More Like this

1. Equivariant Infinite Loop Space Theory: The Space Level Story

   https://doi.org/10.1090/memo/1540  

   May, J; Merling, Mona; Osorno, Angélica (January 2025, Memoirs of the American Mathematical Society)

   We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\mathrm{\Omega <\#comment/>}$- $G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\mathrm{\Omega <\#comment/>}$- $G$-spectra when $G$is finite. We also show what is and is not true when $G$is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature. 

   more » « less
2. Stratified Noncommutative Geometry

   https://doi.org/10.1090/memo/1485  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (May 2024, Memoirs of the American Mathematical Society)

   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

   more » « less
3. On flat manifold bundles and the connectivity of Haefliger’s classifying spaces

   https://doi.org/10.1090/proc/16941  

   Nariman, Sam (November 2024, Proceedings of the American Mathematical Society)

   We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston’s conjecture predicts that every $M$-bundle over a manifold $B$where $\dim <\#comment/>\left(B\right)\le <\#comment/>\dim <\#comment/>\left(M\right)$is cobordant to a flat $M$-bundle. In particular, we study the bordism class of flat $M$-bundles over low dimensional manifolds, comparing a finite dimensional Lie group $G$with ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0\left(G\right)$. 

   more » « less
4. Unavoidable structures in infinite tournaments

   https://doi.org/10.1090/proc/16940  

   Benford, Alistair; DeBiasio, Louis; Larson, Paul (October 2024, Proceedings of the American Mathematical Society)

   We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$such that $G⊈K$, or (ii) every countably-infinite tournament contains aspanningcopy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $\mathrm{\kappa <\#comment/>}$; however, we are only able to provide a concise characterization in the case when $\mathrm{\kappa <\#comment/>}={\mathrm{\aleph <\#comment/>}}_1$. 

   more » « less
5. Non-vanishing of geometric Whittaker coefficients for reductive groups

   https://doi.org/10.1090/jams/1051  

   Færgeman, Joakim; Raskin, Sam (April 2025, Journal of the American Mathematical Society)

   We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $G{L}_n$to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are $t$-exact on the category of tempered $D$-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for $G{L}_n$. We also show that Whittaker coefficient functors are $t$-exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be $t$-exact. We apply our results to show that for suitably irreducible local systems, Whittaker-normalized Hecke eigensheaves are perverse sheaves that are irreducible on each connected component of ${\mathrm{Bun}}_G$. 

   more » « less