48 research outputs found
Seiberg duality, quiver gauge theories, and Ihara's zeta function
We study Ihara’s zeta function for graphs in the context of quivers arising from gauge theories, especially under Seiberg duality transformations. The distribution of poles is studied as we proceed along the duality tree, in light of the weak and strong graph versions of the Riemann Hypothesis. As a by-product, we find a refined version of Ihara’s zeta function to be the generating function for the generic superpotential of the gauge theory
Graph Zeta Function and Gauge Theories
Along the recently trodden path of studying certain number theoretic
properties of gauge theories, especially supersymmetric theories whose vacuum
manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large
classes of quiver theories and periodic tilings by bi-partite graphs. In
particular, we examine issues such as the spectra of the adjacency and whether
the gauge theory satisfies the strong and weak versions of the graph
theoretical analogue of the Riemann Hypothesis.Comment: 35 pages, 7 Figure
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
Arbeitsgemeinschaft: The Kadison-Singer Conjecture
The solution to the Kadison–Singer conjecture used techniques that intersect a number of areas of mathematics. The goal of this Arbeitsgemeinschaft was to bring together people from each of these fields to support interactions between these areas. While the majority of the talks centered around topics in polynomial geometry, combinatorics, and real algebraic geometry, participants came from areas such as harmonic analysis, convex geometry, and frame theory
Low-dimensional Topology and Number Theory
The workshop brought together topologists and number theorists with the intent of exploring the many tantalizing connections between these areas