The Gauss circle problem asks for an approximation to the number of lattice points of \(\mathbb{Z}^2\) contained in \(B_r\), the disk of radius \(r\) centered at the origin. Upper, lower, and average bounds have been established for this number-theoretic problem and have been generalized to any lattice in any dimension. We extend this problem to a more general class of structures known as Fourier quasicrystals. Recent work from Alon, Kummer, Kurasov, and Vinzant provides an upper bound of the form \(\#(\Lambda \cap B_r) = c_0\textrm{Vol}_d(B_r) + O (r^{d-1})\) for any Fourier quasicrystal \(\Lambda \subset \mathbb{R}^d\) of density \(c_0\), where \(B_r\) is the \(d\)-dimensional ball of radius \(r\). In this paper, we improve the upper bound for any uniformly discrete Fourier quasicrystal, by showing we can write \(\# (\Lambda \cap B_r ) = c_0\textrm{Vol}_d(B_r ) + O \left(r^{\theta(\Lambda)}\right)\), where \((d-1)/2 < \theta(\Lambda) < d-1\) is some exponent depending only on the dimension \(d\) and the growth rate of the spectrum \(S\) of \(\Lambda\). In the special case \(d = 2\), we also prove lower and upper bounds for the average of the error.
Let \(\mathfrak{G}_m\) be the symmetric group of permutations of \(m\) symbols, let \(t(m,k)\) be the number of elements in \(\mathfrak{G}_m\) with \(k\) peaks, and let \(T(m,k)\) be the number of elements in \(\mathfrak{G}_m\) with \(k\) descents of length greater than \(1\). For \(z \in \mathbb{C} \setminus \mathbb{Z}\) we give identities expressing the functions \(\sin^m(z)\sum_{n\in\mathbb{Z}}{(n+z)^{-m}}\) and \(\sin^m(z)\sum_{n\in\mathbb{Z}}{(-1)^n(n+z)^{-m}}\) as finite sums of powers of cosine functions where the coefficients are essentially \(t(m,k)\) and \(T(m,k)\), respectively. As applications we obtain explicit sums involving polygamma functions and the Lerch transcendent, and we evaluate certain weighted sums involving the numbers \(t(m,k)\) and \(T(m,k)\). We also obtain closed expressions for \(t(m,k)\) and \(T(m,k)\).
We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.
As a Lab Assistant for 6.191 Computational Structures, I tested lab assignments and held office hours for students seeking help with concepts or their labs.
As an associate advisor, I complement Prof. Mike Sipser, Department of Mathematics, Massachusetts Institute of Technology as faculty advisor to provide academic and social support as well as resources to first-year students.
As the head counselor for the Mathematics Department for the Freshmen Preorientation Program, I lead a team of eight counselors in planning the Freshmen Preorientation Program for incoming freshmen.
As a counselor for the Mathematics Department for the Freshmen Preorientation Program, I lead incoming freshmen at the Massachusetts Institute of Technology and introduce them to the Math Department, MIT, and Boston.
I grade problem sets and answer student questions for 18.901 Introduction to Topology.
I assisted students in the preparation of their papers and presentations detailing their mathematical research during the final week
of the Research Science Institute. I coordinated with tutors to oversee academic efforts. I also helped moderate and keep time in the
Final Symposium as well as evaluate final papers and presentations for the Compendium.
I also served as a Tutor for the first two weeks of the program. I monitor and guide the progress of students in their research during the
Research Science Institute, especially in monitoring milestones. In these milestones, I give constructive feedback on students' presentations and papers.
I chaired the Diversity, Inclusion, and Equity Committee of the Undergraduate Math Association. I hosted talks and events about resources and opportunities in the Mathematics Department. I also led the Mentorship Program in the Mathematics Department that paired undergraduate students in proof-based classes with appropriate mentors.
Tutored classmates in understanding the underlying intuition in their math courses, including Trigonometry, Calculus, Linear Algebra, and more advanced courses.