We investigate quasilinear discrete PDEs ∂ t u = Δ N φ ( u ) + K f ( u ) \partial _t u = \Delta ^N \varphi (u)+ Kf(u) of reaction-diffusion type with nonlinear diffusion term defined on an n n -dimensional unit torus discretized with mesh size 1 N \tfrac 1N for N ∈ N N\in {\mathbb {N}} , where Δ N \Delta ^N is the normalized discrete Laplacian, φ \varphi is a strictly increasing C 5 C^5 function and f f is a C 1 C^1 function. We establish L ∞ L^\infty bounds and space-time Hölder estimates, both uniform in N N , of the first and second spatial discrete derivatives of the solutions. In the equation, K > 0 K>0 is a large constant and we show how these estimates depend on K K . The motivation for this work stems originally from the study of hydrodynamic scaling limits of interacting particle systems. Our method is a two steps approach in terms of the Hölder estimate and Schauder estimate, which is known for continuous parabolic PDEs. We first show the discrete Hölder estimate uniform in N N for the solutions of the associated linear discrete PDEs with continuous coefficients, based on the Nash estimate. We next establish the discrete Schauder estimate for linear discrete PDEs with uniform Hölder coefficients. The link between discrete and continuous settings is given by polylinear interpolations. Since this operation has a non-local nature, the method requires proper modifications. We also discuss another method based on the study of the corresponding fundamental solutions.

Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications. We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.

In this paper, we prove a version of Conn’s linearization theorem for the Lie algebra s l 2 ( C ) ≃ s o ( 3 , 1 ) \mathfrak {sl}_2(\mathbb {C})\simeq \mathfrak {so}(3,1) . Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) is linearizable. In the first part, we calculate the Poisson cohomology associated to s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.

We prove that an infinitesimally Hilbertian C D ( 0 , N ) CD(0,N) space containing a line splits as the product of R \mathbb {R} and an infinitesimally Hilbertian C D ( 0 , N − 1 ) CD(0,N-1) space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space W 1 , 2 ( X , d , m ) W^{1,2}(X,\mathsf {d},\mathfrak {m}) , which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

Let Y Y be a symmetric Borel right process with locally compact state space T ⊆ R 1 T\subseteq R^{1} and potential densities u ( x , y ) u(x,y) with respect to some σ \sigma -finite measure on T T . Let g g and f f be finite excessive functions for Y Y . Set \[ u g , f ( x , y ) = u ( x , y ) + g ( x ) f ( y ) , x , y ∈ T . u_{g, f}(x,y)= u(x,y)+g(x)f(y),\qquad x,y\in T. \] In this paper we take Y Y to be a symmetric Lévy process, or a diffusion, that is killed at the end of an independent exponential time or the first time it hits 0. Under general smoothness conditions on g g , f f , u u and points d ∈ T d\in T , laws of the iterated logarithm are found for X k / 2 = { X k / 2 ( t ) , t ∈ T } X_{k/2} =\{X_{k/2}(t), t\in T \} , a k / 2 − k/2- permanental process with kernel { u g , f ( x , y ) , x , y ∈ T } \{u_{g, f}(x,y),x,y\in T \} , of the following form: For all integers k ≥ 1 k\geq 1 , \[ lim sup x → 0 | X k / 2 ( d + x ) − X k / 2 ( d ) | ( 2 σ 2 ( x ) log ⁡ log ⁡ 1 / x ) 1 / 2 = ( 2 X k / 2 ( d ) ) 1 / 2 , a . s . , \limsup _{x \to 0}\frac {| X_{k/2}( d+x)- X_{k/2} (d)|}{ \left ( 2 \sigma ^{2}\left (x\right )\log \log 1/x\right )^{1/2}}= \left ( 2 X _{k/2} (d)\right )^{1/2}, \qquad a.s. , \] where, \[ σ 2 ( x ) = u ( d + x , d + x ) + u ( x , x ) − 2 u ( d + x , x ) . \sigma ^2(x)=u(d+x,d+x)+u(x,x)-2u(d+x,x). \] Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the iterated logarithm are found for the local times of certain Markov processes with potential densities that have the form of { u g , f ( x , y ) , x , y ∈ T } \{u_{g, f}(x,y),x,y\in T \} or are slight modifications of it.

In this memoir, we study the counterpart of Grothendieck’s projectivization construction in the context of derived algebraic geometry. Our main results are as follows: First, we define the derived projectivization of a connective complex, study its fundamental properties such as finiteness properties and functorial behaviors, and provide explicit descriptions of their relative cotangent complexes. We then focus on the derived projectivizations of complexes of perfect-amplitude contained in [ 0 , 1 ] [0, 1] . In this case, we prove a generalized Serre’s theorem, a derived version of Beilinson’s relations, and establish semiorthogonal decompositions for their derived categories. Finally, we show that many moduli problems fit into the framework of derived projectivizations, such as moduli spaces that arise in Hecke correspondences. We apply our results to these situations.

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry.

We construct a geometric decomposition of the convex cores of ϵ \epsilon -thick hyperbolic 3 3 -manifolds M M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M M and on the radius of any embedded ball in the convex core of M M .

Consider a Schrödinger operator on an asymptotically Euclidean manifold X X of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy’s second 2nd-microlocal approach, “the Lagrangian approach,” we analyze – uniformly, all the way down to E = 0 E=0 – the output of the limiting resolvent R ( E ± i 0 ) = lim ϵ → 0 + R ( E ± i ϵ ) R(E\pm i 0) = \lim _{\epsilon \to 0^+} R(E\pm i \epsilon ) . The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to E = 0 E=0 . In fact, we will construct a compactification of ( 0 , 1 ] E × X (0,1]_E\times X on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as r → ∞ r\to \infty times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime. So, in summary, we develop the low-energy scattering theory of attractive Coulombic potentials in the time-independent formalism, in the process studying delicate behavior at low energy and large scales.

We introduce a new and extensive theory of noncommutative convexity along with a corresponding theory of noncommutative functions. We establish noncommutative analogues of the fundamental results from classical convexity theory, and apply these ideas to develop a noncommutative Choquet theory that generalizes much of classical Choquet theory. The central objects of interest in noncommutative convexity are noncommutative convex sets. The category of compact noncommutative sets is dual to the category of operator systems, and there is a robust notion of extreme point for a noncommutative convex set that is dual to Arveson’s notion of boundary representation for an operator system. We identify the C*-algebra of continuous noncommutative functions on a compact noncommutative convex set as the maximal C*-algebra of the operator system of continuous noncommutative affine functions on the set. In the noncommutative setting, unital completely positive maps on this C*-algebra play the role of representing measures in the classical setting. The role of noncommutative convex functions is crucial to our theory, and this is a new notion in the theory of noncommutative functions. The convex noncommutative functions determine an order on the set of unital completely positive maps that is analogous to the classical Choquet order on probability measures. We characterize this order in terms of the extensions and dilations of the maps, providing a powerful new perspective on the structure of completely positive maps on operator systems. Finally, we establish a noncommutative generalization of the Choquet-Bishop-de Leeuw theorem asserting that every point in a compact noncommutative convex set has a representing map that is supported on the extreme boundary. In the separable case, we obtain a corresponding integral representation theorem.