Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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)

Number Theory - Brad Rodgers (Queen's University)

Tuesday, March 5th, 2019

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

Speaker: Brad Rodgers (Queen's University)

Title: The distribution of traces of powers of matrices over finite fields.

Abstract: Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this convergence is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.

Department Colloquium - Alessandro Portaluri (University of Turin)

Alessandro Portaluri (University of Turin)

Friday, March 1st, 2019

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

Speaker: Alessandro Portaluri (University of Turin)

Title: Existence and stability results in Celestial Mechanics

Abstract: Is the solar system stable? This is maybe one of the oldest open question in dynamical systems. It is still a lively and very active research field starting from Newton, Lagrange, Maxwell, Poincare and Birkhoff (only to mention a few of them) who proved several astonishing results in this direction.

A lot of useful techniques to tackle this problem, were developed during the last decades: KAM theory, symplectic and contact methods, interval arithmetic, etc. One more (variationally oriented) piece we add to this arsenal: the index theory!

In this talk we introduce some new ideas behind many recent results on this topic and we discuss some new perspectives and challenges on singular (weak force) Lagrangian problems. We show the existence of equivariant periodic orbits and we investigate some (in)stability results for a plethora of periodic motions via symplectic techniques.

Prof. Portaluri (University of Turin) obtained his PhD in 2004 from the University of Turin, then worked at the University of Milano-Bicocca and the University of Salento before going back to Turin in 2012. He is currently on sabbatical at Queen's. His research interests include index theory for ordinary and partial differential operators via Maslov index and spectral flow techniques, and linear stability analysis in singular Lagrangian systems.

Math Club - Neha Prabhu (Queen's University)

Thursday, February 28th, 2019

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

Speaker: Neha Prabhu (Queen's University)

Title: The one-sentence proof.

Abstract:  Which primes can be written as a sum of two squares? The answer to this classical question has been known for centuries and many different elaborate proofs have been discovered. In this talk, we present Don Zagier's sensational "one-sentence" proof.

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