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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 \(T(m,k)\) be the number of elements in \(\mathfrak{G}_m\) with \(k\) descents of length greater than \(1\). There are recursion relations that compute \(t(m,k)\) or \(T(m,k)\) in time exponential in \(m\) and \(k\). We give closed expressions for \(t(m,k)\) and \(T(m,k)\) that can be computed in time polynomial in \(m\).
Let \([q]=\{0,\ldots,q-1\}\), let \(\Delta[q]\) denote the simplex of probability measures on \([q]\), and let \(\gamma\) denote the Lebesgue measure normalized on \(\Delta[q]\). We prove that for any symmetric monotone function \(f\colon [q]^n \to [q]\) and any \(a \in [q]\) we have \[\gamma(\{\mu \in \Delta[q]\;\vert\;\mathbf{Pr}_{x \sim \mu^{\otimes n}}[f(x)=a] \in (\varepsilon,1-\varepsilon)\}) = O(1/\log n)\textrm{.}\] We also show that this bound is tight. This improves Kalai and Mossel’s previous bound of \(O(\log\log n/\log n)\) and answers their question completely.
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 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.
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 trigionometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.
I held office hours and review sessions for students in MATH 170E Introduction to Probability and Statistics 1: Probability and reinforced classroom learning.
I held office hours and review sessions for students in 18.781 Theory of Numbers and reinforced classroom learning. I assisted students in mastering proof techniques and provided individualized support to help students develop intuition for number-theroetic concepts. I consistently monitored Piazza (the class forum) and provided quick and reliable answers. I wrote and organized problem set solutions, graded problem sets and exams, and proctored exam sessions.
I served as a Global Teaching Labs (GTL) Intern through MIT MISTI at Hong Kong International School. Designed and taught a computer science-focused STEM camp for middle school students, led interactive STEM workshops for elementary students, presented guest lectures in number theory and advanced mathematics, and assisted with high school courses in computer science, physics, and mathematics.
I lived in and oversaw the daily residential operations and logistics for the \(\surd\)mathroots program for twenty high-potential high school students in mathematics. I supervised students, organized and led recreational activities, and cultivated supportive relationships. I contributed to holistic student development through targeted educational programming and mentorship outside of academics.
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. I organized small frequent check-ins with advisees throughout the year as well as academic meetings, registration meetings, and lunches with Prof. Sipser.
I mentored two high school students in a semester-long reading project on number theory through the PRIMES Circle program at the Massachusetts Institute of Technology. I provided guidance on advanced mathematical concepts, led regular weekly discussions, and supported the development of a final paper and presentation on the Riemann zeta function.
As the head counselor for the Mathematics Department for the First-Year Pre-Orientation Program, I lead a team of eight counselors in planning the First-Year Pre-Orientation Program for incoming freshmen interested in mathematics. Organized the program and events, trained and delegated tasks to other counselors, led logistics such as meals and travel, and provided social and academic support as well as a warm introduction to incoming freshmen.
As a counselor for the Mathematics Department for the First-Year Pre-Orientation Program, I lead incoming freshmen at the Massachusetts Institute of Technology and introduce them to the Mathematics Department at MIT and Boston. Organized the program and events such as academic talks and trivia and provided social and academic support as well as a warm introduction to incoming freshmen.