Monday, September 25, 2023

Seema Kushwaha (IIIT, Allahabad) - Thursday Sept 28, 2023 - 4:00 PM (IST)

Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week. 

Talk Announcement: 
Title:   Farey-subgraphs and Continued Fractions
Speaker: Seema Kushwaha (IIIT, Allahabad)
When: Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Please send email to the organisers for a link.

Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n}
\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.
The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

  Let $\mathcal{X}$ be a  subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a  notion which is analogous to best rational approximation. 

An element  $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.  
In  this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties. 

Thursday, September 14, 2023

Shashank Kanade (University of Denver) - Thursday Sept 14, 2023 - 6:00 PM (IST)

 Dear all,

We are back after an extended summer break. I hope many of us had an opportunity to meet each other and further our research goals. 

The next talk is by Shashank Kanade, University of Denver. It is a little later in the evening from our usual time.

Talk Announcement: 

On the $A_2$ Andrews--Schilling--Warnaar identities

Speaker: Shashank Kanade (University of Denver)

When: Thursday, Sept 14, 2023, 6:00 PM- 7:00 PM IST (6:30 AM MDT)

Where: Zoom: please write to the organisers for the link

I will give a description of my work with Matthew C. Russell on the $A_2$
Andrews--Schilling--Warnaar identities. Majority of our single variable
sum=product conjectures have been proven by S. O. Warnaar; I will also
explain what remains. Bi-variate versions of our conjectures are largely open.