We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free $\mathbb{F}_2[v_2^4]$-module of rank 52, on explicit generators in stems $-1 \leqslant \ast \leqslant 30$. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn, Raksit and Wilson, extending work of Bhatt, Morrow and Scholze from the case of classical commutative rings to $\mathbb{E}_\infty$-rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.

In an infinitesimal variant of Guo–Jacquet trace formulae, the regular semi-simple terms are expressed as noninvariant weighted orbital integrals on two global infinitesimal symmetric spaces. We prove some relations between the Fourier transforms of invariant weighted orbital integrals on the corresponding local infinitesimal symmetric spaces. These relations should be useful in the noninvariant comparison of the infinitesimal variant of Guo–Jacquet trace formulae.

We prove a conjecture of Manin and Orlov: the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with n marked points admits a full exceptional collection that is equivariant under the action of the symmetric group $S_n$ permuting the marked points. In particular, its $K_0$-group with integral coefficients is a permutation $S_n$-lattice.

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many geometric invariant theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety V by the action of a reductive group G to be a relative Mori dream space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V{{/\!\!/\!}}_\theta G$. If $V{{/\!\!/\!}}_\theta G$ is a crepant resolution of $Y\!\!:= V{{/\!\!/\!}}_0 G$, then every projective crepant resolution of Y is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $\Gamma\subset \operatorname{SL}(3,{{\mathbb{C}}})$ whose non-trivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations.

Let $K/k$ be a finite Galois extension of global function fields. Let E be a Drinfeld module over k. We state and prove an equivariant refinement of Taelman’s analogue of the analytic class number formula for $(E,K/k)$, and derive explicit consequences for the Galois structure of the Taelman class group of E over K.

We study the relationship between higher Du Bois singularities and K-regularity, a notion that measures the $\mathbb A^1$-invariance of the algebraic K-groups. Building on this relationship, we establish a strengthened form of Vorst’s conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of K-regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of Euclidean space. Motivated by this, Veech introduced the Siegel–Veech formula for translation surfaces, describing the average number of saddle connections of bounded length. However, no such formula is known for flat surfaces with cone angles that are irrational multiples of $\pi$. A convex flat cone sphere is the Riemann sphere with a flat cone metric whose cone angles lie in $(0,2\pi)$. In this paper, we extend the Siegel–Veech theory to convex flat cone spheres by introducing generalized transforms that are uniformly bounded on moduli space and give rise to Siegel–Veech measures on the positive real line. We prove that these measures are absolutely continuous and piecewise real analytic. In contrast to translation surfaces, which benefit from $GL(2,\mathbb{R})$ dynamics, our approach instead uses tools from o-minimality. Finally, we analyze the asymptotic behavior of the Siegel–Veech measure near zero, obtaining results that parallel the computation of Siegel–Veech constants in the translation surface case.

We present a comprehensive study of the geometry of Hilbert p-adic eigenvarieties at classical parallel weight 1 intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at p and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $\mathrm{SL}_n(\mathbb{Z}, p^m)$ in degrees $* \lt p-1$, for all large enough n, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $\mathrm{SL}_n(\mathbb{Z}/p)$ with certain twisted coefficients.

We prove the Jacquet–Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof by Zhiwei Yun in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations by the first-named author.

We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a $\mathrm{ku}_\ast$-module using the descent spectral sequence for the map $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which is the motivic spectral sequence for $\mathrm{THH}(\mathrm{ku})$ in the sense of Hahn–Raksit–Wilson. We reduce the computation of homotopy groups to the algebra of the universal formal group law, providing a systematic way to compute THH of quotients of MU. We compute the $E_2$-page of the motivic spectral sequence computing $\operatorname{THH}(\mathrm{ku})$, and we show that it degenerates at the $E_2$-page.

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups $H_1$ and $H_2$, we study the tensor product of the associated representation rings $\mathrm{R}H_1$ and $\mathrm{R}H_2$ over the representation ring $\mathrm{R}G$, and prove a vanishing result for the associated higher Tor groups. This result can be viewed as a natural extension of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over $\mathrm{R}G$. It may also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.

We give a categorical formulation of the p-adic local Langlands correspondence for ${\mathrm{GL}}_2({{{\mathbb{Q}_{p}}}})$ as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of$\mathrm{Gal}(\overline{{\mathbb{Q}}}_p/{{{\mathbb{Q}_{p}}}})$. The Montréal functor appears as the‘Whittaker coefficient’ for the universal Galois representation, in the sense of the geometric Langlands program. Moreover, we relate our version of the p-adic local Langlands correspondence for ${\mathrm{GL}}_2({{{\mathbb{Q}_{p}}}})$ to the cohomology of modular curves through a local–global compatibility formula.

We establish dimension formulae for the Witt-vector affine Springer fibers associated to a reductive group over a mixed characteristic local field, under the assumption that the group is essentially tamely ramified and the residue characteristic is not bad. In addition to the discriminant valuations that show up in classical works on the usual affine Springer fibers, our formulae also involve the Artin conductors and the Kottwitz invariants of the relevant conjugacy classes.

The landmark quantum complexity result MIP$^*$ $=$ RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.

In this paper, we prove the abundance conjecture for threefolds over a perfect field k of characteristic $p \gt 3$ in the case where the numerical dimension is equal to 2. More precisely, we show that, if (X,B) is a projective log canonical pair of dimension 3 over k such that $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$, then $K_{X}+B$ is semi-ample.

We study the classification of $\mathbb{Z}$-DGAs (differential graded algebras) with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert \gt0$, motivated by computations in algebraic K-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x\rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG enhancements and a classification of prime DG division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic K-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$.

We provide a complete classification of when the homeomorphism group of a stable surface, $\Sigma$, has the automatic continuity property: Any homomorphism from $\mathrm{Homeo}({\Sigma})$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable, second-countable Stone space has the automatic continuity property. Under the presence of stability, this answers two questions of Mann.

Colmez, Dospinescu et Nizioł démontrent que les seules représentations p-adiques de $\mathrm{Gal}(\bar{{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ qui apparaissent dans la cohomologie étale p-adique de la tour de revêtements du demi-plan de Drinfeld sont les représentations cuspidales (i.e. potentiellement semi-stables, dont la représentation de Weil-Deligne associée est irréductible) de dimension 2, à poids de Hodge-Tate 0 et 1 et que leur multiplicité est donnée par la correspondance de Langlands p-adique. Nous étendons ce résultat en poids quelconque, en considérant la cohomologie étale p-adique à coefficients dans les puissances symétriques du système local universel sur la tour de Drinfeld. Une différence notable est que toutes les représentations potentiellement semi-stables de dimension 2 non cristabélines, pas seulement les cuspidales, apparaissent avec les multiplicités attendues. Le point clé est que les systèmes locaux que l’on considère sont particulièrement simples : ce sont des ‘opers isotriviaux’ sur une courbe. Nous donnons une recette pour calculer la cohomologie proétale de tels systèmes locaux à partir de la cohomologie de Hyodo-Kato de la courbe et du complexe de de Rham du fibré plat filtré associé au système local.

Given a link of a normal surface singularity with its canonical contact structure, we compare the collection of its Stein fillings to its Milnor fillings (i.e. Milnor fibers of possible smoothings). We prove that, unlike Stein fillings, Milnor fillings of a given link have bounded topology; for links of sandwiched singularities, we further establish that there are only finitely many Milnor fillings. We discuss some other obstructions for a Stein filling to be represented by a Milnor fiber, and for various types of singularities, including simple classes like cusps and triangle singularities, we produce Stein fillings that do not come from Milnor fibers or resolutions.