# 2014 Queen’s Graduate Math Seminar

In ’14-’15 I organized the graduate math seminar at Queen’s – this page is left just for historical interest.

If you’d like to give a talk (please do!), send me an email. The seminar is held on Fridays at 4:00 PM, in the window room (Jeffery Hall 319).

## Talks

### October 10, 2014

This talk will discuss Schoof’s algorithm for counting points on elliptic curves over finite fields. Let $$E$$ be an elliptic curve defined over $$\mathbb{F}_q$$, the finite field with $$q$$ elements. We would like to be able to compute $$\#E(\mathbb{F}_q)$$, the number of points on $$E$$ with coordinates in $$\mathbb{F}_q$$. Knowing the order of this group allows us to, for example, estimate the difficulty of the discrete logarithm problem in $$E(\mathbb{F}_q)$$.

The Hasse bound tells us that $\#E(\mathbb{F}_q) = q + 1 - a_q, \qquad \qquad |a_q| \le 2 \sqrt{q},$ so that there are only finitely many possibilities for $$a_q$$ (and hence $$\#E(\mathbb{F}_q)$$). Schoof’s algorithm gives a way to compute $$a_q$$ in polynomial time $$O((\log q)^8)$$: we use a trick to compute the value of $$a_q$$ modulo many small primes, and reconstruct its value using the Chinese Remainder Theorem.

### October 24, 2014

We will discuss some elementary aspects of the theory of binary quadratic forms in its contemporary avatar but from an historical perspective. We will discuss the notion of a reduced form, genera of quadratic forms and composition of classes of quadratic forms. The key players in our story include Euler, Fermat, Lagrange, Gauss.

### November 7, 2014

Given some data input and respective labels, we will ask the question: can we construct an algorithm that allows us to correctly classify new data?

We will discuss the meaning of “learning” in a computational framework and introduce the principle of Empirical Risk Minimization developed by Vapnik and Chervonenkis. The ERM principle will lead us to study the VC dimension of a class of functions, and finally, to a formulation of our results about learning in terms of the latter.

### November 14, 2014

We will continue discussion from the previous talk.

### January 9, 2015

The concept of the Dirac operator has its origins in the works of E. Cartan (1913), W. Pauli (1927) P. Dirac (1928), H.Weyl (1930), and M. Atiyah & I. Singer (1963).

From the physics viewpoint in order to write the relativistic equation of motion in quantum mechanics one needs to find the square root of the Laplacian i.e. a first order differential operator P such that $$P^2= \Delta$$.

In this seminar we will see the insight of Paul Dirac to this problem and we will dicuss how the notion of the Clifford algebra arises in this contex. We then introduce the concept of Clifford bundle, Dirac operator and and some of its geometric, analytic properties.

At the end we will see how the Bochner theorem reveals the interconnection of the topology and geometry of a Riemanninan manifold.

### February 27, 2015

We will study Weyl’s theorem on equidistribution and the differencing technique introduced by Weyl to prove equidistribution results for specific sequences.

### March 13, 2015

We will discuss the equidistribution of arithmetic sequences in arithmetic progressions. More specifically, we will discuss the equidistribution of primes, the divisor function, squarefree numbers, etc. This will be an expository talk.

### March 20, 2015

We discuss the failure of Chinese Remainder theorem for coprime elements over a ring that is not a PID. We give examples and non-examples of systems of polynomial congruences which have solutions. A general framework to study the problem of existence of a solution to a system of polynomial congruences will be discussed. This is joint work with Kamalakshya Mahatab.