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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 15 Apr 2024 15:51:49 +0000The Pfaffian Schur Process
https://resolver.caltech.edu/CaltechETD:etd-05292009-161729
Authors: {'items': [{'email': 'mirjana.vuletic@umb.edu', 'id': 'Vuletić-Mirjana', 'name': {'family': 'Vuletić', 'given': 'Mirjana'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/MXT5-QN56
<p>This thesis consists of an introduction and three independent chapters.</p>
<p>In Chapter 2, we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced by Tracy-Widom and Matsumoto, and is a shifted version of the Schur process introduced by Okounkov-Reshetikhin. We prove that the shifted Schur process defines a Pfaffian point process. Furthermore, we apply this fact to compute the bulk scaling limit of the correlation functions for a measure on strict plane partitions which is an analog of the uniform measure on ordinary plane partitions. This allows us to obtain the limit shape of large strict plane partitions distributed according to this measure. The limit shape is given in terms of the Ronkin function of the polynomial P(z,w)=-1+z+w+zw and is parameterized on the domain representing half of the amoeba of this polynomial. As a byproduct, we obtain a shifted analog of famous MacMahon's formula.</p>
<p>In Chapter 3, we generalize the generating formula for plane partitions known as MacMahon's formula, as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald's symmetric functions. Our formula is especially simple in the Hall--Littlewood case. We also give a bijective proof of the analog of MacMahon's formula for strict plane partitions.</p>
<p>In Chapter 4, generating functions of plane overpartitions are obtained using various methods: nonintersecting paths, RSK type algorithms and symmetric functions. We give t-generating formulas for cylindric partitions. We also show that overpartitions correspond to domino tilings and give some basic properties of this correspondence. This is a joint work with Sylvie Corteel and Cyrille Savelief.</p>
https://thesis.library.caltech.edu/id/eprint/2280Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues
https://resolver.caltech.edu/CaltechTHESIS:05262010-023753573
Authors: {'items': [{'email': 'annavmaltsev@gmail.com', 'id': 'Maltsev-Anna-Victoria', 'name': {'family': 'Maltsev', 'given': 'Anna Victoria'}, 'orcid': '0000-0003-4139-1004', 'show_email': 'YES'}]}
Year: 2010
DOI: 10.7907/QQGJ-1A69
We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel $S_L$ for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions $u(\xi, x)$ are bounded in $x$ by $e^{\epsilon x}$ uniformly for $\xi$ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval $I$ in the interior of the spectrum with $\xi_0\in I$, then uniformly in $I$
$$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ where $\rho(\xi)d\xi$ is the density of states.
We deduce that the eigenvalues near $\xi_0$ in a large box of size $L$ are spaced asymptotically as $\frac{1}{L\rho}$. We adapt the methods used to show similar results for orthogonal polynomials.
https://thesis.library.caltech.edu/id/eprint/5840Spectral Theory for Generalized Bounded Variation Perturbations of Orthogonal Polynomials and Schrödinger Operators
https://resolver.caltech.edu/CaltechTHESIS:05262011-194849007
Authors: {'items': [{'email': 'milivoje.lukic@gmail.com', 'id': 'Lukic-Milivoje', 'name': {'family': 'Lukic', 'given': 'Milivoje'}, 'show_email': 'YES'}]}
Year: 2011
DOI: 10.7907/JQJ2-X857
<p>The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators.</p>
<p>These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0).</p>
<p>The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L¹ perturbation.</p>
<p>This generalizes both usual bounded variation and expressions of the form λ(x) cos(φx + α) with λ(x) of bounded variation (and, in particular, with λ(x) = x<sup>γ</sup>, Wigner-von Neumann potentials) as well as their finite linear combinations.</p>
<p>Assuming generalized bounded variation and an L<sup>p</sup> condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.</p>https://thesis.library.caltech.edu/id/eprint/6460Vanishing Results for Hall-Littlewood Polynomials
https://resolver.caltech.edu/CaltechTHESIS:06082012-201004802
Authors: {'items': [{'email': 'vidyav@stanfordalumni.org', 'id': 'Venkateswaran-Vidya', 'name': {'family': 'Venkateswaran', 'given': 'Vidya'}, 'show_email': 'NO'}]}
Year: 2012
DOI: 10.7907/682S-3125
It is well-known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent work of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However at q=0 (the Hall-Littlewood level), these approaches do not directly work; this obstruction was the motivation for this thesis. We investigate three related projects in chapters 2-4 (the first chapter consists of an introduction to the thesis). In the second chapter, we develop a combinatorial technique for proving the results of Rains and Vazirani at q=0. This approach allows us to generalize some of those results in interesting ways and leads us to a finite-dimensional analog of a recent result of Warnaar, involving the Rogers-Szego polynomials. In the third chapter, we provide a new construction for Koornwinder polynomials at q=0, allowing these polynomials to be viewed as Hall-Littlewood polynomials of type BC. This is a first step in building the analogy between the Macdonald and Koornwinder families at the q=0 limit. We use this construction in conjunction with the combinatorial technique of the previous chapter to prove some vanishing results of Rains and Vazirani for Koornwinder polynomials at q=0. In the fourth chapter, we provide an interpretation for vanishing results for Hall-Littlewood polynomials using p-adic representation theory; it is an analog of the Schur case. This p-adic approach allows us to generalize our original vanishing results. In particular, we exhibit a t-analog of a classical vanishing result for Schur functions due to Littlewood and Weyl; our vanishing condition is in terms of Hall polynomials and Littlewood-Richardson coefficients. https://thesis.library.caltech.edu/id/eprint/7153Elliptic Combinatorics and Markov Processes
https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939
Authors: {'items': [{'email': 'dan.betea@gmail.com', 'id': 'Betea-Dan-Dumitru', 'name': {'family': 'Betea', 'given': 'Dan Dumitru'}, 'show_email': 'NO'}]}
Year: 2012
DOI: 10.7907/MMZG-5G61
We present combinatorial and probabilistic interpretations of recent results in the theory of elliptic special functions (due to, among many others, Frenkel, Turaev, Spiridonov, and Zhedanov in the case of univariate functions, and Rains in the multivariate case). We focus on elliptically distributed random lozenge tilings of the hexagon which we analyze from several perspectives. We compute the N-point function for the associated process, and show the process as a whole is determinantal with correlation kernel given by elliptic biorthogonal functions. We furthermore compute transition probabilities for the Markov processes involved and show they come from the multivariate elliptic difference operators of Rains. Properties of difference operators yield an efficient sampling algorithm for such random lozenge tilings. Simulations of said algorithm lead to new arctic circle behavior. Finally we introduce elliptic Schur processes on bounded partitions analogous to the Schur process of Reshetikhin and Okounkov (
and to the Macdonald processes of Vuletic, Borodin, and Corwin). These give a somewhat different (and faster) sampling algorithm from these elliptic distributions, but in principle should encompass more than just tilings of a hexagon.https://thesis.library.caltech.edu/id/eprint/7115Asymptotic Properties of Orthogonal and Extremal Polynomials
https://resolver.caltech.edu/CaltechTHESIS:05222012-113808604
Authors: {'items': [{'email': 'briansimanek7@gmail.com', 'id': 'Simanek-Brian-Zachary', 'name': {'family': 'Simanek', 'given': 'Brian Zachary'}, 'show_email': 'YES'}]}
Year: 2012
DOI: 10.7907/B44M-XJ50
This thesis is devoted to asymptotic properties of extremal polynomials in a variety of settings. Special attention is given to the orthonormal and monic orthogonal polynomials. Given a positive real number q and a measure with compact and infinite support in the complex plane, one can define - for every natural number n - a monic polynomial of degree n having minimal L<sup>q</sup>-norm with respect to the given measure among all monic polynomials of the same degree. Dividing this polynomial by its norm produces a normalized extremal polynomial. We will study the asymptotic behavior of these extremal polynomials when the given measure is of a certain very general form. Our results concerning extremal polynomial asymptotics will include Szego asymptotics, ratio asymptotics, and relative asymptotics. We will also study the associated Christoffel functions and the weak asymptotic behavior of sequences of measures derived from the normalized extremal polynomials.https://thesis.library.caltech.edu/id/eprint/7061Lattice Quantum Codes and Exotic Topological Phases of Matter
https://resolver.caltech.edu/CaltechTHESIS:05292013-140541902
Authors: {'items': [{'email': 'jeongwan.haah@gmail.com', 'id': 'Haah-Jeongwan', 'name': {'family': 'Haah', 'given': 'Jeongwan'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/GCYW-ZE58
<p>This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.</p>
<p>In this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.</p>
<p>This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.</p>https://thesis.library.caltech.edu/id/eprint/7763Determinantal Hypersurface from a Geometric Perspective
https://resolver.caltech.edu/CaltechTHESIS:05022013-153358975
Authors: {'items': [{'email': 'jingjing_h@hotmail.com', 'id': 'Huang-Jingjing', 'name': {'family': 'Huang', 'given': 'Jingjing'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/KMK9-6493
In this paper, we give a geometric interpretation of determinantal forms, both in the case of general matrices and symmetric matrices. We will prove irreducibility of the determinantal singular loci and state its dimension. We also provide detailed description of the singular locus for small dimensions.https://thesis.library.caltech.edu/id/eprint/7656Some Constructions, Related to Noncommutative Tori; Fredholm Modules and the Beilinson–Bloch Regulator
https://resolver.caltech.edu/CaltechTHESIS:05212015-124038753
Authors: {'items': [{'email': 'vikasatkin@gmail.com', 'id': 'Kasatkin-Victor', 'name': {'family': 'Kasatkin', 'given': 'Victor'}, 'show_email': 'NO'}]}
Year: 2015
DOI: 10.7907/Z91R6NGH
<p>A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2<sup>n-1</sup> generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.</p>
<p>In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.</p>
<p>A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.</p>
<p>For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.</p>https://thesis.library.caltech.edu/id/eprint/8875Blackbox Reconstruction of Depth Three Circuits with Top Fan-In Two
https://resolver.caltech.edu/CaltechTHESIS:06082016-032155301
Authors: {'items': [{'email': 'sinhagaur88@gmail.com', 'id': 'Sinha-Gaurav', 'name': {'family': 'Sinha', 'given': 'Gaurav'}, 'orcid': '0000-0002-3590-9543', 'show_email': 'YES'}]}
Year: 2016
DOI: 10.7907/Z92N507D
<p>Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In
this thesis we present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over characteristic zero fields F i.e. depth−3 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic zero.</p>
<p>The algorithm needs only a black-box query access to the polynomial f ∈ F[x1,...,xn] of degree d, computable by a ΣΠΣ(2) circuit C. In addition, we assume that the
"simple rank" of this polynomial (essential number of variables after removing the g.c.d. of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time polynomial in n and d and with high probability returns an equivalent ΣΠΣ(2) circuit.</p>
<p>The problem of reconstructing ΣΠΣ(2) circuits over finite fields was first proposed by Shpilka [27]. The generalization to ΣΠΣ(k) circuits, k = O(1) (over finite
fields) was addressed by Karnin and Shpilka in [18]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus
the running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of linear locally decodable codes with 2 queries.</p>
<p>In our setting, such ideas immediately pose a problem and we need new techniques.</p>
<p>Our main techniques are based on the use of quantitative Sylvester Gallai theorems from the work of Barak et.al. [3] to find a small collection of "nice" subspaces to
project onto. The heart of this work lies in subtle applications of the quantitative Sylvester Gallai theorems to prove why projections w.r.t. the "nice" subspaces can
be ”glued”. We also use Brill’s equations from [9] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important
technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [17].</p>https://thesis.library.caltech.edu/id/eprint/9861Quasiparabolic Subgroups of Coxeter Groups and Their Hecke Algebra Module Structures
https://resolver.caltech.edu/CaltechTHESIS:11042016-133007537
Authors: {'items': [{'email': 'ludaodi@gmail.com', 'id': 'Lu-Daodi', 'name': {'family': 'Lu', 'given': 'Daodi'}, 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9J67DXZ
It is well known that the R-polynomial can be defined for the Hecke algebra of Coxeter groups, and the Kazhdan-Lusztig theory can be developed to understand the representations of Hecke algebra. There is also a generalization for the existence of R-polynomial and Kazhdan-Lusztig theory for the Hecke algebra module of standard parabolic subgroups of Coxeter groups. In recent work of Rains and Vazirani, a generalization of standard parabolic subgroups, called quasiparabolic subgroups, are introduced, and the corresponding Hecke algebra module is well-defined. However, the existence of the analogous involution (Kazhdan-Lusztig bar operator) on the Hecke algebra module of quasiparabolic subgroups is unknown in general. Assuming the existence of the bar-operator, the corresponding R-polynomials and Kazhdan-Lusztig polynomials can be constructed. We prove the existence of the bar operator for the corresponding Hecke algebra modules of quasiparabolic subgroups in finite classical Coxeter groups with a case-by-case verification (Chapter 4). As preparation, we classify all quasiparabolic subgroups of finite classical Coxeter groups. The approach is to first find all rotation subgroups of finite classical Coxeter groups (Chapter 2). Then we exclude the non-quasiparabolic subgroups and confirm the quasiparabolic subgroups (Chapter 3).https://thesis.library.caltech.edu/id/eprint/9972Periods of Feynman Diagrams
https://resolver.caltech.edu/CaltechTHESIS:05052017-144643636
Authors: {'items': [{'email': 'emad.nasr@gmail.com', 'id': 'Nasrollahpoursamami-Emad', 'name': {'family': 'Nasrollahpoursamami', 'given': 'Emad'}, 'orcid': '0000-0002-9658-1529', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z9GX48MR
<p>We study differential equations for Feynman amplitudes and show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and that the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.</p>https://thesis.library.caltech.edu/id/eprint/10156Analysis on Vector Bundles over Noncommutative Tori
https://resolver.caltech.edu/CaltechTHESIS:05092019-193947900
Authors: {'items': [{'email': 'jtao@alumni.princeton.edu', 'id': 'Tao-Jim', 'name': {'family': 'Tao', 'given': 'Jim'}, 'orcid': '0000-0002-0751-9273', 'show_email': 'YES'}]}
Year: 2019
DOI: 10.7907/C4QF-GF45
<p>Noncommutative geometry is the study of noncommutative algebras, especially <i>C</i><sup>*</sup>-algebras, and their geometric interpretation as topological spaces. One <i>C</i><sup>*</sup>-algebra particularly important in physics is the noncommutative <i>n</i>-torus, the irrational rotation <i>C</i><sup>*</sup>-algebra <i>A</i><sub>Θ</sub> with <i>n</i> unitary generators <i>U</i><sub>1</sub>, . . . , <i>U<sub>n</sub></i> which satisfy <i>U<sub>k</sub>U<sub>j</sub></i> = <i>e<sup>2πiθj,k</sup>U<sub>j</sub>U<sub>k</sub></i> and <i>U<sub>j</sub></i><sup>*</sup> = <i>U<sub>j</sub></i><sup>-1</sup>, where Θ ∈ <i>M<sub>n</sub></i>(ℝ) is skew-symmetric with upper triangular entries that are irrational and linearly independent over ℚ. We focus on two projects: an analytically detailed derivation of the pseudodifferential calculus on noncommutative tori, and a proof of an index theorem for vector bundles over the noncommutative two torus. We use Raymond's definition of an oscillatory integral with Connes' construction of pseudodifferential operators to rederive the calculus in more detail, following the strategy of the derivations in Wong's book on pseudodifferential operators. We then define the corresponding analog of Sobolev spaces on noncommutative tori, for which we prove analogs of the Sobolev and Rellich lemmas, and extend all of these results to vector bundles over noncommutative tori. We extend Connes and Tretkoff's analog of the Gauss-Bonnet theorem for the noncommutative two torus to an analog of the McKean-Singer index theorem for vector bundles over the noncommutative two torus, proving a rearrangement lemma where a self-adjoint idempotent <i>e</i> appears in the denominator but does not commute with the <i>k</i><sup>2</sup> already there from the rearrangement lemma proven by Connes and Tretkoff.</p>https://thesis.library.caltech.edu/id/eprint/11505Twisted Heisenberg Central Extensions and the Affine ADE Basic Representation
https://resolver.caltech.edu/CaltechTHESIS:05272022-215641742
Authors: {'items': [{'email': 'zhangvict@gmail.com', 'id': 'Zhang-Victor', 'name': {'family': 'Zhang', 'given': 'Victor'}, 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/42v3-ws41
<p>We study various aspects of the representation theory of loop groups, all with the aim of giving geometric constructions, parameterized by conjugacy classes of the Weyl group, of the basic representation of the affine Lie algebras associated to a simply laced simple Lie algebra as a restriction isomorphism on dual sections of the level 1 line bundle on the affine Grassmannian. Along the way, we obtain various results on the structure of loop tori, the definition of a notion of a Heisenberg Central extension as an alternative for twisted modules over the lattice vertex algebra and the determination of their representation theory, some computations on central extensions of a torus over a field by K<sub>2</sub>, and a new proof of the classification of the conjugacy classes of the Weyl group by parabolic induction.</p>https://thesis.library.caltech.edu/id/eprint/14644Quantum Statistical Mechanics, Noncommutative Geometry, and the Boundary of Modular Curves
https://resolver.caltech.edu/CaltechTHESIS:05252022-192257156
Authors: {'items': [{'email': 'jane.panangaden@gmail.com', 'id': 'Panangaden-Jane-Mariam', 'name': {'family': 'Panangaden', 'given': 'Jane Mariam'}, 'orcid': '0000-0002-6214-2130', 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/xrba-8471
<p>The Bost-Connes system is a C*-dynamical system whose partition function, KMS states, and symmetries are related to the explicit class field theory of the field of rational numbers. In particular, its zero-temperature KMS states, when evaluated on certain points in an arithmetic sub-algebra, yield the generators of the maximal abelian extension of the rationals. The Bost-Connes system can be viewed in terms of a geometric picture of 1-dimensional Q-lattices. The GL₂ system is an extension of this idea to the setting of 2-dimensional Q-lattices. A specialization of the GL₂-system introduced in by Connes, Marcolli, and Ramachandran, is related in a similar way to the explicit class field theory of imaginary quadratic extensions.</p>
<p>Inspired by the philosophy of Manin's real multiplication program, we define a boundary version of the GL₂2-system. In this viewpoint we see the projective line under a certain PGL(2,Z) action (which is related to the shift of the continued fraction expansion) as a moduli space characterizing degenerate elliptic curves. These degenerate elliptic curves can be realized as noncommutative 2-tori. This moduli space of the non-commutative tori is interpreted as an invisible boundary of the moduli space of elliptic curves. In fact, we define a family of such boundary GL₂ systems indexed by a choice of continued fraction algorithm. We analyze their partition functions, KMS states, and ground states. We also define an arithmetic algebra of unbounded multipliers in analogy with the GL₂ case. We show that the ground states when evaluated on points in the arithmetic algebra give pairings of the limiting modular symbols introduced by Manin and Marcolli with weight-2 cusp forms.</p>
<p>We also begin the project of extending this picture to the higher weight setting by defining a higher-weight limiting modular symbol. We use as a starting point the Shokurov modular symbols, which are constructed using Kuga modular varieties, which are non-singular projective varieties over the modular curves. We subject these modular symbols to a limiting procedure. We then show, using the coding space setting of Kessenbohmer and Stratmann, that these limiting modular symbols can be written as a Birkhoff ergodic average everywhere.</p>https://thesis.library.caltech.edu/id/eprint/14622On the Hecke Module of GLₙ(k[[z]])\GLₙ(k((z)))/GLₙ(k((z²)))
https://resolver.caltech.edu/CaltechTHESIS:12082023-083025167
Authors: {'items': [{'email': 'yuhuijin1995@gmail.com', 'id': 'Jin-Yuhui', 'name': {'family': 'Jin', 'given': 'Yuhui'}, 'show_email': 'YES'}]}
Year: 2024
DOI: 10.7907/d0bn-5e47
<p>[See Abstract in text of thesis for correct representation of mathematics]</p>
<p>Every double coset in GLₘ(k[[z]])\GLₘ(k((z)))/GLₘ(k((z²))) is uniquely represented by a block diagonal matrix with diagonal blocks in { 1,z, (11 z \\0 zⁱ \\) (i>1) } if char(k) ≠ 2 and k is a finite field. These cosets form a (spherical) Hecke module H(G,H,K) over the (spherical) Hecke algebra H(G,K) of double cosets in K\G/H, where K=GLₘ(k[[z]]) and H=GLₘ(k((z²))) and G=GLₘ(k((z))). Similarly to Hall polynomial hλ,ν^µ from the Hecke algebra H(G,K), coefficients hλ,ν^µ arise from the Hecke module. We will provide a closed formula for hλ,ν^µ, under some restrictions over λ, ν, µ.</p>https://thesis.library.caltech.edu/id/eprint/16257