79 research outputs found
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
The equivariant pair-of-pants product in fixed point Floer cohomology
We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.National Science Foundation (U.S.) (Grant DMS-1005288)Simons Foundation (Simons Investigator grant)Radcliffe Institute for Advanced Stud
The equivariant pair-of-pants product in fixed point Floer cohomology
The Floer cohomology of a symplectic automorphism and that of its square are
related by the pair-of-pants product. For exact symplectic automorphisms, we
introduce an equivariant version of that product, and use it to prove a
Smith-type inequality of ranks between Floer cohomology groups. Under
additional topological assumptions, the same inequality was previously proved
by Hendricks, using a different strategy.Comment: v2: discussion of the situation for monotone symplectic manifolds
slightly expanded, and moved into a separate section at the end; v3:
exposition expanded, one error corrected (in the proof of Lemma 6.2); v4:
typos removed; v5: more typos remove
Non-Deterministic Communication Complexity of Regular Languages
In this thesis, we study the place of regular languages within the
communication complexity setting. In particular, we are interested in the
non-deterministic communication complexity of regular languages.
We show that a regular language has either O(1) or Omega(log n)
non-deterministic complexity. We obtain several linear lower bound results
which cover a wide range of regular languages having linear non-deterministic
complexity. These lower bound results also imply a result in semigroup theory:
we obtain sufficient conditions for not being in the positive variety Pol(Com).
To obtain our results, we use algebraic techniques. In the study of regular
languages, the algebraic point of view pioneered by Eilenberg (\cite{Eil74})
has led to many interesting results. Viewing a semigroup as a computational
device that recognizes languages has proven to be prolific from both semigroup
theory and formal languages perspectives. In this thesis, we provide further
instances of such mutualism.Comment: Master's thesis, 93 page
Dynamical Systems on Spectral Metric Spaces
Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert
space on which A acts and D is a selfadjoint operator with compact resolvent
such that the set of elements of A having a bounded commutator with D is dense.
A spectral metric space, the noncommutative analog of a complete metric space,
is a spectral triple (A,H,D) with additional properties which guaranty that the
Connes metric induces the weak*-topology on the state space of A. A
*-automorphism respecting the metric defined a dynamical system. This article
gives various answers to the question: is there a canonical spectral triple
based upon the crossed product algebra AxZ, characterizing the metric
properties of the dynamical system ? If is the noncommutative analog
of an isometry the answer is yes. Otherwise, the metric bundle construction of
Connes and Moscovici is used to replace (A,) by an equivalent dynamical
system acting isometrically. The difficulties relating to the non compactness
of this new system are discussed. Applications, in number theory, in coding
theory are given at the end
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