Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium - Carolyn Gordon (Dartmouth College)

Carolyn Gordon (Dartmouth College)

Friday, March 29th, 2019

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

Speaker: Carolyn Gordon (Dartmouth College)

Title: Decoding geometry and topology from the Steklov spectrum of orbisurfaces.

Abstract: The Dirichlet-to-Neumann or "voltage-to-current" operator of, say, a surface $M$ with boundary is a linear map $C^\infty(\partial M)\to C^\infty(\partial M)$ that maps the Dirichlet boundary values of each harmonic function $f$ on M to the Neumann boundary values of $f$. The spectrum of this operator is discrete and is called the Steklov spectrum. The Dirichlet-to-Neumann operator also generalizes to the setting of orbifolds, e.g., cones. We will address the extent to which the Steklov spectrum encodes the geometry and topology of the surface or orbifold and, in particular, whether it recognizes the presence of orbifold singularities such as cone points.

This is joint work with Teresa Arias-Marco, Emily Dryden, Asma Hassannezhad, Elizabeth Stanhope and Allie Ray.

Prof. Gordon is an expert in spectral geometry. She obtained her PhD from Washington University in 1979, then went to the Technion institue and held positions at Lehigh University and Washington University before moving to Dartmouth where she is currently the Benjamin Cheney Professor of Mathematics.
Prof. Gordon was awarded an AMS Centennial Fellowship in 1990, the MAA Chauvenet prize in 2001 and was the 2010 Noether Lecturer. In 2012, she became a fellow of both the AMS and the American Association for the Advancement of Science. In 2017, she was selected to be a fellow of the AWM in the inaugural class.

Number Theory - Tariq Osman (Queen's University)

Tuesday, March 26th, 2019

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

Speaker: Tariq Osman (Queen's University)

Title: Counting Integer Points on Vinogradov's Quadric.

Abstract: Consider the variety defined by the pair of equations $x_1 + x_2 + x_3 = y_1 + y_2 + y_3$ and $ x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2$, known as Vinogradov's quadric. Following a brief historical overview and a few motivational remarks, we will derive an asymptotic formula (due to Ragovskya) for the number of integer points on Vinogradov's quadric in a large box.

Dynamics, Geometry, & Groups - Alessandro Portaluri (U of Torino)

Friday, March 22nd, 2019

Time: 10:30 a.m Place: Jeffery Hall 102

Speaker: Alessandro Portaluri (University of Torino, Italy)

Title: Visiting Kepler with a couple of symplectic friends.

Abstract: Starting from the classical planar Kepler problem, by using the conservation law of the angular momentum, we reduce the problem to a one degree of freedom singular problem. Thanks to this reduction and after a suitable time scaling we show that, for negative energy, the orbit is an ellipse. Finally, by using a refined version of the Conley-Zehnder intersection index , we give a homotopic classification of all bounded motions.

Department Colloquium - Maksym Radziwill (Caltech)

Maksym Radziwill (California Institute of Technology)

Friday, March 22nd, 2019

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

Speaker: Maksym Radziwill (California Institute of Technology)

Title: Recent progress in multiplicative number theory.

Abstract: Multiplicative number theory aims to understand the ways in which integers factorize, and the distribution of integers with special multiplicative properties (such as primes). It is a central area of analytic number theory with various connections to $L$-functions, harmonic analysis, combinatorics, probability etc. At the core of the subject lie difficult questions such as the Riemann Hypothesis, and they set a benchmark for its accomplishments. An outstanding challenge in this field is to understand the multiplicative properties of integers linked by additive conditions, for instance $n$ and $n+ 1$. A central conjecture making this precise is the Chowla-Elliott conjecture on correlations of multiplicative functions evaluated at consecutive integers. Until recently this conjecture appeared completely out of reach and was thought to be at least as difficult as showing the existence of infinitely many twin primes. These are also the kind of questions that lie beyond the capability of the Riemann Hypothesis. However recently the landscape of multiplicative number theory has been changing and we are no longer so certain about the limitations of our (new) tools. I will discuss the recent progress on these questions.

Maksym Radziwill graduated from McGill University in Montreal in 2009, and in 2013 took a PhD under Kannan Soundararajan at Stanford University in California. In 2013-2014, he was at the Institute for Advanced Study in Princeton, New Jersey as a visiting member, and in 2014 became a Hill assistant professor at Rutgers University. In 2016, he became an assistant professor at McGill. In 2018, he became Professor of Mathematics at Caltech.

Math Club - Ivan Dimitrov (Queen's University)

Thursday, March 21st, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Ivan Dimitrov (Queen's University)

Title: Why 0.499999999992646 does not equal 12.

Abstract:  We will see why

  • $\int_{0}^\infty \frac{\sin x}{x} \,dx = \pi/2,$
  • $\int_{0}^\infty \frac{\sin x}{x} \cdot \frac{\sin x/3}{x/3}\, dx = \pi/2, $

… and so on all the way to

  • $\int_{0}^\infty \frac{\sin x}{x} \cdot \frac{\sin x/3}{x/3} \cdots \frac{\sin x/13}{x/13}\, dx = \pi/2, $


  • $\int_{0}^\infty \frac{\sin x}{x} \cdot \frac{\sin x/3}{x/3} \cdots \frac{\sin x/13}{x/13} \cdot \frac{\sin x/15}{x/15} \,dx < \pi/2, $

Number Theory - Allysa Lumley (York University)

Tuesday, March 19th, 2019

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

Speaker: Allysa Lumley (York University)

Title: Complex moments and the distribution of values of $L(1, \chi_D)$ over function fields with applications to class numbers.

Abstract: In 1992, Hoffstein and Rosen proved a function field analogue to Gau\ss' conjecture (proven by Siegel) regarding the class number, $h_D$, of a discriminant $D$ by averaging over all polynomials with a fixed degree. In this case $h_D=|\text{Pic}(\mathcal{O}_D)|$, where $\text{Pic}(\mathcal{O}_D)$ is the Picard group of $\mathcal{O}_D$. Andrade later considered the average value of $h_D$, where $D$ is monic, squarefree and its degree $2g+1$ varies. He achieved these results by calculating the first moment of $L(1,\chi_D)$ in combination with Artin's formula relating $L(1,\chi_D)$ and $h_D$. Later, Jung averaged $L(1,\chi_D)$ over monic, squarefree polynomials with degree $2g+2$ varying. Making use of the second case of Artin's formula he gives results about $h_DR_D$, where $R_D$ is the regulator of $\mathcal{O}_D$.

For this talk we discuss the complex moments of $L(1,\chi_D)$, with $D$ monic, squarefree and degree $n$ varying. Using this information we can describe the distribution of values of $L(1,\chi_D)$ and after specializing to $n=2g+1$ we give results about $h_D$ and specializing to $n=2g+2$ we give results about $h_DR_D$.

If time permits, we will discuss similar results for $L(\sigma,\chi_D)$ with $1/2<\sigma<1$.

Geometry & Representation - Colin Ingalls (Carleton University)

Monday, March 18th, 2019

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

Speaker: Colin Ingalls (Carleton University)

Title: McKay quivers

Abstract:  Fix a finite group G and a representation W, the McKay quiver has vertices given by irreducible representations $V_i$ and $\dim\mathrm{Hom}_G(W\otimes V_i,V_j)$ many arrows between $V_i$ and $V_j$. We briefly present the history of McKay quivers and their applications in geometry and representation theory. Then we discuss recent descriptions of McKay quivers of relfection groups by M. Lewis, and work with E. Faber and R. Buchweitz applying results of Lustzig on McKay quivers to understand the relations of the basic model of the skew group ring.

Department Colloquium - Henri Darmon (McGill)

Henri Darmon (McGill)

Friday, March 15th, 2019

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

Speaker: Henri Darmon (McGill)

Title: Kronecker's Jugendtraum: a non-archimedean approach.

Abstract: Kronecker's Jugendtraum ("dream of youth") seeks to construct abelian extensions of a given field through the special values of explicit analytic functions, in much the same way that the values of the exponential function $e(x) = e^{2\pi i x}$ at rational arguments generate all the abelian extensions of the field of rational numbers. Modular functions like the celebrated $j$-function play the same role when the field of rational numbers is replaced by an imaginary quadratic field. An extensive literature, both classical and modern, is devoted to the special values of modular functions at imaginary quadratic arguments, known as singular moduli; and modular functions have also been central to such modern developments as Wiles' proof of Fermat's Last Theorem and significant progress on the Birch and Swinnerton--Dyer conjecture arising through the work of Gross--Zagier and Kolyvagin. I will describe some recent attempts, in colaboration with Jan Vonk, to extend Kronecker's Jugendtraum to real quadratic fields by replacing modular functions by mathematical structures called "$p$-adic modular cocycles". While they are still poorly understood, these objects exhibit many of the same rich arithmetic properties as modular functions.

Prof. Darmon obtained his PhD from Harvard in 1991, then went on to Princeton before coming to McGill University in 94 where he is now a James McGill Professor. Prof. Darmon received many awards and distinctions, including a Sloan Research Award in 96, the Prix Andre Aisenstadt in 97, the Killam Research Fellowship in 2008, and both the Cole prize and the CRM-Fields-PIMS prize in 2017. Prof. Darmon was elected fellow of the Royal Society of Canada in 2003.

Math Club - Richard Gottesman (Queen's University)

Thursday, March 14th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Richard Gottesman (Queen's University)

Title: Gaussian Integers and Sums of Two Squares.

Abstract:  The Gaussian integers consist of the set of all complex numbers whose real and imaginary parts are both integers. We begin by exploring the number theory of the Gaussian integers. For example, we shall see why $3$ is a Gaussian prime but $5 = (2 +i)(2 - i)$ is not.

We will then show how to use the Gaussian integers to prove that if $p$ is a prime number which is one more than a multiple of $4$ then $p$ is a sum of two perfect squares. This proof is very striking and it generalizes to other number systems, such as the Hurwitz quaternions.

In honor of $\pi$ day, we must also mention that $\pi$ is equal to the average number of ways to write an integer as a sum of two squares.