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Global Smooth Solutions for the Inviscid SQG Equation
In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.
Degree Theory of Immersed Hypersurfaces
The authors develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function.
Conformal Graph Directed Markov Systems on Carnot Groups
The authors develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, they develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. They illustrate their results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.
The Irreducible Subgroups of Exceptional Algebraic Groups
This paper is a contribution to the study of the subgroup structure of excep-tional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group G is called irreducible if it lies in no proper parabolic subgroup of G. In this paper we com-plete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of ir...
Localization for $THH(ku)$ and the Topological Hochschild and Cyclic Homology of Waldhausen Categories
The authors resolve the longstanding confusion about localization sequences in $THH$ and $TC$ and establish a specialized devissage theorem.
Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case
The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $epsilon leq c_0mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of ``2.5 dimensional'' streamwise-independent solutions referred to as streaks.
Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees
First, there are sets with minimal weak truth table degree which bound noncomputable computably enumerable sets under Turing reducibility. Second, no set with computable enumerable Turing degree can have minimal weak truth table degree. Third, no $Delta^0_2$ set which Turing bounds a promptly simple set can have minimal weak truth table degree.
The Mother Body Phase Transition in the Normal Matrix Model
In this present paper, the authors consider the normal matrix model with cubic plus linear potential.
Theory of Fundamental Bessel Functions of High Rank
In this article, the author studies fundamental Bessel functions for $mathrm{GL}_n(mathbb F)$ arising from the Voronoí summation formula for any rank $n$ and field $mathbb F = mathbb R$ or $mathbb C$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of this study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. The author proves the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.
Double Affine Hecke Algebras and Congruence Groups
The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by auto-morphisms of a finite index subgroup of the Artin group of type A2, which descends to a faithful outer action of a congruence subgroup of SL(2, Z)or PSL(2, Z). This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. It turns out that the structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, even by adjoint data; we prove our main result by...
