Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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, $

However,

  • $\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.

Number Theory - Arpita Kar (Queen's University)

Tuesday, March 12th, 2019

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

Speaker: Arpita Kar (Queen's University)

Title: On the normal number of prime factors of Euler Phi function at shifts of prime arguments.

Abstract: Let $\omega(n)$ and $\Omega(n)$ denote the number of prime factors of a natural number $n$ counted without and with multiplicity respectively. Let $\phi(n)$ denote the Euler totient function. In 1984, R.Murty and K. Murty defined a certain class of multiplicative functions and computed the normal order of $\omega(f(p))$ and $\omega(f(n))$ for $f$ belonging in that class. An example of functions in this class is $\phi(n)$. In this talk, we will discuss the normal number of prime factors of $\phi(n)$ at shifts of prime arguments, that is, $\Omega(\phi(p+a))$, for primes $p$ and any non-zero integer $a$.
This is joint work with Prof. Ram Murty.

Department Colloquium - Maria Chudnovsky (Princeton University)

Fields Queen's Distinguished Lecture Series, Maria Chudnovsky (Princeton University)

Friday, March 8th, 2019

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

Speaker: Maria Chudnovsky (Princeton University)

Title: Detecting odd holes.

Abstract: A hole in a graph is an induced cycle of length at least four; and a hole is odd if it has an odd number of vertices. In 2003 a polynomial-time algorithm was found to test whether a graph or its complement contains an odd hole, thus providing a polynomial-time algorithm to test if a graph is perfect. However, the complexity of testing for odd holes (without accepting the complement outcome) remained unknown. This question was made even more tantalizing by a theorem of D. Bienstock that states that testing for the existence of an odd hole through a given vertex is NP-complete. Recently, in joint work with Alex Scott, Paul Seymour and Sophie Spirkl, we were able to design a polynomial time algorithm to test for odd holes. In this talk we will survey the history of the problem and the main ideas of the new algorithm.

Prof. Chudnovsky (Princeton) is a leading researcher in graph theory and combinatorics. She received her B.A. and M.Sc. form the Technion, and her PhD from Princeton University in 2003. Before returning to Princeton, she was a member of the IAS, a Clay Math. Inst. research fellow, and a Liu Family Professor of IEOR at Columbia University. For her joint work proving the strong perfect graph theorem, she was awarded the Ostrowski foundation research stipend in 2003 and the Fulkerson prize in 2009. In 2012, she was awarded the MacArthur Foundation Fellowship, and was an invited speaker at the 2014 ICM. Prof. Chudnovsky was also named one of the "brilliant ten" young scientists by the Popular Science magazine.

Fields Lecture Series - Maria Chudnovsky (Princeton University)

Fields Queen's Distinguished Lecture Series, Maria Chudnovsky (Princeton University)

Thursday, March 7th, 2019

Time: 5:30 p.m.  Place: Jeffery Hall 127

Speaker: Maria Chudnovsky (Princeton University)

Title: Parties, Doughnuts and Coloring: Some Problems in Graph Theory

Abstract: A graph is a mathematical construct that represents information about connections between pairs of objects. As a result, graphs are widely used as a modeling tool in engineering, social sciences, and other fields. The paper written by Leonhard Euler in 1736 on the Seven Bridges of Konigsberg is often regarded as the starting point of graph theory; and we have come a long way since. This talk will survey a few classical problems in graph theory, and explore their relationship to the fields of research that are active today. In particular, we will discuss Ramsey theory, graph coloring, perfect graphs, as well as some more recent research directions.

Prof. Chudnovsky (Princeton) is a leading researcher in graph theory and combinatorics. For her joint work proving the strong perfect graph theorem, she was awarded the Ostrowski foundation research stipend in 2003 and the Fulkerson prize in 2009. In 2012, she was awarded the MacArthur Foundation Fellowship, and was an invited speaker at the 2014 ICM. Prof. Chudnovsky was also named one of the "brilliant ten" young scientists by the Popular Science magazine.

Images from Dr. Maria Chudnovsky's Lecture - Mar. 7th, 2019

Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)
Fields@Queen's Distinguished Lecture Series - Maria Chudnovsky (Princeton University)

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