(\mathbb C_p)$. The quasi-compactness assumption is necessary even when the source is a strict rigid analytic curve, while a quasi-compact perfectoid source can still have an orbit closure with nonempty interior. For every finite extension $F/\mathbb Q_p$ and $d\geq3$, we construct a closed smooth irreducible non-quasi-compact curve in $\Omega_F^{d-1}$ with dense orbit under a fixed diagonal one-parameter subgroup. We also construct a $\mathbb Q_p$-defined analytic map from the open unit disc to $\mathbf P^2$ with dense $\operatorname{PGL}_3(\mathbb Q_p)$-orbit, and an analytic map from a one-dimensional affinoid perfectoid torus to $\mathbf P^2$ whose $\operatorname{PGL}_3(\mathbb Q_p)$-orbit closure has nonempty interior.

Consider the Mathieu operator $L=-d^2/dx^2+2a\cos(2x)$ for fixed $a\in\mathbb{C}\setminus\{0\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,\pi]$ has exactly two algebraically simple eigenvalues $\lambda_n^-,\lambda_n^+$ in the disc $|\lambda-n^2|<1$. We prove that, as $n\to\infty$, $\lambda_n^+-\lambda_n^-=\pm\frac{8(a/4)^n}{[(n-1)!]^2}\exp\left[-\frac{a^2(n+2)}{4(n^2-1)^2}-\frac{15a^4}{128n^7}+O(n^{-8})\right]$, where the sign depends on the labelling. For real $a$ with $\lambda_n^-<\lambda_n^+$, this difference is the length of the $n$th spectral gap. Expanding the exponential determines the relative corrections of orders $n^{-4}$ to $n^{-7}$, beyond the correction $-a^2/(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $ce^{-2ix}+de^{2ix}$ with $a^2=cd\ne0$.

We study algebraic vector bundles on cluster varieties. Over a perfect field of characteristic different from two, every algebraic vector bundle is trivial on a cluster variety with principal coefficients at an acyclic seed of irreducible finite Dynkin type. The proof is based on integral contractions of cellular $\mathbb A^1$-chain complexes and successive freezing. We also construct smooth cluster varieties admitting indecomposable rank-two vector bundles, whose nontriviality is detected by quadratic Euler classes. The cellular calculation also gives a criterion for stable freeness and computes integral real cohomology and the images of cycle class maps.

We develop a neural transport framework for nonlinear, nonlocal aggregationdiffusion equations arising in chemotaxis, polymer dynamics, and related applications. Based on the Onsager variational principle, we reformulate the governing equation as a constrained optimal control problem, taking the velocity field as the control and evolving the density via the associated continuity equation. The velocity field is parameterized by a neural network trained on the variational objective evaluated along particle trajectories. By coupling the density and generalized score equations into a single system solved via the Euler method, the score function propagates efficiently for loss evaluation. Furthermore, we establish a rigorous error analysis that decomposes the total error into approximation, optimization, Monte Carlo sampling, and time-discretization components, and prove corresponding error bounds. Numerical experiments across several representative models demonstrate the accuracy and effectiveness of the proposed approach.

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected $r$-th moment assumptions. Let $N=n\wedge m$, where $n$ and $m$ are two sample sizes. We establish the minimax rates in the noncritical regimes. When $p<r<2p$, the minimax risk is of order $N^{-(r-p)/r}$, while for $r>2p$ it is of order $N^{-1/2}$. In both cases, the natural plug-in estimator is minimax rate-optimal. At the endpoint $r=p$, the minimax risk is bounded away from zero, so uniformly consistent estimation is impossible. At the critical order $r=2p$, we obtain the exact worst-case rate of the plug-in estimator and a lower bound for the minimax risk, leaving a logarithmic gap between the lower and upper bounds. The exact minimax rate in this critical case remains open.

For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$. By means of a Grothendieck-Riemann-Roch theorem of Toën we give explicit representation theoretic formulas of $\dim(\mathcal{X}^{[\rho]})$ for all $\rho \in \operatorname{Rep}(G)$ which are similar to the McKay correspondence. We also give a classification of all such quotient stacks according to an associated quadratic form called the Euler form. For $[\mathbb{A}^2/\mu_n]$ given by the cyclic action $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$, we give an interpretation of the corresponding Euler form in terms of border strips which may be removed by a diagram, an equational characterization of $n$-core partitions, and the number of significant arrows of a monomial ideal as defined in "Smooth and Irreducible Multigraded Hilbert Schemes" by D. Maclagan and G. Smith.

Parents seeking childcare often face long waitlists. At the same time, vacancies at childcare centers can remain unfilled for extended periods. We trace this paradox to two persistent features of childcare services. First, families consider multiple child care options due to enrollment uncertainties, resulting in phantom listings that remain on waitlists even after families have found care elsewhere or no longer need a slot. Second, filling a slot is time-consuming: centers learn whether a family is actively seeking enrollment only after making an offer and waiting for a response. As families pursue alternatives while they wait and their children age, these delays and feedback effects amplify the mismatch between demand and available capacity. We study two operational remedies. First, we examine regular verification that removes phantom listings before offers are made, improving the efficiency of enrollment and reducing vacancies. Second, we study mixed-age classrooms, in which some slots designated for older children can be flexibly used by younger children. This flexibility increases utilization, shortens waits, and reduces wasted offers. Its benefits extend beyond conventional pooling: as children move between age groups, classrooms can be filled through internal flows rather than costly external search, further reducing enrollment delays. Together, verification and mixed use offer a way to expand effective capacity in early childhood classrooms, where regulatory and operational constraints make adding physical capacity particularly costly.

In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form $[A,\tau_{s,t} \, B]$ in terms of self-avoiding paths of sets, where $A$ and $B$ are local operators and $\tau_{s,t}$ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.

We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.

The strong Szegő theorem and the Fisher-Hartwig formula describe the asymptotic behavior of determinants obtained by restricting translationally invariant operators to large finite regions. In particular, Szegő's theorem states that, beyond the leading extensive contribution to such a determinant, there is a universal finite correction, $\sum_{r>0} rV_rV_{-r}$ where $V_r$ are the Fourier coefficients of the logarithm of the symbol. Although this formula appears throughout analysis and mathematical physics, the origin of its quadratic form and of the weight $r$ is often obscured in its standard analytic derivations. We develop an elementary picture in which the strong Szegő term arises from boundary leakage: paths contributing to the matrix exponential leave the finite interval and subsequently return. Counting the shortest such excursions immediately produces the factor $r$. We then show that an isolated boundary contributes exactly one half of the strong Szegő correction, while the two boundaries of a long interval decouple. For finite-range matrices this gives a rigorous and intuitive elementary derivation of the asymptotics, with extensions to sufficiently rapidly decaying interactions. A complementary free-fermion argument provides a physical interpretation of the same cancellation mechanism. Numerical examples illustrate the asymtotics in the well controlled regime, the Fisher Hartwig logarithmic correction. The boundary formulation does not itself require translation invariance and therefore suggests natural extensions to disordered systems, higher dimensions, and symbols with internal matrix structure. To illustrate this we numerically go beyond the formal applicability region of the Szegő formula, considering slow decay and disordered cases, and showing in some cases a robustness of the approach even for slowly decaying but disordered potentials.

We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(\Sigma=\operatorname{Var}(S_n)\), and let \(G\sim N(0,\Sigma)\). Define the effective rank and the normalized squared spectral mass by \[ r(\Sigma)=\frac{\operatorname{tr}\Sigma}{\|\Sigma\|}, \qquad v(\Sigma)=\frac{\operatorname{tr}(\Sigma^2)}{\|\Sigma\|^2}. \] Under a uniform directional fourth-moment condition, covariance-balance assumptions, and the spectral conditions \(r(\Sigma)\le C_\Sigma v(\Sigma)\) and \(v(\Sigma)\ge6\), we prove \[ \sup_{z\ge0} \left| \mathbb P\{\|S_n\|^2\le z\} - \mathbb P\{\|G\|^2\le z\} \right| \le C\,\frac{r(\Sigma)^{3/2}}{n}. \] The constant depends only on the fixed parameters in the assumptions and is independent of the ambient dimension and individual small eigenvalues of the covariance matrix. The bound is uniform at the endpoint \(v(\Sigma)=6\), without a logarithmic loss. Its main feature is the combination of the \(n^{-1}\) rate with dependence on effective rank, including this endpoint. Consequently, the bound can vanish when \(r(\Sigma)=o(n^{2/3})\), even when \(p\) is much larger than \(n\). The same estimate extends to random elements of a separable Hilbert space with trace class covariance. The proof builds on the Fourier and symmetrization methods of \citet{GoetzeZaitsev2014}, combined with moment matching, truncation, and comparison estimates that retain the covariance structure.

A stable matching in a bipartite preference system is a matching that has no unmatched pair whose endpoints both prefer each other to their current assignments. Rothblum gave a linear description of the stable marriage polytope, the convex hull of the incidence vectors of stable matchings. We prove that this polytope is box-totally dual integral. The proof uses the affine representation of the stable marriage polytope as an order polytope together with the matricial characterization of box-TDI polyhedra. Since Kiraly and Pap proved that Rothblum's description is totally dual integral, our result further implies that Rothblum's system is box-totally dual integral.

Hybrid-abandonment systems, where jobs abandon from certain stages of service but not others, occur frequently in medical, service, and computing systems. Abandonment affects the heavy-traffic behavior of both the stages with and without abandonment. In contrast to existing heavy-traffic works that focus on cases where there is abandonment at either all stages or no stages, we focus on the heavy-traffic behavior for these hybrid-abandonment systems. We provide the first heavy-traffic characterization of the second queue length in a two-stage tandem queue with abandonment only in the first stage. We characterize the scaled mean of the second stage queue length and prove that the scaled second queue length is asymptotically exponentially distributed and asymptotically independent of the first stage queue length. We derive our results by analyzing Markovian arrival process queues with countably infinite phases. We do this by extending techniques used to analyze queues with Markov-modulated arrivals to the broader class of infinite Markovian arrival process queues. We prove asymptotic results for a more general infinite phase process condition, an infinite birth-death phase process, and the hybrid-abandonment tandem case. We also empirically demonstrate that our heavy-traffic approximations are good approximations across various system settings.

Stochastic dual dynamic programming (SDDP) and its variants are run in practice with forward passes that sample the noise at random. Recent iteration-complexity results favour another choice: when the trial points are selected deterministically, for instance along the scenario with the largest gap, the number of iterations is polynomial in the horizon $T$, whereas every bound we know of for randomly sampled forward passes is exponential in $T$. We show that this is not intrinsic to random sampling, for a broad class of trajectory-following dynamic programming algorithms, which includes SDDP, stochastic dual dynamic integer programming and stochastic Lipschitz dynamic programming. With random sampling, and with Lipschitz constants, diameters and per-stage loss ranges bounded independently of the stage and of the horizon, the expected number of iterations whose policy is not $T\varepsilon$-optimal is $O(T\varepsilon^{-(d+1)})$, where $d$ is the state dimension: linear in $T$, as for deterministic selection rules. The proof needs no particular scenario to be sampled: the gap between the expected cost of the current policy and the current lower bound equals the expected sum of one-step Bellman residuals along the sampled path, and Lipschitz cuts that are tight at the visited states turn the positive residuals into permanent increases of a bounded potential. The results extend to noise without finite support, to backward passes on a sample average approximation, and to stationary and periodic infinite-horizon SDDP with geometric pass lengths.

We combine the universal polynomial matrix identities of Larsen–Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of $\text{SL}_2(\mathcal{O}_{K,S})$ for every global function field $K$ and every nonempty finite set of places $S$. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over $\mathbb{Z}$ that works simultaneously for all these rings.

We prove that if the first Lyapunov exponent of a compactly supported random matrix product is simple, then it is pointwise Hölder continuous for the Wasserstein–Hausdorff distance. By passing to exterior powers, we obtain pointwise Hölder continuity of the first $k$ exponents whenever they are simple.

This paper studies an open network design problem of constructing graph topologies with maximum algebraic connectivity under fixed numbers of vertices and edges. A density-dependent approach is proposed that decomposes the design task into two stages: a high-quality, low-cost structural backbone construction and a reinforcement learning completion stage. The method selects the backbone based on density, using complete multipartite graphs in dense regimes and Cayley graphs in low-to-intermediate regimes, and then refines the remaining edges through a learning-based procedure. Numerical results show that the proposed method achieves high algebraic connectivity with favorable runtime scalability, outperforming existing methods. These results support a structural-family-first strategy for scalable network design.

This paper studies formal Laurent series, with particular emphasis on their continued-fraction and Laurent-series expansions. Knowledge of both expansions is important, for example, in the study of Kronecker-type sequences in uniform distribution theory and in the search for potential counterexamples to variants of Littlewood's conjecture over fields of formal Laurent series. Our approach investigates Hankel matrices over a field and relates them to Laurent-series expansions over the same field. We describe such matrices via a block $\mathcal{L}\mathcal{B}\mathcal{U}$ decomposition, using recursive algorithms based on the continued-fraction coefficients of the corresponding Laurent series. The upper-triangular factor $\mathcal{U}$ has several remarkable properties. In particular, it is constructed columnwise from the so-called Ostrowski-type representations of powers of $X$. Applying $\mathcal{U}$ in Niederreiter's digital construction yields an Ostrowski-type van der Corput sequence that can be related to the corresponding Kronecker-type sequence. For ordinary Kronecker sequences, such a relation is known in only one case, that of the $(n\varphi)$-sequence. The original motivation for our study of Hankel matrices was to recast the search for counterexamples to variants of Littlewood's conjecture over fields of Laurent series as the search for specific Kronecker-type sequences. Interestingly, the method developed for the study of Kronecker-type sequences also allows a new perspective on the celebrated Sobol' construction. Our approach yields a short and elegant proof of its excellent distribution properties.

We prove that every finite ordered set of width at most three and cardinality at least four is reconstructible from its deck. This settles the width-three part of an open problem posed by Schröder in Ordered Sets; the separate question concerning ordered sets of height one is not addressed here. The new ingredient is a criterion for recovering the marked maximal deck of an ordered set with exactly three maximal elements from its unmarked maximal deck and the isomorphism type of the suborder obtained by deleting all maximal elements. We express the criterion as an orbital balance condition on the attachment ideals in this common base. Necessity follows by double counting; sufficiency uses the structure of automorphism actions in width at most three. The full-deck theorem then follows from reconstruction results of Schröder concerning maximal cards and marked maximal cards. A local reattachment argument correcting one intermediate step in the two-maximal-element case is included in an appendix.

We revisit the construction of compact liquid drop equilibria from Delaunay unduloids in [Arch. Ration. Mech. Anal. 249 (2025), Paper No. 74]. We evaluate exactly the coefficient arising in the final solvability condition of the Lyapunov-Schmidt reduction, identifying it as the product of the axial period and the normalised Delaunay flux. Using this identity and the monotonicity of the period, we establish the conjectured positivity of the coefficient and prove that its product with the period strictly increases with the neck radius. We also derive an exact relation between the volume and the surface area of one period and prove that both strictly decrease with the neck radius. These results extend the construction throughout the embedded unduloid family and yield compact equilibria of every sufficiently large mass, with reference neck radii converging to any prescribed interior value as the mass tends to infinity.

Let \(X\) be a smooth connected complex projective variety with an effective algebraic \(\mathbb C^{\times}\)-action, a nonempty finite fixed locus whose weights are \(\pm1\). We show that if its Hattori line bundles are nef, then \(X\) is algebraically isomorphic to \((\mathbb{P}^1)^{\dim X}\), and these bundles become the hyperplane bundles of the factors. The product statement follows from the nef-complexity classification of Enwright and collaborators, and we describe the isomorphism through global sections. To test the required nefness we specialise the orbit-graph criterion of Romano and Wiśniewski. The Hirzebruch surface \(\mathbb F_2\) is an example why the topological hypotheses alone do not imply either the product conclusion or the Fano property.

The standard Yee finite-difference time-domain (FDTD) algorithm uses Maxwell's equations to replace first order temporal derivatives with spatial curls. In this work, the relation is extended to arbitrary orders: a Time-Space Theorem establishes the correspondence between higher-order temporal derivatives and successively higher-order spatial curl operators. Based on this relation, an arbitrary-order framework for electromagnetic field approximations is developed. The framework constructs higher temporal orders through higher curl orders and permits different spatial discretizations and curl-estimation techniques, thereby providing a general basis for developing numerical algorithms of arbitrary orders. As one implementation, an arbitrary-order FDTD algorithm is constructed using staggered time and space stepping together with recursive curl estimations. Numerical simulations of an initial-value problem are performed for temporal approximation orders from second to tenth order while the spatial and temporal step sizes are kept fixed. Comparisons with the analytical solution show that the errors decrease rapidly and monotonically with increasing approximation order. These results demonstrate the practical potential of the framework for high-accuracy electromagnetic computation and provide a basis for further development of arbitrary-order numerical methods.

We obtain the $L^3$ reverse square function estimate for the light cone in four dimensions from the critical local-smoothing and round-cone propagation statements in the September 2026 OpenAI manuscript. We state these inputs, together with the plate maximal theorem of Di Plinio and Parissis, in a preliminaries section. The proof passes from geometric concentration of packet coefficients to regularity of the analyzed wave, and then from free waves to arbitrary functions supported near the cone. As a consequence, we obtain the corresponding $L^3$ estimate for the sphere and elliptic paraboloids in three dimensions.

We study products of independent, identically distributed random matrices with an exponential moment and a simple top Lyapunov exponent, without assuming irreducibility. Outside an exceptional invariant subspace called the equator, we establish contraction on average for a weighted projective distance and a spectral gap for the associated Markov operator. We obtain a central limit theorem, a Berry–Esseen bound, deviation estimates near the mean, and an almost sure invariance principle both for the growth of a fixed vector outside the equator and for the operator norm. For fixed vectors, a non-arithmeticity assumption also yields a local limit theorem and a renewal theorem. Our contraction criterion applies more generally to mostly contracting random maps on locally compact spaces with a Lyapunov function, including certain random interval maps.

In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg–Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.

Do lattice triviality, suspected Borel non-summability and the perturbative Landau pole exclude an interacting four-dimensional theory satisfying the Osterwalder–Schrader axioms and reproducing every renormalized coefficient of positive-coupling $O(N)$ $\phi^4$ theory? An example involving one real function separates vanishing cutoff limits along its allowed bare trajectories from a nonzero result obtained by renormalizing the coefficients first; an exponentially small correction preserves its expansion and removes the pole in its running coupling. For the massless two-point expansion of the singlet composite $\Phi_a\Phi_a$ at leading order in $1/N$, ordinary Borel summation fails even for Gaussian smearing. The arithmetic mean of two Borel integrals along opposite imaginary rays defines a real distribution on every four-dimensional Schwartz momentum or position test, with finite-order remainder bounds uniform on bounded test sets. Two alternative covariances with nonnegative mass spectral weights remove the Euclidean pole and construct generalized free fields. After a fixed contact subtraction, their Euclidean extensions preserve every coefficient of the collective correlations obtained by Wick pairing. The interacting finite-$N$ family remains open. We also state a conditional interacting route through the physical scalar correlations of a larger theory (the parent) with heavy particles. A concrete $N=1$ gauge–Higgs–fermion candidate has a tuned one-loop trajectory on which all interaction couplings tend to zero in the ultraviolet, and a stable tree vacuum with additional masses comparable to the heavy scale. Constructing the parent and its corrected ultraviolet trajectory, and proving coefficient decoupling and the exact remainder bounds as the quartic coupling tends to zero and the family's heavy masses grow, remain open.

The so-called “100% rule” is a somewhat classical and yet somehow obscure rule that gives some local sensitivity-analysis information based on the very limited local sensitivity-analysis information that is often made available by linear-programming software, derived from an optimal basis that is hidden from the user. Introduced in a classic undergraduate textbook in 1977, half a century ago, it was only picked up on by researchers in a very limited way, and while it appears in some undergraduate textbooks, it tends not to appear in most modern graduate-level textbooks. Our contributions are to celebrate and revive interest in the 100% rule and to extend it in a meaningful way. We introduce a cone rule that certifies primal feasibility for more right-hand sides than the 100% rule does, using the same information, and that can be applied in linear time. We then study when this cone rule is tight, that is, when it captures the entire cone of right-hand sides for which the basis remains feasible. We characterize tightness in every dimension for right-hand sides interior to that cone, make the characterization explicit for m=2 and m=3, and settle the question for tridiagonal bases in every dimension: tightness is impossible once m>=4, and for m=3 we give an explicit criterion in the entries of the basis. We give further results for symmetric, integer and unimodular bases. Finally, when tightness fails, we measure the fraction of the feasibility cone captured by the rule using normalized solid angles. For the autoregressive family at m=3, we obtain a closed-form candidate for the infimum of this fraction over all right-hand sides, whose optimality is supported numerically. We also obtain a closed-form expression for the best case over most of the correlation range, subject to two numerically verified properties, and estimate the best-case value numerically more generally.

We introduce a new stochastic model of telomere length dynamics during hematopoiesis. The numerous cell divisions occurring in this setting makes telomere length dynamics, and in particular their attrition, a crucial aspect of hematopoiesis. After recalling the main aspect of the biological setting, we describe an individual based model of hematopoietic population cells. We then study its behaviour in the large population limit, as well as its fluctuations.

We study multiblock convex–concave saddle-point problems of the form $\min_x\max_y\{f(x)+H(x,y)-g(y)\}$, with separately charged partial-gradient oracles for all three objective components. We impose joint smoothness and strong convexity bounds on the functions $f$ and $g$, together with blockwise Lipschitz conditions on the partial gradients of $H$, so that the resulting oracle complexities reflect the smoothness and dependency parameters specific to each oracle. Under strong convexity, we develop a randomized lifted extragradient method that achieves linear convergence while providing simultaneous complexity guarantees for all oracles. The partial-gradient complexities for $f$ and $g$ exhibit the expected square-root dependence on the corresponding block condition numbers, whereas the coupling-gradient complexities depend on the associated curvature-normalized smoothness parameters. Finite-dimensional bilinear subclasses provide lower-complexity benchmarks within a randomized blockwise linear-span model. When strong convexity of $f$ and $g$ is not assumed, a regularization argument yields expected restricted-gap guarantees with accuracy dependence $\varepsilon^{-1/2}$ for the objective-gradient oracles and $\varepsilon^{-1}$ for the coupling-gradient oracles, up to logarithmic factors. Experiments show observed oracle savings and runtime improvements in regimes of poor objective conditioning.