Deswegen zurück zum echten Leben. In dem, wenn uns Ruhe umgibt, eine Erinnerung hochkommen kann. Und die kündigt sich leise an, wiederholt sich ein paar Mal und man fragt sich: ist das jetzt wirklich so gewesen oder doch anders? Aber es war schmerzhaft, diese Erinnerung. Und die kommt immer näher, und wird immer erlebbarer, und dann ist es plötzlich so, als wäre es ganz aktuell, als würde es wieder

Analogous to subfactor theory, employing Watatani's notions of index and C ∗ -basic construction of certain inclusions of C ∗ -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C ∗ -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the t = q specialisation of the genus two spherical double affine

As is well-known, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper, we show that ℓ 1 -bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer

We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.

It is established that for every pair of additive forms f , g of degree k in more than 2 k 2 variables, the equations f = g = 0 have a non-trivial p -adic solution for all odd primes p .

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogs of local and global LYM inequalities, the Erdős–Ko–Rado theorem, and the Ahlswede–Khachatrian

In this note we present upper bounds for the variational eigenvalues of the p -Laplacian on smooth domains of complete n -dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given metric for 1 < p ⩽ n , and upper bounds for all p > 1 when we fix a metric. To do so, we

We propose an algorithm that calculates isogenies between elliptic curves defined over an extension K of Q 2 . It consists in efficiently solving with a logarithmic loss of 2-adic precision the first-order differential equation satisfied by the isogeny.

Let S be an orientable, finite-type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this paper, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates

We study the space of all d -tuples of unitaries u = ( u 1 , … , u d ) using dilation theory and matrix ranges. Given two such d -tuples u and v generating, respectively, C*-algebras A and B , we seek the minimal dilation constant c = c ( u , v ) such that u ≺ c v , by which we mean that there exist faithful ∗ -representations π : A → B ( H ) and ρ : B → B ( K ) , with H ⊆ K , such that for all i

We show that group actions on irreducible CAT ( 0 ) cube complexes with no free faces are uniquely determined by their ℓ 1 length function. Actions are allowed to be non-proper and non-cocompact, as long as they are ℓ 1 -minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception

We describe the second order obstruction to deformation for nearly G 2 structures on compact manifolds. Building on work of Alexandrov and Semmelmann, this allows proving rigidity under deformation for the proper nearly G 2 structure on the Aloff–Wallach space N ( 1 , 1 ) .

For n -dimensional Riemannian manifolds M with Ricci curvature bounded below by − ( n − 1 ) , the volume entropy is bounded above by n − 1 . If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD ∗ ( − ( N − 1 ) , N ) spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure

Kobayashi recently proved that the generalized Heegner cycles of Bertolini–Darmon–Prasanna can be interpolated along the anticyclotomic tower, giving rise to distribution valued cohomology classes with expected growth rate. We interpolate these classes along Coleman families. This construction plays a role in the proofs of p -adic Gross–Zagier formulae at for non-ordinary eigenforms and consequentially

We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result, showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.

We study G L ( 2 ) -structures on differential manifolds. A G L ( 2 ) -structure is a smooth field of rational normal curves in the tangent bundle of a manifold. We provide an explicit construction of a canonical connection for any G L ( 2 ) -structure in any dimension greater than 3. Moreover, we prove that any G L ( 2 ) -structure on an even-dimensional manifold defines a certain almost-complex structure

We consider the Pfaffian–Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi–Yau threefolds, and verifications of Kimura's finite‐dimensionality conjecture, Voevodsky's smash conjecture

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on n vertices can be decomposed into at most ⌈ n 2 ⌉ paths, while a conjecture of Hajós states that any Eulerian graph on n vertices can be decomposed into at most ⌊ n − 1 2 ⌋ cycles. The Erdős–Gallai conjecture states that any graph on n vertices

For a connected reductive group G over R , we study cohomological A ‐parameters, which are Arthur parameters with the infinitesimal character of a finite‐dimensional representation of G ( C ) . We prove a structure theorem for such A ‐parameters, and deduce from it that a morphism of L ‐groups which takes a regular unipotent element to a regular unipotent element respects cohomological A ‐parameters

Suppose F is a finite extension of Q p , G is the group of F ‐points of a connected reductive F ‐group, and I 1 is a pro‐ p ‐Iwahori subgroup of G . We construct two spectral sequences relating derived functors on mod‐ p representations of G to the analogous functors on Hecke modules coming from pro‐ p ‐Iwahori cohomology. More specifically: (1) using results of Ollivier–Vignéras, we provide a link

We show that the Quot scheme Q L n = Quot A 3 ( I L , n ) parameterising length n quotients of the ideal sheaf of a line in A 3 is a global critical locus, and calculate the resulting motivic partition function (varying n ), in the ring of relative motives over the configuration space of points in A 3 . As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot

In this work, we compare Lévy space‐time white noises and cylindrical Lévy processes. Lévy space‐time white noises are defined as infinitely divisible independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes. It is shown that Lévy space‐time white noises correspond to an entire sub‐class of cylindrical Lévy processes, which is

A partial linear space is a pair ( P , L ) where P is a non‐empty set of points and L is a collection of subsets of P called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms G of a proper partial linear space acts transitively on

We establish L p 1 × ⋯ × L p k → L r and ℓ p 1 × ⋯ × ℓ p k → ℓ r type bounds for multilinear maximal operators associated to averages over isometric copies of a given non‐degenerate k ‐simplex in both the continuous and discrete settings. These provide natural extensions of L p → L p and ℓ p → ℓ p bounds for Stein's spherical maximal operator and the discrete spherical maximal operator, with each of

We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients

We classify n ‐dimensional geometric graph manifolds with non‐negative scalar curvature by first showing that if n > 3 , the universal cover splits off a codimension 3‐Euclidean factor. We then proceed with the classification of the 3‐dimensional case, where the condition is equivalent to the eigenvalues of the Ricci tensor being ( λ , λ , 0 ) with λ ⩾ 0 . In this case we prove that such a manifold

In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color.

Given a sequence ξ : Z + → C , we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series F ξ ( z ) = ∑ n ⩾ 0 ξ ( n ) z n n ! . This condition yields practically all known instances of random and pseudo‐random sequences ξ with this property (due to Nassif, Littlewood, Chen–Littlewood, Levin, Eremenko–Ostrovskii, Kabluchko–Zaporozhets, Borichev–Nishry–Sodin)

In this paper we generalize classical results regarding minimal realizations of non‐commutative (nc) rational functions using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. We prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number 1. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is

This paper studies the virtual χ − y ‐genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea–Pandharipande. We first prove a structural result expressing the equivariant virtual χ − y ‐genera of Quot schemes universally in terms of the Seiberg–Witten invariants. The formula is simpler for curve classes of Seiberg–Witten length

In this paper we prove that certain points in the Hecke orbit of a CM point in the Siegel modular variety do not lie in the open Torelli locus under suitable conditions on the field of definition and the CM factors that arise in the corresponding CM abelian variety. The proof is a combination of properties of stable Faltings height and known cases of the Sato–Tate equidistribution, and is motivated

We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly generated ∞ ‐categories. The first theorem says that the construction of higher derivatives for spectrum‐valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so‐called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi‐isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck

We prove that synthetic lower Ricci bounds for metric measure spaces — both in the sense of Bakry–Émery and in the sense of Lott–Sturm–Villani — can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality.

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R 3 . The motion of the fluid is modeled by the Navier–Stokes equations, whereas the motion of the rigid body is described by the conservation law of linear and angular momentum. Under the assumption that the diameter of the rigid body tends to zero and that the density of the rigid body goes

In this paper, we develop the theory of abstract crystals for quantum Borcherds–Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of B ( ∞ ) and B ( λ ) as its application, where B ( ∞ ) and B ( λ ) are the crystals of the negative half part of the quantum Borcherds–Bozec algebra U q ( g ) and its

This paper considers the following fundamental problem about intersections in projective space: When is the intersection of a (varying) curve with a (fixed) hypersurface a general set of points on the hypersurface?

We consider topological dynamical systems ( X , T ) , where X is a compact metrizable space and T denotes an action of a countable amenable group G on X by homeomorphisms. For two such systems ( X , T ) and ( Y , S ) and a factor map π : X → Y , an intermediate factor is a topological dynamical system ( Z , R ) for which π can be written as a composition of factor maps ψ : X → Z and φ : Z → Y . In

In this paper, we study transition density functions for pure jump unimodal Lévy processes killed upon leaving an open set D . Under some mild assumptions on the Lévy density, we establish two‐sided Dirichlet heat kernel estimates when the open set D is C 1 , 1 . Our result covers the case that the Lévy densities of unimodal Lévy processes are regularly varying functions whose indices are equal to

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated

The deformed Hermitian Yang‐Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper, we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both

The affine diffeomorphism group Aff ( S , q ) of a half‐translation surface ( S , q ) comprise the self‐diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space. We prove that any finitely generated subgroup of Aff ( S , q ) is undistorted in the

In this paper, we construct incomplete versions of the equivariant stable category; that is, equivariant stabilizations of the category of G ‐spaces with respect to incomplete systems of transfers encoded by an N ∞ operad O . These categories are built from the categories of O ‐algebras in G ‐spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties

Using a variant of the Boardman–Vogt tensor product, we construct an action of the Grothendieck–Teichmüller group on the completion of the little n ‐disks operad E n . This action is used to establish a partial formality theorem for E n with mod p coefficients and to give a new proof of the formality theorem in characteristic zero.

We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary ( M , g ) of dimension n = 2 , 3 . We prove results of unique recovery of the nonlinear term F ( t , x , u ) , appearing in the equation ∂ t 2 u − Δ g u + F ( t , x , u ) = 0 on ( 0 , T ) × M with T > 0 , from partial knowledge of

Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1‐parameter family of homomorphisms f r , from the knot concordance group to R . Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points

For convex domains with C 1 , ε boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non‐compact connected simple Lie group

We study the compactness properties of metrics of prescribed fractional Q -curvature of order 3 in R 3 . We will use an approach inspired from conformal geometry, seeing a metric on a subset of R 3 as the restriction of a metric on R + 4 with vanishing fourth-order Q -curvature. We will show that a sequence of such metrics with uniformly bounded fractional Q -curvature can blow up on a large set (roughly

We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a class of differential and pseudo‐differential operators L , both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by L . We cover the cases of fractional Laplacians and Ornstein–Uhlenbeck operators with fractional diffusion in finite

Let f be a Bianchi modular form, that is, an automorphic form for GL ( 2 ) over an imaginary quadratic field F , and let p be a prime of F at which f is new. Let K be a quadratic extension of F , and L ( f / K , s ) the L -function of the base-change of f to K . Under certain hypotheses on f and K , the functional equation of L ( f / K , s ) ensures that it vanishes at the central point. The Bloch–Kato

We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson–Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles-quivers correspondence for rational tangles to algebraic tangles by developing

We develop the theory of weighted Ricci curvature in a weighted Lorentz–Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz–Finsler

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U ⊕ ⟨ − 2 k ⟩ -polarized K3 surfaces. Such moduli spaces are proved to be of general type for k ⩾ 220 . The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli

We study the behaviour of quasi-geodesics in Out ( F n ) . Given an element ϕ in Out ( F n ) , there are several natural paths connecting the origin to ϕ in Out ( F n ) ; for example, paths given by Stallings' folding algorithm and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these paths is, in general, a quasi-geodesic in Out ( F n ) . In fact, in contrast

We study simply connected Lie groups G for which the hull-kernel topology of the primitive ideal space Prim ( G ) of the group C ∗ -algebra C ∗ ( G ) is T 1 , that is, the finite subsets of Prim ( G ) are closed. Thus, we prove that C ∗ ( G ) is AF-embeddable. To this end, we show that if G is solvable and its action on the centre of [ G , G ] has at least one imaginary weight, then Prim ( G ) has

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ -norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement;

In this paper, we compute the cohomology class of certain ‘special maximal-rank loci’ originally defined by Aprodu and Farkas. By showing that such classes are non-zero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg

We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the Γ -limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies