Department of Mathematics and Statistics

Tuesday, January 15th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: Dedekind zeta function at odd positive integers.

Abstract: Let $\zeta(s)$ denote the Riemann zeta-function. Thanks to Euler's evaluation and Lindemann's theorem on transcendence of $\pi$, we understand that $\zeta(2k)$ is transcendental for any positive integer $k$. However, the arithmetic nature of the values of $\zeta(s)$ at odd positive integers remains a mystery. Recently, significant progress was made concerning the irrationality of these values, with perhaps the most remarkable theorem being that infinitely many of $\zeta(2k+1)$ are irrational, which was shown by T. Rivoal in 2000.

Similarly, one can inquire regarding the arithmetic nature of values of the Dedekind zeta-function $\zeta_K(s)$ attached to a number field $K$. When $K$ is totally real, the values $\zeta_K(2k)$ were proven to be algebraic multiples of powers of $\pi$ by Siegel and Klingen, independently. But this question remains unsolved in all other cases. In this talk, we discuss how our current knowledge allows us to deduce certain irrationality results for $\zeta_K(2k+1)$, where $K$ is an imaginary quadratic field. This is joint work with Prof. M. Ram Murty.

Monday, January 7th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Alistair Savage (University of Ottawa)

Title: Universal categories

Abstract:  Universal constructions are ubiquitous in mathematics.  For example, the polynomial ring is uniquely characterized by a universal property for commutative rings.  Other examples include free monoids, free groups, and tensor algebras.  In this mainly expository talk we will discuss an analogous, but somewhat less well known, concept on the level of categories.  In particular, we will see how one can define categories that are determined by universal properties.  Examples include the Temperley--Lieb category (the free monoidal category on a self-dual object), the Brauer category (the free symmetric monoidal category on a self-dual object), and the oriented Brauer category (the free symmetric monoidal category on a pair of dual objects).  We will discuss intuitive diagrammatic descriptions of these categories and how these universal constructions allow one to easily find deep symmetries in a wide range of categories.

Monday, December 3rd, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Kaveh Mousavand (UQAM)

Title: $\tau$-tilting finiteness of special biserial algebras

Abstract:  $\tau$-tilting theory, recently introduced by Adachi-Iyama-Reiten, is an elegant generalization of the classical tilting theory which fixes the deficiency of the tilting modules with respect to the notion of mutation. In this talk, I view $\tau$-tilting finiteness of algebras as a natural generalization of the representation finiteness property. The natural question then becomes: For which families of algebras does $\tau$-tilting finiteness imply representation finiteness?

First I introduce a reductive method that can be applied to certain families of algebras to reduce this, a priori, intractable problem to a subfamily with nice features. Then, as an interesting class of algebras, I consider the special biserial algebras and for every minimal representation infinite member of this family, I give a full answer to the above question and show. As a corollary, we conclude that a gentle algebra is $\tau$-tilting finite if and only if it is representation finite.

Friday, November 30th, 2018

Time: 10:30 a.m Place: Jeffery Hall 422

Speaker: Jing Tao (University of Oklahoma/Fields Institute)

Title: Big Torelli groups

Abstract: I will discuss some joint work with Aramayona, Ghaswala, Kent, McLeay, and Winarski on the Torelli subgroup of big mapping class groups.

Tuesday, November 27th, 2018

Time: 2:00-3:30 p.m Place: Jeffery Hall 116

Speaker: Mike Roth (Queen's University)

Title: Beauville’s results on classification of Kähler manifolds with $c_1(K_X)=0$, II.

Abstract: We will continue the discussion of Beauville’s paper, focussing on the structure theorem for simply connected Kähler manifolds with $c_1(K_X)=0$.