ALT Let p be a prime and n≥ 1. We prove that the Tate-Frobenius of the K(n)-local sphere induces an equivalence between Lₙ₋₁LK(ₙ)Y and (LK(ₙ)Y)ᵗΣₚ ≅ (LₙY)ᵗΣₚ for any finite p-local type n-1 spectrum Y with trivial Σₚ-action.

ALT We prove that the deformation functor associated to an algebra over a sufficiently nice Koszul operad can be lifted to an operadic formal moduli problem over the Koszul dual operad. We further show that this operadic formal moduli problem corresponds to the derived center of the algebra under the operadic Lurie-Pridham correspondence. As an application, we prove that the classical Gerstenhaber algebra structure on polyvector fields of a commutative algebra agrees with the derived center of the algebra as an abelian Poisson algebra.

ALT Fundamental to the field of topological signal processing (TSP) is the persistence diagram (PD), or barcode, of a signal, which gives noise-resistant measures of topologically significant features as distinct from noise. Existing literature proves that the total persistence computed from these diagrams is equal to the total variation, for certain continuous signals on compact domains. We provide a closed formula for this shared quantity, and generalize to all continuous and discrete functions on bounded domains, and to certain functions on non-compact 1D manifolds. Extending the identification of continuous and discrete persistence, we investigate the similarities of functions with the same PDs, and prove that, for any n, the persistence diagram can be calculated as a function of the sequence of critical values; we compute these formulas for n ≤ 6. Finally, we examine which function transformations cause predictable changes in the PD of the transformed function, characterizing affine a

ALT Background: Clinical influenza vaccine effectiveness (VE) has often been low over the past two decades, highlighting the need for preseason VE forecasting. Timely forecasts could help health authorities plan prevention strategies and allocate healthcare resources. However, preseason forecasting of clinical influenza VE remains relatively uncommon. Methods: We obtained hemagglutinin (HA) sequences from GISAID for U.S. specimens collected in August–September before each influenza season from 2010–2011 through 2024–2025. After quality screening, we compared each virus with the corresponding vaccine target strain for the same season and subtype or lineage using two mathematical distances based on amino acid patterns in the HA protein. We calculated mean distances for each season and subtype or lineage and paired them with published clinical VE estimates, yielding 26 observations across 13 seasons. Using a mathematical AI approach, we fitted linear regression models of VE against each dista

ALT We study the negative gradient flow of analytic maps f:X→mathbbR defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category \cL – called the landfall category – whose classifying space B\cL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse–Conley spectral sequence over a

ALT We construct and describe the moduli stack Isog of isogenies between oriented elliptic curves. We show that Isog is a spectral Deligne-Mumford stack and a relative spectral algebraic space over ℳₑₗₗᵒʳ, and give a fiber product decomposition of Isog corresponding to a connected-étale factorization on oriented elliptic curves. In the process of proving these results, we construct a theory of duality and quotients for spectral abelian varieties, prove a higher-algebraic version of Nakayama's lemma, and lift the rigidity theorem for morphisms of elliptic curves to the setting of spectral algebraic geometry.

ALT Discrete homotopy theory is a combinatorial homotopy theory defined for discrete objects. Barcelo–Severs–White conjectured that the homotopy groups of complements of k-parabolic subspace arrangements are isomorphic to the discrete homotopy groups of graphs associated with the Coxeter complexes. We show that this correspondence fails in general by giving a counterexample in degree two for the 4-parabolic subspace arrangement of type A₅.

ALT We define a Nielsen coincidence number for pairs of maps (f,g) where f is n-valued and g is m-valued. The classical definition of the Nielsen number does not work in this setting, but we propose a theory where coincidence classes are allowed to overlap. We invoke a notion of multiplicity of a coincidence point, which indicates to how many classes it can belong, and we show that our Nielsen number is a homotopy-invariant lower bound for the number of coincidence points with multiplicity. We show a computational result in terms of single-valued lift factors, and using this, a Wecken property for self-maps of the circle.

ALT We freely enrich the pointed homotopy category of CW complexes over abelian groups and equip the resulting functor category with the enriched Day convolution induced by the smash product. After identifying the convolution unit and recording the coherence maps, we compute the convolution of the homotopy-group functors. For integers n,m≥ 2 we construct a natural isomorphism π_n⋆π_mcongπ_n+m induced by the external smash product; equivalently, the external smash product is the universal natural pairing of homotopy groups. Under this identification, the symmetry interchanging the two factors acts on π_n+m by multiplication by (-1)^nm.

ALT Segal computed π₀ of the equivariant sphere spectrum as the Burnside ring by a Pontryagin-Thom construction, transversality, and marks. We carry out the same argument in the motivic setting over a field k: the Burnside ring is replaced by the Grothendieck ring Aᵘ(Γₖ) of finite étale k-schemes decorated with a unit, the marks by the Dress map d(L,u)=trL/ₖ〈u〉 into GW(k), and a Pontryagin-Thom map Aᵘ(Γₖ)→π₀(Sₖ)(k) is shown to be surjective with kernel \ker(d). This gives π₀(Sₖ)(k)congGW(k) for infinite perfect fields of characteristic ≠ 2 without Morel's computation. We then identify \ker(d), by a reciprocity law on ℙ¹, as the group of principal quadratic divisors of rational functions, so that π₀ of the motivic sphere is the unit-decorated Burnside ring of Γₖ modulo principal quadratic divisors. The transversality step is proved by an elementary "re-topping" argument.

ALT We define a moduli space of continuous maps from smooth hypersurfaces varying in an ample linear system. To study its topology, we construct a scanning map to an appropriate section space and investigate its effect on homology. Our main innovation is a general argument based on Goodwillie calculus and applicable to various moduli spaces of maps. Along the way, we compute the Goodwillie tower of the section space functor based at any retractive space.

ALT This book presents calculations of the 2-primary components of the homotopy groups πᵢ(𝑆𝑂(n)) for all 23≤ i≤ 28 and all n, as well as some π₂₉(𝑆𝑂(n)), with explicit generators.

ALT We show that for almost every finite dimensional complex representation W of SO(2) the associated complex projective space ℂP(W)₊ does not have the rational stable homotopy type of a specific wedge of representation spheres constructed from a chosen filtration of ℂP(W)₊. This contrasts with the case for representations of finite Abelian groups where such an equivalence does hold. In the finite group case, the splitting result provides us with a classification result for projective spaces up to rational stable equivalence. In light of the non-splitting as a particular wedge of spheres in the case of projective spaces of SO(2)-representations, we also prove a separate classification theorem.

ALT We compute the 2-completed integral motivic homology, effective algebraic K-theory, and very effective Hermitian K-theory of the geometric classifying spaces Bμ₂ⁿ, n≥ 2, over algebraically closed and real closed fields. We first carry out the corresponding classical calculations and, then obtain the motivic results using motivic spectral sequences and comparison techniques. The results reveal the classical 2-torsion together with motivic τ-periodicity and ρ-torsion.

ALT In this note, we elaborate on a result of Krannich–Kupers–Mehzer on Disc-presheaves. We show that equivalences of 2-truncated Disc-presheaves of manifolds yields homotopy equivalences with trivial normal invariant. As an application, we show that two closed, simply connected manifolds are homeomorphic if and only if their 2-truncated Disc-presheaves are equivalent.

ALT We exhibit the p-complete ℂ-motivic Eilenberg-Mac Lane spectra MFₚ and (Mℤ)ₚ∧ as E₃-MGLₚ∧-motivic Thom spectra over the motivic Gₘ-loop spaces ΩGₘBGL and ΩGₘBSL, respectively. Furthermore, for p=2, we compute the MGL₂∧-relative Gₘ-Hochschild Homology Gₘ ⊙ᴹᴳᴸ₂^⁽∧) -, recently introduced by Bachmann-Burklund-Dundas-Østvær, of MF₂ and (Mℤ)₂∧ to be MF₂[BGL] and (Mℤ)₂∧[BSL] as E₂-MF₂-algebras and E₂-(Mℤ)₂∧-algebras, respectively.

ALT We introduce the notion of A¹-n-coverings and begin the development of their theory over an arbitrary base scheme S, in analogy to that of higher coverings in an ∞-topos. We show that for n=0, our notion recovers Morel's A¹-coverings between simplicial sheaves. Under suitable assumptions, we then provide a Galois-theoretic classification of A¹-n-coverings. Finally, we answer a question of Morel concerning a geometric criterion for A¹-coverings in the negative.

ALT We give a direct Fox-calculus proof of the known rational asphericity theorem for every finite presentation complex with one nontrivial relator, including proper powers. The proof uses completed Fox derivatives and the absence of zero divisors in the completed algebra of a prounipotent group. Exact scalar extension to the finite adeles then yields adelic asphericity and a product decomposition of the higher homotopy groups of integral Bousfield–Kan completion. We also prove rational and pro-p analogues of Mannan's formula for finite two-complexes, with their natural topologies and fundamental group actions. The rational cohomological Whitehead criterion is proved directly. For a discrete one-relator group with relator not a proper power, we show that the tower L₂νₙ(G) is Mittag–Leffler. Consequently limoneₙL₂νₙ(G)=0, and π₂(ℤ∞ BG) is naturally the inverse limit of the nilpotent Baer invariants. We identify this Baer limit canonically and topologically with the kernel of the integral au

ALT In the paper Structure and realizability for rational maps, Wei proved the following. Let k nontrivial partitions of an integer d≥ 3 be given, satisfying the Riemann–Hurwitz formula for the case of maps between two spheres. If k>ℓ+1, where ℓ=minᵢ|πᵢ|is the shortest partition length, then there exists a rational map whose branch datum is given by these partitions. In this paper we give a new proof of Wei's theorem on realizing a tuple of partitions as the ramification data of a rational function. The proof is inductive on the shortest partition length. Two auxiliary results are established: a two-cut partition theorem and a gluing theorem for marked spherical coverings. These reduce any counterexample with shortest length ℓ≥3 to auxiliary data with either fewer factors or smaller shortest length, while preserving the total defect and the marked conditions. The base case ℓ≤2 is quoted from the literature.

ALT In general, it is interesting to investigate the complexity of simplicial maps under various topological constraints. In this paper, we consider maps between (triangulated oriented rational homology) manifolds with non-zero Hopf invariant and establish bounds on the simplicial complexity of the fiber of a regular point. The lower bound, which is cubic in the dimension, is proved via a refinement of a theorem of S. Eilenberg. The upper bound is notable in that it is uniform with respect to the Hopf invariant.