Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Curves Seminar - Mike Roth (Queen's University)

Tuesday, November 6th, 2018

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

Speaker: Mike Roth (Queen's University)

Title: Local description of the Hilbert scheme when $n=2$, II.

Abstract: We will finish the discussion of local descriptions of the quotient by a finite group, and use them to give local pictures of $X^{[n]}$, $\operatorname{Sym}^n(X)$, the universal family over $X^{[n]}$, and the maps between them when $X$ is a smooth surface and $n=2$.

Number Theory - Brad Rodgers (Queen's University)

Tuesday, November 6th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Brad Rodgers (Queen's University)

Title: Integers in short intervals representable as sums of two squares.

Abstract: Consider the set S of integers that can be represented as a sum of two squares. How are the elements of S distributed? In particular, how many elements fall into a random "short interval". (The definition of short interval will be given in the talk.) For very short intervals elements of S seem to be laid down at random, but I will discuss evidence that this ceases to be the case for longer short intervals. In particular, I will discuss a function field analogue of this problem and a connection to z-measures, an object first investigated in the context of asymptotic representation theory. This is joint work with O. Gorodetsky.

Geometry & Representation - Steven Spallone (Indian Institute)

Monday, November 5th, 2018

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

Speaker: Steven Spallone (Indian Institute of Science Education and Research)

Title: Spinoriality of Orthogonal Representations of Reductive Groups

Abstract:  Let G be a connected semisimple complex Lie group and $\pi$ an orthogonal representation of G. We give a simple criterion for whether $\pi$ lifts to the spin group Spin(V), in terms of its highest weights. This is joint work with Rohit Joshi.

Department Colloquium - Thomas Koberda (University of Virginia)

Thomas Koberda (University of Virginia)

Friday, November 2nd, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Thomas Koberda (University of Virginia)

Title: Algebra versus regularity for group actions on one-manifolds.

Abstract: I will survey some results concerning the algebraic structure of finitely generated groups which admit faithful actions on compact one-manifolds. I will concentrate on continuous, $C^1$, and $C^2$ actions, and on the various algebraic restrictions imposed by regularity requirements. Of particular interest will be nilpotent groups, right-angled Artin groups, mapping class groups of surface, and Thompson's groups F and T. Time permitting, I will indicate some recent progress.

After obtaining his undergraduate at the University of Chicago, Thomas Koberda got his Ph.D.~from Harvard in 2012, then went to Yale as an NSF and Gibbs assistant professor before joining the University of Virginia in 2015. Thomas achievements have been recognized by a Sloan Research Fellowship and the Kamil Duszenko Prize of 2017.

Curves Seminar - Mike Roth (Queen's University)

Tuesday, October 30th, 2018

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

Speaker: Mike Roth (Queen's University)

Title: Local description of the Hilbert scheme when n=2

Abstract: Given a smooth surface $X$, we have the Hilbert scheme of points $X^{[n]}$, the associated Hilbert-Chow morphism to $\operatorname{Sym}^n(X)$, and the universal family over $X^{[n]}$. We will study what these look like when $n=2$.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, October 30th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: Central limit theorems in number theory.

Abstract: In this talk, I will report on joint work in progress with Ram Murty, where we obtain central limit theorems for sums of quadratic characters, and eigenvalues of Hecke operators acting on spaces of holomorphic cusp forms.

Geometry & Representation - Yin Chen (NE Normal University, China)

Monday, October 29th, 2018

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

Speaker: Yin Chen (Northeast Normal University, China)

Title: On vector invariant fields for finite classical groups

Abstract:  In the recent work (https://doi.org/10.1016/j.jpaa.2018.07.015) with David Wehlau, we found a minimal polynomial generating set for the vector and covector invariant field of the general linear group over finite fields. Our method relied on some relations between the generators for the invariant ring of one vector and one covector. The remaining case (without covectors) is more complicated. In this talk, I will present an approach to find polynomial generating sets for the vector invariant fields of the most of finite classical groups. This is a joint work with Zhongming Tang.

Number Theory - Payman Eskandari (University of Toronto)

Tuesday, October 23rd, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Payman Eskandari (University of Toronto)

Title: On the transcendence degree of the field genarated by quadratic periods of a smooth curve over a number field.

Abstract: Grothendieck's period conjecture predicts the transcendence degree of the field generated by the periods of a smooth projective variety (or more generally, a pure motive) over a number field, in terms of the dimension of its Mumford-Tate group. For mixed motives a similar conjecture was made by Andre. The upper bound part of Grothendieck's conjecture was proved in the case of abelian varieties by Deligne (as a consequence of his "Hodge implies absolute Hodge" theorem for abelian varieties). The lower bound part of Grothendieck's conjecture is known for a CM elliptic curve, thanks to a theorem of G. V. Chudnovsky.

This talk is a report on an aspect of a work in progress with Kumar Murty, in which we use Hodge theoretic methods and Tannakian formalism to study quadratic and higher periods of a punctured curve. We start by some background material and motivation. In the end, we prove the upper bound part of Andre's conjecture for quadratic periods of a punctured elliptic curve, defined over a subfield of $\mathbb{R}$. The argument is quite formal, and in fact, applies to any extension of $H^1\otimes H^1$ by $H^1$.

Department Colloquium - Camille Horbez (CNRS-Universite Paris Sud)

Camille Horbez (CNRS-Universite Paris Sud)

Friday, October 19th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Camille Horbez (CNRS – Universite Paris Sud)

Title: Mostow-type rigidity and normal subgroups of automorphisms of free groups.

Abstract: The talk is based on a recent joint work with Richard D. Wade. Let Out(Fn) be the outer automorphism group of a finitely generated free group. In 2007, Farb and Handel proved that when n is at least 4, every isomorphism between two finite-index subgroups of Out(Fn) extends to an inner automorphism of Out(Fn). This rigidity statement, asserting that Out(Fn) has no more symmetries than the obvious ones, can be viewed as an analogue of the Mostow rigidity theorem for lattices in Lie groups, or of a result of Ivanov for mapping class groups of surfaces. We recently gave a new proof of Farb and Handel's theorem, which enabled us to also understand the symmetries of some natural normal subgroups of Out(Fn). In my talk, I will emphasize the analogies between Out(Fn), arithmetic groups and mapping class groups, and will present some general ideas behind these rigidity phenomena.

Camille Horbez obtained his Ph.D in at the Universite de Rennes in 2014, and after a year at the University of Utah, he became a Charge de Recherches for the CNRS at the Universite de Paris Sud (Orsay). In 2017, Camille was selected to give a Cours Peccot, a semester long course given at the College de France by a mathematician less than 30 year old.

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