983 research outputs found
Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity
This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.Comment: 70 pages, abstract corrected and modifie
Gender Nonconformity and the Unfulfilled Promise of Price Waterhouse v. Hopkins
The Supreme Court has articulated a doctrinal framework that, if construed and applied properly, provides the lower federal courts with the analytical tools necessary to identify and proscribe workplace rules that compel individuals to adhere to appearance, attire, and behavioral norms that operate to reinforce gendered expectations.1 Since the Supreme Court has ruled that penalizing an individual for failing to conform to gendered norms of behavior constitutes a form of sex-based discrimination,2 one would expect that employees would have achieved some measure of success in challenging such policies
Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem
For each graph on two vertices, and each divisor on the graph in the sense of
Baker-Norine, we describe a sheaf of vector spaces on a finite category whose
zeroth Betti number is the Baker-Norine "Graph Riemann-Roch" rank of the
divisor plus one. We prove duality theorems that generalize the Baker-Norine
"Graph Riemann-Roch" Theorem
The Relativized Second Eigenvalue Conjecture of Alon
We prove a relativization of the Alon Second Eigenvalue Conjecture for all
-regular base graphs, , with : for any , we show that
a random covering map of degree to has a new eigenvalue greater than
in absolute value with probability .
Furthermore, if is a Ramanujan graph, we show that this probability is
proportional to , where
is an integer depending on , which can be computed by a finite algorithm for
any fixed . For any -regular graph, , is
greater than .
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
-regular graphs, for certain values of
- β¦