Tuesday, August 14th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Arpita Kar (Queen's University)

Title: ON A THEOREM OF HARDY AND RAMANUJAN.

Abstract: In 1917, G.H Hardy and S. Ramanujan coined the phrase ``normal order" and proved that $\omega(n)$ has normal order $\log \log n$. (Here, $\omega(n)$ denotes the number of distinct prime factors of $n$.) In other words, they showed that $\omega(n) \approx \log \log n$ for all but $o(x)$ many integers $n \leq x$, as $x \to \infty$. In this talk, we will show that the size of this exceptional set is, in fact $O(\frac{x}{(\log x)^A)})$ for any $A>0$, improving upon the work of Hardy, Ramanujan and Tur\'an.

Wednesday, August 1st, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: Adjoint of the Serre derivative, Poincaré series and special values of shifted Dirichlet series.

Abstract: The Serre derivative is a linear differential operator from the space of modular forms of weight k to the space of modular forms of weight k+2. In this talk, we compute the adjoint of the Serre derivative with respect to the Petersson inner product, as done by A. Kumar. Using this as a pretext, we will highlight the connection between the Serre derivative and special values of shifted Dirichlet series via the tool of Poincaré series.

Wednesday, July 18th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Kalyan Chakraborty (Harish-Chandra Research Institute, Allahabad, India)

Title: ON CLASS NUMBER OF CERTAIN QUADRATIC FIELDS.

Abstract: Class number of a number field and in particular that of a quadratic number field, is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.

Wednesday, July 4th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: THE SELBERG TRACE FORMULA.

Abstract: Let $G$ be a locally compact group and $\Gamma$ be a discrete subgroup of $G$ such that $\Gamma \backslash G$ is compact. In 1956, A. Selberg introduced the trace formula relating the traces of irreducible unitary representations of G on $L^2(\Gamma \backslash G)$ to orbital integrals.

In this talk, we will give a brief exposition on Tamagawa's formulation of the Selberg trace formula. Further, we will emphasize that the Poisson summation formula as well as the Frobenius reciprocity for finite groups are special cases of the Selberg trace formula.

Wednesday, June 20th, 2018

Time: 2:00-3:20p.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: EXTREMAL PRIMES FOR NON-CM ELLIPTIC CURVES.

Abstract: For an elliptic curve E over the field of rationals, the distribution of the number of points on E modulo a prime p has been well-studied over the last few decades. A relatively recent study is that of extremal primes for a given curve E. These are the primes p of good reduction for which the number of points on E mod p is either maximal or minimal. If E is a curve with CM, an asymptotic for the number of extremal primes was determined by James and Pollack. The talk will discuss the non-CM case and focus on obtaining upper bounds. This is joint work with C. David, A. Gafni, A. Malik and C. Turnage-Butterbaugh