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You searched for `+publisher:"Cornell University" +contributor:("Aguiar, Marcelo")`

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Cornell University

1. Gallagher, Joseph. On conjectures related to character varieties of knots and Jones polynomials .

Degree: 2018, Cornell University

URL: http://hdl.handle.net/1813/64984

It is well known that the Kauffman Bracket Skein Module of a knot complement K_{q}(S^{3} K) is canonically a module over the Z_{2-invariants} of the quantum torus, A_{q}^{Z2}, and this module determines the colored Jones polynomials J_{n}(K; q) of the knot K. Berest and Samuelson identified a conjecture for knots under which a close variant of K_{q}(S^{3} K) canonically becomes a module over a certain Double Affine Hecke Algebra, from which they defined a family of polynomials J_{n}(K; q; t_{1}; t_{2}) generalizing the classical polynomials of Jones. In this thesis an analogue of Habiro’s cyclotomic equation for the J_{n}(K; q) is discovered for J_{n}(K; q; t_{1}; t_{2}). An integrality result for the coefficients in this equation is found as a corollary, offering evidence for the conjecture of Berest and Samuelson for all knots. Separately, the conjecture of Berest and Samuelson is studied at the particular value q = -1 where it is known to relate to properties of SL_{2}(C)-character varieties of knots. Computational methods are used to establish that the conjecture holds for some non-invertible knots, which was not previously known.
*Advisors/Committee Members: Manning, Jason F. (committeeMember), Aguiar, Marcelo (committeeMember).*

Subjects/Keywords: Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Gallagher, J. (2018). On conjectures related to character varieties of knots and Jones polynomials . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/64984

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Gallagher, Joseph. “On conjectures related to character varieties of knots and Jones polynomials .” 2018. Thesis, Cornell University. Accessed December 06, 2019. http://hdl.handle.net/1813/64984.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Gallagher, Joseph. “On conjectures related to character varieties of knots and Jones polynomials .” 2018. Web. 06 Dec 2019.

Vancouver:

Gallagher J. On conjectures related to character varieties of knots and Jones polynomials . [Internet] [Thesis]. Cornell University; 2018. [cited 2019 Dec 06]. Available from: http://hdl.handle.net/1813/64984.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Gallagher J. On conjectures related to character varieties of knots and Jones polynomials . [Thesis]. Cornell University; 2018. Available from: http://hdl.handle.net/1813/64984

Not specified: Masters Thesis or Doctoral Dissertation

Cornell University

2. Saied, Amin. Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups .

Degree: 2018, Cornell University

URL: http://hdl.handle.net/1813/64873

This work deals with the stable representation theory of categories related to various families of symmetric groups. In particular, we study the categories FB, FI, and introduce the new category \PD. The notion of representation stability is recast in the setting of FB-bimodules. The first part of the present work is an application of theory of FI-modules to a family of groups Γ_{n,s} arising in the study of free group automorphisms. We observe that the cohomology of these groups determines an FI-module H^{i}(Γ_{n,\bullet}) which we show is finitely generated of stability degree n and weight i. It follows that the sequence {H^{i}(Γ_{n,s})}_{s} is representation stable in the range s ≥ i +n, an improvement on the previously known stable range. Another consequence of this finitely generated FI-module structure is the existence of character polynomials which determine the stable characters of H^{i}(Γ_{n,s}). In particular, this implies that the dimension of H^{i}(Γ_{n,s}) is given by a single polynomial in s for s ≥ i+n. We compute explicit examples of such character polynomials to demonstrate this phenomenon. Next we provide an algorithm that computes certain structural coefficients c_{λμ} related to the n-th tensor power of the free associative algebra on a vector space \mathcal{T}(V)^{\otimes n}. By extending the known range of computation by a factor of over 750 we reveal striking patterns that motivate our recasting of representation stability to families of bimodules. Finally, we develop the theory of \PD-modules. Our main result is that finitely generated \PD-modules give rise to representation stable families of bimodules over symmetric groups. We provide two main examples of this framework. First we show that the coefficients c_{λμ} determine a finitely generated \PD-module and thus provide a first example of this new representation stability. Second, we introduce the extended Whitney homology of the lattice of set partitions, and show that it determines a finitely generated \PD-module. It is known that the ordinary Whitney homology of the lattice of set partitions forms a finitely generated FI-module, and we are able to recover, and generalize this result.
*Advisors/Committee Members: Riley, Timothy R. (committeeMember), Aguiar, Marcelo (committeeMember).*

Subjects/Keywords: Representation; Category; Mathematics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Saied, A. (2018). Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/64873

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Saied, Amin. “Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups .” 2018. Thesis, Cornell University. Accessed December 06, 2019. http://hdl.handle.net/1813/64873.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Saied, Amin. “Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups .” 2018. Web. 06 Dec 2019.

Vancouver:

Saied A. Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups . [Internet] [Thesis]. Cornell University; 2018. [cited 2019 Dec 06]. Available from: http://hdl.handle.net/1813/64873.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Saied A. Stable Representation Theory of Categories and Application to Families of (Bi)Modules over Symmetric Groups . [Thesis]. Cornell University; 2018. Available from: http://hdl.handle.net/1813/64873

Not specified: Masters Thesis or Doctoral Dissertation

Cornell University

3. Chan, Swee Hong. Nonhalting abelian networks .

Degree: 2019, Cornell University

URL: http://hdl.handle.net/1813/67434

An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. The foundational theory for abelian networks was laid down in a series of papers by Bond and Levine (2016), which mainly focused on networks that halt on all inputs. In this dissertation, we extend the theory of abelian networks to nonhalting networks (i.e., networks that can run forever). A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the \emph{torsion group} of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as [ ∑_{n ≥ 1} r_{n} z^{n} = \det (\frac{1}{1-z}D - A ) ] where r_{n} is the number of recurrent chip-and-rotor configurations with n chips; D is the diagonal matrix of outdegrees, and A is the adjacency matrix. A consequence is that the sequence (r_{n})_{n ≥ 1} completely determines the spectrum of the simple random walk on the network.
*Advisors/Committee Members: Aguiar, Marcelo (committeeMember), Meszaros, Karola (committeeMember).*

Subjects/Keywords: Critical group; rotor-routing; Mathematics; abelian distributed processors; abelian networks; chip-firing

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chan, S. H. (2019). Nonhalting abelian networks . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/67434

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Chan, Swee Hong. “Nonhalting abelian networks .” 2019. Thesis, Cornell University. Accessed December 06, 2019. http://hdl.handle.net/1813/67434.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Chan, Swee Hong. “Nonhalting abelian networks .” 2019. Web. 06 Dec 2019.

Vancouver:

Chan SH. Nonhalting abelian networks . [Internet] [Thesis]. Cornell University; 2019. [cited 2019 Dec 06]. Available from: http://hdl.handle.net/1813/67434.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chan SH. Nonhalting abelian networks . [Thesis]. Cornell University; 2019. Available from: http://hdl.handle.net/1813/67434

Not specified: Masters Thesis or Doctoral Dissertation