Zelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

given a compact $n$-dimensional immersed riemannian manifold $m^n$ in some euclidean space we prove that if the hausdorff dimension of the singular set of the gauss map is small, then $m^n$ is homeomorphic to the sphere $s^n$.

also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with a small set of points of zero gauss–kronecker curvature are topologically the sphere minus a finite number of points. a characterization of the $2n$-catenoid is obtained.

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

We give a motivic proof of the fact that for nonsingular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This was originally proved by Itenberg in the case of surfaces in $\mathbb {C}P^{3}$, and has been successively generalized by Bertrand and by Bihan and Bertrand. Our proof, different from previous approaches, is an application of the motivic nearby fiber of semistable degenerations. In particular, it extends the original result by Itenberg, Bertrand, and Bihan to real analytic families admitting a $\mathbb {Q}$-nonsingular tropical limit.

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

We consider operators $ - \Delta + X $, where $ X $ is a constant vector field, in a bounded domain, and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter $h$, which should be thought of as the reciprocal of the Péclet constant. This instability is due to the presence of the boundary: just as in the case of $ - \Delta + X $, some of our operators are normal when considered on $\mathbb{R}^d$. We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated with these elliptic operators.

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

Let $\sigma $ be a stability condition on the bounded derived category $D^b({\mathop{\mathrm {Coh}}\nolimits } W)$ of a Calabi–Yau threefold W and $\mathcal {M}$ a moduli stack parametrizing $\sigma $ -semistable objects of fixed topological type. We define generalized Donaldson–Thomas invariants which act as virtual counts of objects in $\mathcal {M}$ , fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne–Mumford stack $\widetilde {\mathcal {M}}^{\mathbb {C}^{\ast }}$ , called the $\mathbb {C}^{\ast }$ -rigidified intrinsic stabilizer reduction of $\mathcal {M}$ , with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized Donaldson–Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde {\mathcal {M}}]^{\mathrm {vir}} \in A_0(\widetilde {\mathcal {M}})$ . This stays invariant under deformations of the complex structure of W. Applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

The reduced peripheral system was introduced by Milnor [18] in the 1950s for the study of links up to link-homotopy, that is, up to homotopies leaving distinct components disjoint; this invariant, however, fails to classify links up to link-homotopy for links of four or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. The main result gives indeed a complete characterization of links having the same reduced peripheral system, in terms of ribbon solid tori in 4–space up to ribbon link-homotopy. The proof relies on an intermediate characterization given in terms of welded diagrams up to self-virtualization, hence providing a purely topological application of the combinatorial theory of welded links.

Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

We formulate a conjecture on the special values of zeta functions of regular arithmetic schemes in terms of Weil-étale cohomology…

In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections.

We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$: It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$. In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$, and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

We construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$-modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$, T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism.

We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the Bousfield–Kan total complex of $BG$ for all cosimplicial groupoids $G$. The $k$-invariants for the Postnikov tower of a cosimplicial space $X$ are naturally elements of stack cohomology for the stack associated to the fundamental groupoid ${\it\pi}(X)$ of $X$. Cocycle-theoretic ideas and techniques are used throughout the paper.

We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor’s proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin’s work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.