Related papers: Geometric Complexity Theory VI: the flip via satur…
We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the…
Copositive optimization is a special case of convex conic programming, and it consists of optimizing a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
Structured reinforcement learning and stochastic optimization often involve parameters evolving on matrix Lie groups such as rotations and rigid-body transformations. We establish a representation-optimization dichotomy for…
In this paper, we study the weighted difference substitutions from geometrical views. First, we give the geometric meanings of the weighted difference substitutions, and introduce the concept of convergence of the sequence of substitution…
Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of…
In the building of a finite group of Lie type we consider the incidence relations defined by oppositeness of flags. Such a relation gives rise to a homomorphism of permutation modules (in the defining characteristic) whose image is a simple…
Let F be a nonarchimedean locally compact field with residue characteristic p and G(F) the group of F-rational points of a connected reductive group. Following Schneider and Stuhler, one can realize, in a functorial way, any smooth complex…
This article describes a {\em nonstandard} quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson…
Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class…
The complexity class NP of decision problems that can be solved nondeterministically in polynomial time is of great theoretical and practical importance where the notion of polynomial-time reductions between NP-problems is a key concept for…
We give an introduction to some of the recent ideas that go under the name "geometric complexity theory". We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the…
Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular…
We prove that the vanishing of the functoriality morphism for the \'etale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups…
Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…
The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the…
An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the…
Motivated by the idea that consistent quantum field theories should admit a finite description, we investigate the complexity of effective field theories using the framework of effective o-minimality. Our focus is on quantifying the…
We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an…
We explain how recent developments in the fields of realisability models for linear logic -- or geometry of interaction -- and implicit computational complexity can lead to a new approach of implicit computational complexity. This…