This work is devoted to the study of the foundations of quantum $K$-theory, a $K$-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary $K$-ring $K(X)$ of a smooth projective variety $X$, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.

We show that every algebraic group of type $E_8$ over any field becomes split over some field extension of degree dividing ${\text{2}}^{\text{6}}\cdot 3^2\cdot 5=2880$. This improves a bound by Tits and, in fact, is optimal.

The amoeba of a holomorphic function $f$ is, by definition, the image in $R^n$ of the zero locus of $f$ under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function $N_f$ with the property that its Monge-Ampére mass is concentrated to the amoeba of $f$ We obtain results of two kinds; by approximating $N_f$ with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of $f$; by computing the Monge-Ampére measure, we find sharp bounds for the area of amoebas in $\text{ℝ}$. We also consider systems of functions $f_1,\dots ,f_n$ and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semisimple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral that arises in the fundamental lemma is equal to the Lefschetz trace of the Frobenius acting on the étale cohomology of a related variety.