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1. The topological complexity of pure graph braid groups is stably maximal

   https://doi.org/10.1017/fms.2022.83  

   Knudsen, Ben (January 2022, Forum of Mathematics, Sigma)

   Abstract We prove Farber’s conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity. 

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2. Robertson’s conjecture and universal finite generation in the homology of graph braid groups

   https://doi.org/10.1007/s00029-024-00971-1  

   Knudsen, Ben; Ramos, Eric (November 2024, Selecta Mathematica)

   Abstract We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest. 

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3. The local sum conjecture in two dimensions

   https://doi.org/10.1142/S1793042120500888  

   Fraser, Robert; Wright, James (September 2020, International Journal of Number Theory)

   The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko’s treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective. 

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4. On Supercompactness of \omega_1

   Ikegami, Daisuke; Trang, Nam (January 2020, Springer Proceedings in Mathematics & Statistics)

   This paper studies structural consequences of supercompact- ness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ω1 is supercompact”. “ω1 is supercompact” also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equiv- alent to ADR. It is shown that “ω1 is supercompact” does not imply AD. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from “ω1 is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness con- ditions on the universe of sets. “ω1 is supercompact” on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ω1 is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property. 

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5. Fourier restriction above rectangles

   https://doi.org/10.1007/s00208-021-02202-w  

   Schwend, Jeremy; Stovall, Betsy (May 2021, Mathematische Annalen)

   null (Ed.)

   In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure. 

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