ALT We study Eₙ-algebras with ramification in their π₀ and in particular show that the existence of ramification places strong limitations on their chromatic behavior. In particular, we show that π₀ of nonzero K(1)-local E₄-KU-algebras cannot have a map from the ring of integers in a ramified extension of Qₚ, and we show that a p-complete E₂-ring with a map from a perfectoid ring into its π₀ is an associative Zₚ-algebra.

ALT For an E∞-ring spectrum R of chromatic height k, the trace method and the chromatic redshift imply that Tⁿ-fixed point of n-fold iterated topological Hochschild homology of R has chromatic height at most k+n. In this paper, we compute these heights for HFₚ, Hℤ and the p-complete connective image-of-J spectrum jₚ for all primes and n≥1, which are exactly n-1, n and n+1, respectively. Our method is based on properties of Wilson spaces and relative topological Hochschild homology over MU.

ALT Motivated by Teleman's study of gauged topological quantum field theories, our main result is an analog of Elmendorf's theorem in the setting of simplicial group actions on ∞-categories. In particular, we prove that if G is a simplicial group, there is a Quillen equivalence between the homotopy category of ∞-categories with G-action and the homotopy category of ∞-category-valued presheaves on the orbit category of G. We also show an analog of Whitehead's theorem in this setting.

ALT Every nonempty connected finite-dimensional CW topological semigroup admits a continuous associative multiplication on its universal cover that lifts the multiplication of the base. Such a multiplication is unique up to a deck transformation: in fact, any two associative lifts are conjugate whenever they exist. We show that the finite-dimensional hypothesis cannot be dropped in general: a connected, countable, infinite-dimensional CW topological semigroup is constructed for which continuous lifts of multiplication exist, but none is associative. The counterexample has nonabelian fundamental group isomorphic to Thompson's group F. Thus finite dimensionality is sufficient for associative lifting in the CW setting, whereas the unrestricted infinite-dimensional statement fails.

ALT We study higher complements of pseudohyperplane arrangements and the specialization maps between them, induced by degenerations. We develop a presheaf formalism on oriented matroids and weak maps, which model degenerations. Our main constructions are the posets of weighted and weightwise topes, which provide functorial combinatorial models for higher complements and their specialization morphisms. Their presheaf structure yields a deletion-contraction homotopy pushout, from which we derive a uniform combinatorial proof of the integral cohomology presentation for all n ≥ 2. This presentation is natural with respect to weak maps and identifies cohomology rings with higher Orlik-Solomon algebras extending the known formulas for linear arrangements to non-realizable oriented matroids. Finally, we equip Deligne groupoids with a presheaf structure and identify them functorially with the fundamental groupoids of the complexified complements.

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.