Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics (4)"

Time: 14:30

Room: MC 107

By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

Analysis Seminar

Speaker: Dayal Dharmasena (Syracuse University)

"Holomorphic Fundamental Semigroup of Riemann Domains"

Time: 15:30

Room: MC 108

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\partial\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\overline {\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ are called {\it holomorphically homotopic or $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into ${\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. \par There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity and show that $\eta_1(W,M,w_0)$ is an algebraic biholomorphic invariant of Riemann domains. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. When $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is the identity, we show that $\iota_1$ is an isomorphism of $\eta_1(W,M,w_0)$ onto the minimal subsemigroup of $\pi_1(W,w_0)$ containing holomorphic generators and invariant with respect to the inner automorphisms. In particular, we show for a general domain $W\subset\mathbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$. This is a joint work with Evgeny Poletsky.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Localization in equivariant cohomology and index formula"

Time: 14:30

Room: MC 107

The path integral formula for the index of the Dirac operator can be interpreted as a localization formula for U(1)-equivariant cohomology of the free loop space of the manifold. In this lecture I shall first recall the Cartan model of equivariant differential forms of a finite dimensional manifold and the localization formula of Berline-Vergne. We shall then see that the loop space analogue of this result will give the A hat genus. This can be regarded as the bosonic component of the index formula. The corresponding localization formula in the supersymmetric case gives the full index formula.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Applications of the Atiyah-Singer Index theorem 4: the Hirzebruch-Riemann-Roch Theorem"

Time: 10:30

Room: MC 107

Following the previous talks on the Atiyah-Singer index theorem by Masoud, we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.

Algebra Seminar

Speaker: David Riley (Western)

"On the behaviour of the Frobenius map in a noncommutative world"

Time: 14:30

Room: MC 108