Commutative Algebra seminar
Speaker: Kaushik Khamari, IIT Bombay
Host: Tony Puthenpurakal
Title: On licci monomial ideals II
Time, day and date: 11:30:00 AM - 12:30:00 PM, Monday, October 26
Venue: Ramanujan Hall
Abstract: In this talk, we discuss basic notions of complete-intersection liaison and licci ideals, which are ideals in the liaison class of a complete intersection. We then introduce universal linking sequences corresponding to an arbitrary sequence of links. As an application, we give a characterization of finite colength licci monomial ideals in a polynomial ring over a field, due to Huneke and Ulrich.

Number Theory Seminar
Speaker: Arshay Seth, School of Mathematics, TIFR
Host: Ravi Raghunathan
Title: Euler products inside the critical strip
Time, day and date: 11:30:00 AM – 12:30:00 PM, Tuesday, October 27
Venue: Room 113
Abstract: Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip.  Indeed, the behaviour of Euler products inside the critical strip is very closely related to both the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture.  In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip.  We will end by giving applications of these results to questions related to Chebyshev's bias, and also discuss a new conjectural local-global principle for higher genus curves. This talk is partly based on joint works with Alina Bucur, Kiran Kedlaya and Shin-ya Koyama.

Mathematics Colloquium
Speaker: Jean-Marc Deshouillers, University of Bordeaux
Host: Keshav Aggarwal
Title: Coprimality of consecutive elements in regular sequences
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 28
Venue: Ramanujan Hall
Abstract: ``Regular sequences'' are integral sequences $ \mathcal{A}_f$ of the shape $\{ \lfloor f(n) \rfloor, n \in \mathbb{N}\}$, where $f$ is a smooth function from $[1, + \infty)$ to itself with polynomial growth (for example the Segal-Piatetski-Shapiro sequences where $f(n)=n^c$). We shall study the following questions: are there infinitely many arbitrarily large blocks of consecutive elements of $ \mathcal{A}_f$ which are pairwise coprime? If so, do they occur often? To illustrate the results in the case of SPS sequences, we show that the sequence $SPS(c)$ contains arbitrary large blocks of pairwise coprime elements when $1<c<2$. Moreover, those blocks of given length occur with positive density. For $c>2$ although we expect the same result to hold, we can prove it only for $c$ rational less than $111/55$. Proofs involve combinatorial constructions and harmonic analysis.\\ \textit{From joint works with M. Drmota and C. M\"{u}llner, with S. Naik and a work in progress with H. Iwaniec.}