Analytic number theory and random matrix theory

我的研究将集中在与许多数论对象相联系的L函数上。我感兴趣的是探索L几个家庭中的瞬间-非常详细的功能。这些时刻是理解L函数的值分布的关键，并揭示了支撑所有数论的丰富织锦的一瞥。从统计学的角度来看，看似完全不同的数论对象之间的相互联系变得显而易见。我还将学习L函数的恒等式和算法，以及与随机矩阵统计的联系。Florea研究了与超椭圆曲线相关的函数域上的L函数临界点的矩。第一次，她发现了一个额外的立方根大小的词作为主词。我将调查在二次狄里克莱特L函数族的数域设置中是否存在类似的项。一个相关问题涉及有限域上椭圆曲线的L函数的10次及更高阶矩。这里给出了完全模群的所有Hecke算子的迹。对于数域上的二次狄里克莱特L函数，黑克算子是否进入高阶矩？多重Dirichlet级数理论在方法得到证明的情况下，为矩产生了已知最尖锐的余项。多重Dirichlet级数方法的一个有趣的特点是，它从二次Dirichlet L函数的三次矩和椭圆曲线L函数的二次扭曲的二次矩开始，在矩中预测额外的低项。和我的研究生库马尔一起，我们计划研究椭圆曲线的二次扭曲的二次矩，L函数，以求这样的项。我和同事一起研究了酉阵的特征多项式的对数导数的矩，以及它与Riemann Zeta函数的相同统计量的联系。我们已经发现了非凡的身份，我们想要更好地理解它们的组合。我们和Peter Sarnak一起研究了计算L函数零点的Weil显式公式的潜力和局限性。我们在随机矩阵理论的背景下对问题进行建模，并提出相关问题。给定矩阵A的特征值的前k个矩，关于该矩阵的特征值的位置可以确定什么？我们相信，通过利用我们问题所固有的高度非凸几何，我们可以提高数论中某些算法的复杂性，我们计划在随机矩阵设置和L函数设置下进行研究。我的研究将为数论领域做出贡献，提供基础和重要的知识进步，并培养高素质的人才。

My research will focus on L-functions which connect with many number theoretic objects. I am interested in exploring the moments in several families of L-functions at a very detailed level. The moments are key to understanding the value distribution of L-functions, and reveal glimpses of a rich tapestry that underlies all of number theory. The interconnectedness of seemingly disparate number theoretic objects becomes apparent when looking at them from a statistical point of view. I will also study identities and algorithms for L-functions, and connections with random matrix statistics. Florea has studied moments at the critical point of L-functions over function fields associated to hyperelliptic curves. For the first moment, she found an extra term of size cube root the main term. I will investigate whether a similar term exists in the number field setting, in the family of quadratic Dirichlet L-functions. A related problem concerns the 10-th and higher moments of the L-functions associated to elliptic curves over finite fields. Here the traces of all the Hecke operators for the full modular group. Do the Hecke operators enter into the higher moments in the case of quadratic Dirichlet L-functions over number fields? The theory of multiple Dirichlet series produces the sharpest known remainder terms for the moments in cases where the method yields proofs. An interesting feature of the multiple Dirichlet series approach is that it predicts extra lower terms in the moments, starting with the cubic moment of quadratic Dirichlet L-functions, and with the second moment for quadratic twists of an elliptic curve L-function. With my grad student Kumar, we plan to examine the second moment of quadratic twists of an elliptic curve L-functions for such terms. With colleagues, I have studied the moment of the logarithmic derivative of characteristic polynomials of unitary matrices and its connection to the same statistic for the Riemann zeta function. We have uncovered remarkable identities whose combinatorics we would like to better understand. With Peter Sarnak, we have been studying the potential and the limitations in the Weil explicit formula for computing the zeros of L-functions. We model the problem in a random matrix theory setting and ask related questions. Given the first k moments of the eigenvalues of a matrix A, what can be determined about the location of the eigenvalues of the matrix? We believe, by exploiting the highly non-convex geometry inherent in our problem, that we can improve the complexity of certain algorithms in number theory and we plan to investigate this both in the random matrix setting and in the L-function setting. My research will contribute to the field of number theory, providing fundamental and important advances in knowledge, and also resulting in training of highly qualified personnel.