Mathematics

The evolution and maintenance of cooperation is a fundamental problem in evolutionary biology. Because cooperative behaviors impose a cost, Cooperators are vulnerable to exploitation by Defectors that do not pay the cost to cooperate but still benefit from the cooperation of others. The bacteriophage $\Phi_6$ exhibits cooperative and defective phenotypes in infection: during replication, phages produce essential proteins in the host cell cytoplasm. Coinfection between multiple phages is possible. A given phage cannot guarantee exclusive access to its own proteins, so Cooperators contribute to the common pool of proteins while Defectors contribute less and instead appropriate proteins from Cooperators. Previous experimental work found that $\Phi_6$ was trapped in a prisoner's dilemma, predicting that the cooperative phenotype should disappear. Here we propose that environmental feedback, or interplay between phage and host densities, can maintain cooperation in $\Phi_6$ populations by modulating the rate of co-infection and shifting the advantages of cooperation vs. defection. We build and analyze an ODE model and find that for a wide range of parameter values, environmental feedback allows Cooperation to survive.

Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to obtain the Weyl bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound for quadratic characters obtained by Iwaniec-Conrey in 2000. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.

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.