Let $G$ be a semisimple linear algebraic group defined over an algebraically closed field $k$. Fix a smooth projective curve $X$ defined over $k$, and also fix a closed point $x\in X$. Given any strongly semistable principal $G$-bundle $E_G$ over $X$, we construct an affine algebraic group scheme defined over $k$, which we call the monodromy of $E_G$. The monodromy group scheme is a subgroup scheme of the fiber over $x$ of the adjoint bundle $E_G\times {}^{G}G$ for $E_G$. We also construct a reduction of structure group of the principal $G$-bundle $E_G$ to its monodromy group scheme. The construction of this reduction of structure group involves a choice of a closed point of $E_G$ over $x$. An application of the monodromy group scheme is given. We prove the existence of strongly stable principal $G$-bundles with monodromy $G$

We consider a model describing premixed combustion in the presence of fluid flow: a reaction-diffusion equation with passive advection and ignition-type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the solution turns out to be surprisingly subtle. In this article we provide a sharp characterization of quenching for shear flows; the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant. The efficiency of quenching depends strongly on the geometry and scaling of the flow. We discuss the cases of slowly and quickly varying flows, proving rigorously scaling laws that have been observed earlier in numerical experiments. The results require new estimates on the behavior of the solutions to the advection-enhanced diffusion equation (also known as passive scalar in physical literature), a classical model describing a wealth of phenomena in nature. The technique involves probabilistic and partial-differential-equation (PDE) estimates, in particular, applications of Malliavin calculus and the central limit theorem for martingales

Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c={\mathrm{eH}}_{c}e$. Then $U_c$ is filtered by order of differential operators with associated graded ring $\mathrm{gr}{U}_c=\mathbb{C}[\mathfrak{h}\oplus {\mathfrak{h}}^{*}]{}^W$, where $W$ is the nth symmetric group. Using the $\mathbb{Z}$-algebra construction from [GS], it is also possible to associate to a filtered $H_c$- or $U_c$-module $M$ a coherent sheaf $\stackrel{\wedge }{\Phi }(M)$ on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of $U_c$ and $H_c$, and we relate it to Hilb(n) and to the resolution of singularities $\tau :\mathrm{Hilb}(n)\to \mathfrak{h}\oplus {\mathfrak{h}}^{*}/W$. For example, we prove the following.

• If $c=1/n$ so that $L_c(\mathsf{triv})$ is the unique one-dimensional simple $H_c$-module, then $\stackrel{\wedge }{\Phi }({\mathrm{eL}}_c(\mathsf{triv}))\cong {\mathcal{O}}_{Z_n}$, where $Z_n={\tau }^{-1}(0)$ is the punctual Hilbert scheme.

• If $c=1/n+k$ for $k\in \mathbb{N}$, then under a canonical filtration on the finite-dimensional module $L_c(\mathsf{triv})$, $\mathrm{gr}{\mathrm{eL}}_c(\mathsf{triv})$ has a natural bigraded structure that coincides with that on $H^0(Z_n,{\mathcal{L}}^k)$, where $\mathcal{L}\cong O_{\mathrm{Hilb}(n)}(1)$; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].

• Under mild restrictions on $c$, the characteristic cycle of $\stackrel{\wedge }{\Phi }(e{\Delta }_c(\mu ))$ equals $\sum _{\lambda }{K}_{\mu \lambda }[Z_{\lambda }]$, where $K_{\mu \lambda }$ are Kostka numbers and the $Z_{\lambda }$ are (known) irreducible components of ${\tau }^{-1}(\mathfrak{h}/W)$

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose $L$-groups have classical derived groups. The important transfer from ${\mathrm{GSp}}_4$ to ${\mathrm{GL}}_4$ follows from our result as a special case