We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\text{Spec}\,H_{T}^{*}\left( X;\,\mathbb{C} \right)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_{2}^{T}\left( X;\,\mathbb{C} \right)$. In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

Let $G$ be a simple, compact, simply-connected Lie group localized at an odd prime $p$. We study the group of homotopy classes of self-maps $\left[ G,\,G \right]$ when the rank of $G$ is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H\left[ G,\,G \right]$. The low rank condition gives $G$ certain structural properties which make calculations accessible. Several examples and applications are given.

Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{L}\left( \mathcal{H} \right)$ denote the collection of bounded linear operators on $\mathcal{H}$. In this paper, we show that for any operator $A\,\in \,\mathcal{L}\left( \mathcal{H} \right)$, there exists a stably finitely $\left( \text{SI} \right)$ decomposable operator ${{A}_{\epsilon }}$, such that $\left\| A-{{A}_{\epsilon }} \right\|\,<\,\epsilon$ and ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)/\text{rad}\,{{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is commutative, where rad ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is the Jacobson radical of ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\,\mapsto \,\text{Det}\,{{\text{D}}^{2}}u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function $u$ on a bounded set $\Omega \,\subset \,{{\mathbb{R}}^{2}}$ satisfies the equation $\text{Det}\,{{D}^{2}}u\,=\,0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration ${{C}_{14}}$. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.

Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph $G$ is $S$-edge-colourable for a Steiner triple system $S$ if its edges can be coloured with points of $S$ in such a way that the points assigned to three edges sharing a vertex form a triple in $S$. Among others, we show that all cubic graphs are $S$-edge-colourable for every non-projective non-affine point-transitive Steiner triple system $S$.

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras ${{\mathfrak{L}}_{q}}$. The center of ${{\mathfrak{L}}_{q}}$ now is generally infinite dimensional.

In this paper, $\mathbb{Z}$-graded Verma modules $\tilde{V}\left( \varphi \right)$ over ${{\mathfrak{L}}_{q}}$ and their corresponding irreducible highest weight modules $V\left( \varphi \right)$ are defined for some linear functions $\varphi $. Necessary and sufficient conditions for $V\left( \varphi \right)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\tilde{V}\left( \varphi \right)$ to be irreducible are obtained.

We study $p$-indivisibility of the central values $L\left( 1,\,{{E}_{d}} \right)$ of quadratic twists ${{E}_{d}}$ of a semi-stable elliptic curve $E$ of conductor $N$. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants $d$ splits naturally into several families ${{\mathcal{F}}_{S}}$, indexed by subsets $S$ of the primes dividing $N$. Let ${{\delta }_{S}}={{\gcd }_{d\in {{\mathcal{F}}_{S}}}}L{{(1,{{E}_{d}})}^{\text{alg}}}$, where $L{{(1,{{E}_{d}})}^{\text{alg}}}$ denotes the algebraic part of the central $L$-value, $L(1,\,{{E}_{d}})$. Our main theorem relates the $p$-adic valuations of ${{\delta }_{S}}$ as $S$ varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation between Waldspurger packets on $\widetilde{\text{S}{{\text{L}}_{2}}}$ and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of $p$-indivisibility of $L$-values of quadratic twists.

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

In this paper, we reinterpret the Colmez conjecture on the Faltings height of $\text{CM}$ abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a $\text{CM}$ abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for $\text{CM}$ abelian surfaces is equivalent to the cuspidality of this modular form.

We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\mathcal{F}{{\ell }_{G}}$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\mathcal{F}{{\ell }_{G}}$ lie on a paraboloid.