# Calendar

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Tuesday, January 24, 2023

Posted January 16, 2023

6:00 pm - 6:50 pm Zoom

Xin Wan, Chinese Academy of Sciences
[NOTE UNUSUAL TIME] Iwasawa main conjecture for universal families

We formulate and prove the Iwasawa main conjecture for the universal family for ${\rm GL}_2/\mathbb{Q}$ in the $p$-adic Langlands program. As a consequence we prove the Iwasawa main conjecture and rank 0 BSD formula at bad primes. This is joint work with Olivier Fouquet.

Tuesday, February 7, 2023

Posted January 27, 2023

3:10 pm - 4:00 pm Lockett 233 and Zoom

Jeffrey Lagarias, University of Michigan
The Floor Quotient Partial Order

We say that a positive integer $m$ is a floor quotient of $n$ if $m = [n/k]$ for some integer $k$, where $[\, \cdot \,]$ denotes the floor function. We show this relation between $m$ and $n$ defines a partial order on the positive integers. This partial order refines the multiplicative divisor order on the positive integers and is refined by the additive total order. We describe results on the internal structure of this partial order, especially on its initial intervals. We study the (two-variable) Möbius function of this partial order. This is joint work with David Harry Richman (see $\texttt{arXiv:2212.11689}$).

Tuesday, February 28, 2023

Posted January 16, 2023

3:10 pm - 4:00 pm Lockett 233 and Zoom

Pranabesh Das, Xavier University of Louisiana
Perfect Powers in Power Sums

Let $k \geq 1$, $n \geq 2$ be integers. A power sum is a sum of the form $x_1^k+x_2^k+\cdots +x_n^k$ where $x_1, x_2, \cdots, x_n$ are all integers. Perfect powers appearing in power sums have been well studied in the literature and are an active field of research. In this talk, we consider the Diophantine equation of the form $$(x+r)^k + (x+2r)^k + \cdots + (x+nr)^k = y^m \ \ \ \ \ \ \ \ \ (1)$$ where $x, y, r \in \mathbb{Z}$, $n, k \in \mathbb{N}$, and $m \geq 2$. We begin with discussing the literature on the Diophantine equation (1). Then we consider explicit solutions for a particular case of the Diophantine equation (1); more precisely, we consider the Diophantine equation $$\ \ \ (x-r)^5 + x^5 + (x+r)^5 = y^n, \ \ \ \ n \geq 2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ where $r, x, y \in \mathbb{Z}$ and $r$ is composed of certain fixed primes. The talk is based on a joint work with Dey, Koutsianas, and Tzanakis where we determine the integral solutions of the Diophantine equation (2) as an application of the modular method.

Tuesday, March 7, 2023

Posted January 27, 2023

3:10 pm - 4:00 pm Lockett 233 and Zoom

Wijit Yangjit, University of Michigan
On the Montgomery–Vaughan weighted generalization of Hilbert's inequality

Hilbert's inequality states that $$\left\vert\sum_{m=1}^N\sum_{\substack{n=1\\n\neq m}}^N\frac{z_m\overline{z_n}}{m-n}\right\vert\le C_0\sum_{n=1}^N\left\vert z_n\right\vert^2,$$ where $C_0$ is an absolute constant. In 1911, Schur showed that the optimal value of $C_0$ is $\pi$. In 1974, Montgomery and Vaughan proved a weighted generalization of Hilbert's inequality and used it to estimate mean values of Dirichlet series. This generalized Hilbert inequality is important in the theory of the large sieve. The optimal constant $C$ in this inequality is known to satisfy $\pi\le C \lt \pi+1$. It is widely conjectured that $C=\pi$. In this talk, I will describe the known approaches to obtain an upper bound for $C$, which proceed via a special case of a parametric family of inequalities. We analyze the optimal constants in this family of inequalities. A corollary is that the method in its current form cannot imply an upper bound for $C$ below $3.19$.

Tuesday, April 4, 2023

Posted January 16, 2023