Venkatesh, Akshay 1981
Overview
Works:  26 works in 31 publications in 2 languages and 109 library holdings 

Roles:  Thesis advisor, Author 
Publication Timeline
.
Most widely held works by
Akshay Venkatesh
Periods and harmonic analysis on spherical varieties by
Yiannis Sakellaridis(
Book
)
5 editions published in 2017 in English and held by 83 WorldCat member libraries worldwide
This volume elaborates the idea that harmonic analysis on a spherical variety X is intimately connected to the Langlands program. In the local setting, the key conjecture is that the spectral decomposition of L^2(X) is controlled by a dual group attached to X. Guided by this, the authors develop a Plancherel formula for L^2(X), formulated in terms of simpler spherical varieties which model the geometry of X at infinity. This local study is then related to global conjecturesnamely, conjectures about period integrals of automorphic forms over spherical subgroups.Publisher's website
5 editions published in 2017 in English and held by 83 WorldCat member libraries worldwide
This volume elaborates the idea that harmonic analysis on a spherical variety X is intimately connected to the Langlands program. In the local setting, the key conjecture is that the spectral decomposition of L^2(X) is controlled by a dual group attached to X. Guided by this, the authors develop a Plancherel formula for L^2(X), formulated in terms of simpler spherical varieties which model the geometry of X at infinity. This local study is then related to global conjecturesnamely, conjectures about period integrals of automorphic forms over spherical subgroups.Publisher's website
Limiting forms of the trace formula (Robert P. Langlands) by
Akshay Venkatesh(
Book
)
2 editions published in 2002 in English and held by 2 WorldCat member libraries worldwide
We carry out the first nontrivial cases of the limiting process proposed by Langlands in his manuscript Beyond Endoscopy, with technical variations that enable us to treat the limit unconditionally. This gives an elementary proof, on GL(2), of the classification of forms such that the symmetric square Lfunction has a pole (including, implicitly, the construction of these forms). The result of this may be seen as one of the simplest cases of the "pipedream" Langlands proposes. We also apply similar methods to derive a converse theorem, and to produce a result that generalizes Duke's estimate on the dimension of weight 1 forms to arbitrary number fieldsbut is sharper, even over Q, than Duke's original estimate
2 editions published in 2002 in English and held by 2 WorldCat member libraries worldwide
We carry out the first nontrivial cases of the limiting process proposed by Langlands in his manuscript Beyond Endoscopy, with technical variations that enable us to treat the limit unconditionally. This gives an elementary proof, on GL(2), of the classification of forms such that the symmetric square Lfunction has a pole (including, implicitly, the construction of these forms). The result of this may be seen as one of the simplest cases of the "pipedream" Langlands proposes. We also apply similar methods to derive a converse theorem, and to produce a result that generalizes Duke's estimate on the dimension of weight 1 forms to arbitrary number fieldsbut is sharper, even over Q, than Duke's original estimate
Controlling ramification in number fields by Simon RubinsteinSalzedo(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
This thesis focuses on two aspects of limited ramification and is split up into two independent sections. The first section (which comprises the second and third chapters) is on the distribution of class groups of cyclic cubic fields. We propose an explanation for the discrepancy between the observed number of cyclic cubics whose 2class group is C_2 x C_2 and the number predicted by the CohenLenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute 10^5 examples. The second section (which comprises the fourth and fifth chapters) discusses branched covers of algebraic curves, especially covers of elliptic curves with one branch point. We produce some techniques that allow us to write down explicit equations for such maps, and then we give examples of number fields which arise from such covers. Finally, we present some possibilities for future works that the author hopes to pursue
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
This thesis focuses on two aspects of limited ramification and is split up into two independent sections. The first section (which comprises the second and third chapters) is on the distribution of class groups of cyclic cubic fields. We propose an explanation for the discrepancy between the observed number of cyclic cubics whose 2class group is C_2 x C_2 and the number predicted by the CohenLenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute 10^5 examples. The second section (which comprises the fourth and fifth chapters) discusses branched covers of algebraic curves, especially covers of elliptic curves with one branch point. We produce some techniques that allow us to write down explicit equations for such maps, and then we give examples of number fields which arise from such covers. Finally, we present some possibilities for future works that the author hopes to pursue
Dichotomy between structure and randomness in combinatorial number theory by
Xuancheng Shao(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
This dissertation is about exploring the dichotomy between structure and randomness in analytic number theory and additive combinatorics. We explore various problems including bounds for character sums, Goldbachtype questions, and a Freimantype result
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
This dissertation is about exploring the dichotomy between structure and randomness in analytic number theory and additive combinatorics. We explore various problems including bounds for character sums, Goldbachtype questions, and a Freimantype result
Periods and harmonic analysis on spherical varieties by
Yiannis Sakellaridis(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Given a spherical variety X for a group G over a nonarchimedean local field k, the Plancherel decomposition for L^2(X) should be related to "distinguished" Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo center) spectra of its "boundary degenerations", certain Gvarieties with more symmetries which model X at infinity. Along the way, we discuss the asymptotic theory of subrepresentations of C^{infty}(X), and establish conjectures of IchinoIkeda and LapidMao. We finally discuss global analogues of our local conjectures, concerning the period integrals of automorphic forms over spherical subgroups
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Given a spherical variety X for a group G over a nonarchimedean local field k, the Plancherel decomposition for L^2(X) should be related to "distinguished" Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo center) spectra of its "boundary degenerations", certain Gvarieties with more symmetries which model X at infinity. Along the way, we discuss the asymptotic theory of subrepresentations of C^{infty}(X), and establish conjectures of IchinoIkeda and LapidMao. We finally discuss global analogues of our local conjectures, concerning the period integrals of automorphic forms over spherical subgroups
Some problems in multiplicative number theory by
Junsoo Ha(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is msmooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are msmooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n <= m <= n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is msmooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are msmooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n <= m <= n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields
Statistics of random integral matrices by
Iurie Boreico(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
This thesis consists of two separate projects. The first part investigates the asymptotic behavior of the number of integral mbyn matrices, with entries bounded by T, whose cokernel is isomorphic to a fixed abelian group G. We answer this question by building on work of Katznelson, who obtained asymptotics on the number of such matrices of given rank. In particular, we show that if G has torsion B for a finite abelian group B, then a positive proportion of matrices of rank r have cokernel isomorphic to G, and we compute this proportion explicitly (as an infinite product over primes). The corresponding problem for symmetric matrices is also discussed, with a different answer. Part two of this thesis deals with the infinitesimal frequency of a monic polynomial appearing as the characteristic polynomial of an nbyn matrix with coefficients in the padic integers. Relying on the concept of rational singularities, we prove that this frequency is described by a continuous density on the space of monic polynomials, and show that the normalized density function is multiplicative, thus reducing its computation to the case of a monic irreducible polynomial. In the monic irreducible case, we express the density function as a finite sum over modules in the ring of integers of a finite extension of Q_p, and compute it in the case of degree <=3. For the general case, we conjecture bounds on the size of this function, as well as conjectures on the underlying geometric structures. In the end, we study a modification of this question as n goes to infinity. As an aside, we also use CohenLenstra measures to compute the distribution of the Jordan blocks of matrices with coefficients in a fixed finite field as the size goes to infinity
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
This thesis consists of two separate projects. The first part investigates the asymptotic behavior of the number of integral mbyn matrices, with entries bounded by T, whose cokernel is isomorphic to a fixed abelian group G. We answer this question by building on work of Katznelson, who obtained asymptotics on the number of such matrices of given rank. In particular, we show that if G has torsion B for a finite abelian group B, then a positive proportion of matrices of rank r have cokernel isomorphic to G, and we compute this proportion explicitly (as an infinite product over primes). The corresponding problem for symmetric matrices is also discussed, with a different answer. Part two of this thesis deals with the infinitesimal frequency of a monic polynomial appearing as the characteristic polynomial of an nbyn matrix with coefficients in the padic integers. Relying on the concept of rational singularities, we prove that this frequency is described by a continuous density on the space of monic polynomials, and show that the normalized density function is multiplicative, thus reducing its computation to the case of a monic irreducible polynomial. In the monic irreducible case, we express the density function as a finite sum over modules in the ring of integers of a finite extension of Q_p, and compute it in the case of degree <=3. For the general case, we conjecture bounds on the size of this function, as well as conjectures on the underlying geometric structures. In the end, we study a modification of this question as n goes to infinity. As an aside, we also use CohenLenstra measures to compute the distribution of the Jordan blocks of matrices with coefficients in a fixed finite field as the size goes to infinity
The behaviour of Lfunctions at the edge of the critical strip and applications by
Xiannan Li(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
A large number of problems in number theory can be reduced to statements about Lfunctions. In this thesis, we study Lfunctions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
A large number of problems in number theory can be reduced to statements about Lfunctions. In this thesis, we study Lfunctions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest
Padic Hodge theory in rigid analytic families by Rebecca Michal Bellovin(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
In this thesis, we study padic Hodge theory in rigid analytic families. Roughly speaking, padic Hodge theory is the study of padic representations of padic Galois groups. One introduces certain padic period rings B, such as B_{HT}, B_{dR}, B_{st}, and B_{cris}, and uses them to define functors D_B(.) from the category of padic Galois representations to various categories of linear algebra data. In the first half of this thesis, we study generalizations of these functors to families of padic Galois representations with rigid analytic coefficients. We prove that the functors D_{HT}(.) and D_{dR}(.) are coherent sheaves, and we prove that the Badmissible locus is a closed subspace of the base. In the second half of this thesis, we study the linear algebra data which arises from families of potentially semistable Galois representations valued in a connected reductive group G. We prove that for any G, the moduli space of linear algebra data is reduced and locally a complete intersection, and we deduce that potentially semistable deformation rings are generically smooth and equidimensional
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
In this thesis, we study padic Hodge theory in rigid analytic families. Roughly speaking, padic Hodge theory is the study of padic representations of padic Galois groups. One introduces certain padic period rings B, such as B_{HT}, B_{dR}, B_{st}, and B_{cris}, and uses them to define functors D_B(.) from the category of padic Galois representations to various categories of linear algebra data. In the first half of this thesis, we study generalizations of these functors to families of padic Galois representations with rigid analytic coefficients. We prove that the functors D_{HT}(.) and D_{dR}(.) are coherent sheaves, and we prove that the Badmissible locus is a closed subspace of the base. In the second half of this thesis, we study the linear algebra data which arises from families of potentially semistable Galois representations valued in a connected reductive group G. We prove that for any G, the moduli space of linear algebra data is reduced and locally a complete intersection, and we deduce that potentially semistable deformation rings are generically smooth and equidimensional
Upper bounds and moments of Lfunctions by Vorrapan Chandee(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
Lfunctions are some of the most studied objects in number theory. Although many crucial properties of Lfunctions remain mysterious, central conjectures such as the generalized Riemann hypothesis (GRH). This thesis concerns properties of Lfunctions. In particular, we focus on studying upper bounds and moments of $L$functions. Assuming GRH, we give effective explicit upper bounds for Lfunctions on the critical line and apply these bounds to determine what numbers are represented by a given ternary quadratic form. Moreover the best known version of the Lindelof hypothesis from the Riemann hypothesis (RH) is also derived. Another important way of understanding LH is through moments of Lfunctions. Information about moments sheds light on the distribution of values of \zeta(1/2 + it). We try to understand the joint distribution of quantities like \zeta(1/2 + it) and \zeta(1/2 + it + i). To study these we consider "shifted moments" of the zeta function and obtain good upper and lower estimates for such moments
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
Lfunctions are some of the most studied objects in number theory. Although many crucial properties of Lfunctions remain mysterious, central conjectures such as the generalized Riemann hypothesis (GRH). This thesis concerns properties of Lfunctions. In particular, we focus on studying upper bounds and moments of $L$functions. Assuming GRH, we give effective explicit upper bounds for Lfunctions on the critical line and apply these bounds to determine what numbers are represented by a given ternary quadratic form. Moreover the best known version of the Lindelof hypothesis from the Riemann hypothesis (RH) is also derived. Another important way of understanding LH is through moments of Lfunctions. Information about moments sheds light on the distribution of values of \zeta(1/2 + it). We try to understand the joint distribution of quantities like \zeta(1/2 + it) and \zeta(1/2 + it + i). To study these we consider "shifted moments" of the zeta function and obtain good upper and lower estimates for such moments
Highperformance heterogeneity/ energyaware communication for multipetaflop HPC systems by
Akshay Venkatesh(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
In view of these developments, communication runtimes must aim to take into consideration the abundance of heterogeneity that abounds in modern HPC systems and leverage advancements in network design to achieve low latency, high throughput and to increase computation/communication overlap possibilities. At the same time, they must also aim to reduce energy expended by CPUs during communication phases with minimal or without affecting overall performance. As Message Passing Interface (MPI) is the de facto standard for communication calls in scientific applications, this dissertation proposes high performance designs while addressing heterogeneity and energy challenges for the MPI communication routines on modern HPC systems. In particular, this dissertation attempts to address heterogeneity challenges in communication algorithms for dense collective operations through the design of novel heterogeneity aware collective designs. The proposed designs are considerate of the cost differences in communication paths and leverage delegation mechanisms, propose adaptations to clas sic algorithms, and use novel heuristics to achieve large benefits in collectives (such as MPI Alltoall and MPI Allgather) at scale. The dissertation also takes advantage of specialcapability network subsystems that have peertopeer access with other PCIe devices, RDMA and multicast capabilities, offload mechanisms, and the ability to service network requests without explicit CPU intervention. These are used to design pointtopoint and collective operations that overcome the challenges posed by heterogeneous memory subsystems and yield high throughput, overlap, and reduced couplinginduced synchronization overheads. Finally, the dissertation proposes rules for applicationoblivious energy savings during 2sided and 1sided MPI routines through intimate knowledge of the underlying protocols used to realize communication routines. This results in applications automatically saving CPU and memory energy during communication phases with minimal performance degradation and without needing application code changes. The proposed designs for MPI have been integrated into the MVAPICH2 MPI library and are already widely deployed on HPC clusters
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
In view of these developments, communication runtimes must aim to take into consideration the abundance of heterogeneity that abounds in modern HPC systems and leverage advancements in network design to achieve low latency, high throughput and to increase computation/communication overlap possibilities. At the same time, they must also aim to reduce energy expended by CPUs during communication phases with minimal or without affecting overall performance. As Message Passing Interface (MPI) is the de facto standard for communication calls in scientific applications, this dissertation proposes high performance designs while addressing heterogeneity and energy challenges for the MPI communication routines on modern HPC systems. In particular, this dissertation attempts to address heterogeneity challenges in communication algorithms for dense collective operations through the design of novel heterogeneity aware collective designs. The proposed designs are considerate of the cost differences in communication paths and leverage delegation mechanisms, propose adaptations to clas sic algorithms, and use novel heuristics to achieve large benefits in collectives (such as MPI Alltoall and MPI Allgather) at scale. The dissertation also takes advantage of specialcapability network subsystems that have peertopeer access with other PCIe devices, RDMA and multicast capabilities, offload mechanisms, and the ability to service network requests without explicit CPU intervention. These are used to design pointtopoint and collective operations that overcome the challenges posed by heterogeneous memory subsystems and yield high throughput, overlap, and reduced couplinginduced synchronization overheads. Finally, the dissertation proposes rules for applicationoblivious energy savings during 2sided and 1sided MPI routines through intimate knowledge of the underlying protocols used to realize communication routines. This results in applications automatically saving CPU and memory energy during communication phases with minimal performance degradation and without needing application code changes. The proposed designs for MPI have been integrated into the MVAPICH2 MPI library and are already widely deployed on HPC clusters
Periods and harmonic analysis on spherical varieties by
Yiannis Sakellaridis(
Book
)
1 edition published in 2017 in French and held by 1 WorldCat member library worldwide
1 edition published in 2017 in French and held by 1 WorldCat member library worldwide
Zerodistribution and size of the Riemann zetafunction on the critical line by Maksym Radziwill(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We investigate the analytic properties of the Riemann zetafunction. We focus on the average size of the Riemann zetafunction on the critical line and on the vertical distribution of its zeros on the critical line
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We investigate the analytic properties of the Riemann zetafunction. We focus on the average size of the Riemann zetafunction on the critical line and on the vertical distribution of its zeros on the critical line
On the structure and complex analysis of dirichlet series by Ralph Furmaniak(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
Dirichlet series, such as the Riemann zeta function, serve to encode useful number theoretic information. We investigate the general analytic theory of these Dirichlet series, in particular those with analytic continuation, given growth rates, or functional equations
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
Dirichlet series, such as the Riemann zeta function, serve to encode useful number theoretic information. We investigate the general analytic theory of these Dirichlet series, in particular those with analytic continuation, given growth rates, or functional equations
Part I, Symplectic ice, Part II, Global and local Kubota symbols by
Dmitriy Ivanov(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
My Ph. D. thesis consists of two parts. In the first part I construct an icemodel (a sixvertex model). I show the partition function of this ice model is equal to the product of an irreducible character of the symplectic group and a deformation of the Weyl denominator. A similar result was originally proved by Hamel and King, but the Boltzmann weights (for the vertices at the caps) that I use are different then the ones that are used by Hamel and King. Also, my proof of this result uses YangBaxter equation (while the proof of Hamel and King does not). This gives us a 6vertex models for characters of the symplectic group, but this result can also be interpreted as an example of an exactly solved model, that is, an ice model whose partition function can be computed explicitly (this is of interest to people who work in statistical mechanics). In the second part of my thesis I work with the global and local Kubota symbols. I introduce a local Kubota symbol, and show that it satisfies some properties that are satisfies by the global Kubota symbol; in particular, it satisfies the reciprocity law. Using the parallel between the properties of the global and local Kubota symbols, I give a new proof of Kubota's theorem (for the case when p does not divide n), and then generalize this theorem to the case when p divides n. In the last chapter of my thesis I work with the global Kubota symbol. I investigate the levels for the global Kubota symbols for particular fields, and I obtain new levels for these fields which improve the results of Kubota, and of Bass, Milnor and Serre
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
My Ph. D. thesis consists of two parts. In the first part I construct an icemodel (a sixvertex model). I show the partition function of this ice model is equal to the product of an irreducible character of the symplectic group and a deformation of the Weyl denominator. A similar result was originally proved by Hamel and King, but the Boltzmann weights (for the vertices at the caps) that I use are different then the ones that are used by Hamel and King. Also, my proof of this result uses YangBaxter equation (while the proof of Hamel and King does not). This gives us a 6vertex models for characters of the symplectic group, but this result can also be interpreted as an example of an exactly solved model, that is, an ice model whose partition function can be computed explicitly (this is of interest to people who work in statistical mechanics). In the second part of my thesis I work with the global and local Kubota symbols. I introduce a local Kubota symbol, and show that it satisfies some properties that are satisfies by the global Kubota symbol; in particular, it satisfies the reciprocity law. Using the parallel between the properties of the global and local Kubota symbols, I give a new proof of Kubota's theorem (for the case when p does not divide n), and then generalize this theorem to the case when p divides n. In the last chapter of my thesis I work with the global Kubota symbol. I investigate the levels for the global Kubota symbols for particular fields, and I obtain new levels for these fields which improve the results of Kubota, and of Bass, Milnor and Serre
Moments of automorphic Lfunctions and related problems by Ian Petrow(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We present in this dissertation several theorems on the subject of moments of automorphic Lfunctions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of Lfunctions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is 1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number 1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the offdiagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of fulllevel GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of RankinSelberg Lfunctions
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We present in this dissertation several theorems on the subject of moments of automorphic Lfunctions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of Lfunctions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is 1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number 1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the offdiagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of fulllevel GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of RankinSelberg Lfunctions
Conic degeneration and the determinant of the laplacian by David Alexander Sher(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We consider a family of smooth Riemannian manifolds which degenerate to a manifold with a conical singularity. Such families arise in various settings in spectral theory, including the study of the isospectral problem. We investigate the behavior of the determinant of the Laplacian under the degeneration. Our main result is an approximation formula for the determinant, including all terms which do not vanish in the limit. The key idea is a uniform parametrix construction for the heat kernel on the degenerating family of manifolds, which enables us to analyze the determinant via the heat trace. It becomes clear in the construction that we need to understand both the shorttime and longtime behavior of the heat kernel on an asymptotically conic manifold. Using techniques of Melrose and building on previous work of Guillarmou and Hassell, we give a complete description of the asymptotic structure of this heat kernel in all spatial and temporal regimes
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We consider a family of smooth Riemannian manifolds which degenerate to a manifold with a conical singularity. Such families arise in various settings in spectral theory, including the study of the isospectral problem. We investigate the behavior of the determinant of the Laplacian under the degeneration. Our main result is an approximation formula for the determinant, including all terms which do not vanish in the limit. The key idea is a uniform parametrix construction for the heat kernel on the degenerating family of manifolds, which enables us to analyze the determinant via the heat trace. It becomes clear in the construction that we need to understand both the shorttime and longtime behavior of the heat kernel on an asymptotically conic manifold. Using techniques of Melrose and building on previous work of Guillarmou and Hassell, we give a complete description of the asymptotic structure of this heat kernel in all spatial and temporal regimes
Small solutions to linear congruences and Hecke equidistribution by
Andreas Strömbergsson(
)
1 edition published in 2005 in English and held by 1 WorldCat member library worldwide
1 edition published in 2005 in English and held by 1 WorldCat member library worldwide
On the shape of a highdimensional random lattice by Seungki Kim(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
The statistics concerning lattices in high dimensions (call "the geometry of numbers" in the literature) used to be studied intensively in the mid20th century, especially by C.A. Rogers and Wolfgang Schmidt. By perhaps an accident of history, it has become forgotten by mathematicians for almost 50 years. Yet there is still plenty of wonderful mathematics left there to develop, and furthermore, recently people in various sectors  computer science, cryptography, number theory, and even dynamics  are starting to realize the need and benefits of understanding highdimensional lattices and related topics. This thesis is largely in three parts. In Chapter 1, we give an exposition on the Rogers integration formula and its variants, the main technical device in the study of highdimensional lattices. In Chapter 2, inspired by the ideas of Schmidt, we prove that the lengths of the first O(n^(1/2)) shortest vectors of an ndimensional random lattice exhibit a Poisson distribution as n → ∞, improving a recent result of Sodergren. In Chapters 3 and 4, we prove a few previously unknown facts on the statistics of the LLL bases that help answer a few questions that have been raised regarding the peculiar behavior of the LLL algorithm, and then present an experimental result that supports our theoretical conclusions and suggests where future research could be headed to
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
The statistics concerning lattices in high dimensions (call "the geometry of numbers" in the literature) used to be studied intensively in the mid20th century, especially by C.A. Rogers and Wolfgang Schmidt. By perhaps an accident of history, it has become forgotten by mathematicians for almost 50 years. Yet there is still plenty of wonderful mathematics left there to develop, and furthermore, recently people in various sectors  computer science, cryptography, number theory, and even dynamics  are starting to realize the need and benefits of understanding highdimensional lattices and related topics. This thesis is largely in three parts. In Chapter 1, we give an exposition on the Rogers integration formula and its variants, the main technical device in the study of highdimensional lattices. In Chapter 2, inspired by the ideas of Schmidt, we prove that the lengths of the first O(n^(1/2)) shortest vectors of an ndimensional random lattice exhibit a Poisson distribution as n → ∞, improving a recent result of Sodergren. In Chapters 3 and 4, we prove a few previously unknown facts on the statistics of the LLL bases that help answer a few questions that have been raised regarding the peculiar behavior of the LLL algorithm, and then present an experimental result that supports our theoretical conclusions and suggests where future research could be headed to
Gvalued flat deformations and local models by Brandon William Allen Levin(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
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 Sakellaridis, Yiannis Author
 Stanford University Department of Mathematics
 Soundararajan, Kannan 1973 Thesis advisor
 Conrad, Brian 1970 Thesis advisor
 Bump, Daniel 1952 Thesis advisor
 Vasy, András Thesis advisor
 Mazzeo, Rafe Thesis advisor
 Diaconis, Persi Thesis advisor
 Vakil, Ravi Thesis advisor
 Brendle, Simon 1981 Thesis advisor
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