Number Theory

Let E be an elliptic curve defined over ℚ. Let p > 3 be a prime such that p - 1 is not divisible by 3, 4, 5, 7, 11. In this article, we classify the groups that can arise as E(ℚ(ζp))tors up to isomorphism. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of E(ℚab)tors.

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

There are two ways to compute moments in families of L-functions: one uses the approximation by Dirichlet polynomials, and the other is based on multiple Dirichlet series. We will introduce the two methods to study the family of real Dirichlet L-functions, compare them and show that they lead to the same results. We will then focus on obtaining the meromorphic continuation of the associated multiple Dirichlet series.

We will present a matching upper and lower bound for the right tail
probability of the maximum of a random model of the Riemann zeta function over
short intervals. In particular, we show that the right tail interpolates
between that of log-correlated and IID random variables as the interval varies
in length. We will also discuss a new normalization for the moments over short
intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over
short intervals.

Let $G$ be a graph with adjacency matrix $A$. A continuous quantum walk on $G$ is determined by the complex unitary matrix $U(t)=\exp(itA)$, where $i^2=−1 and $t$ is a real number. Here, $G$ represents a quantum spin network, and its vertices and edges represent the particles and their interactions in the network. The propagation of quantum states in the quantum system determined by $G$ is then governed by the matrix $U(t)$. In particular, $|U(t)_{u,v}|^2$ may be interpreted as the probability that the quantum state assigned at vertex $u$ is transmitted to vertex $v$ at time $t$. Quantum walks are of great interest in quantum computing because not only do they produce algorithms that outperform classical counterparts, but they are also promising tools in the construction of operational quantum computers. In this talk, we give an overview of continuous quantum walks, and discuss old and new results in this area with emphasis on the concepts and techniques that fall under the umbrella of discrete mathematics.