Given a chiral vertex operator algebra satisfying a suitable finiteness condition with semisimplicity of the zero-mode algebra as well as a regularity condition for induced modules, we construct conformal field theories over the projective line and prove the factorization theorem. We appropriately generalize the arguments in [TUY] so that we are able to define sheaves of conformal blocks and study them in detail.

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups $S_3$, ${\left(\mathbb{Z}/n\mathbb{Z}\right)}^2$, and $\mathbb{Z}/2\mathbb{Z}$. The last symmetry is a certain \emph{curious duality} of the quantum cohomology which inverts the quantum parameter $q$. Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter $q$ which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of $q$ with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.

Let $X$ be a normal projective variety defined over an algebraically closed field $k$ of positive characteristic. Let $G$ be a connected reductive group defined over $k$. We prove that some Frobenius pullback of a principal $G$-bundle admits the canonical reduction $E_P$ such that its extension by $P\to P/R_u\left(P\right)$ is strongly semistable (see Theorem 5.1).

Then we show that there is only a small difference between semistability of a principal $G$-bundle and semistability of its Frobenius pullback (see Theorem 6.3). This and the boundedness of the family of semistable torsion-free sheaves imply the boundedness of semistable (rational) principal $G$-bundles.

We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding ``zeta function'' ${\zeta }_G\left(s\right)=\sum _{\mathrm{M\; max}G}{∣ G:M∣ }^{-s}$, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups $G$ and for any fixed real number $s>1$, ${\zeta }_G\left(s\right)\to 0$ as $∣ G∣ \to \infty$. This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343, that the symmetric group $S_n$ has $n^{o\left(1\right)}$ conjugacy classes of primitive maximal subgroups.

Consider a convex domain $B$ of ${ℝ}^3$. We prove that there exist complete minimal surfaces that are properly immersed in $B$. We also demonstrate that if $D$ and $D\text{'}$ are convex domains with $D$ bounded and the closure of $D$ contained in $D\text{'}$, then any minimal disk whose boundary lies in the boundary of $D$ can be approximated in any compact subdomain of $D$ by a complete minimal disk that is proper in $D\text{'}$. We apply these results to study the so-called type problem for a minimal surface: we demonstrate that the interior of any convex region of ${ℝ}^3$ is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in [9].