9 |
10 |
11 Geometry and Topology
Geometry and Topology Speaker: Eli Hawkins (University of York, UK) "Operations on the Hochschild Complex of a Functor" Time: 14:30 Room: MC 107 In order to study deformations of associative algebras, in 1963, Gerstenhaber constructed operations on the Hochschild complex of an algebra. These operations induce a product and bracket on cohomology, which generalize those on the space of multivector fields over a manifold. This is now known as a Gerstenhaber algebra structure. Now consider a functor from a small category to a category of algebras. In 1988, Gerstenhaber and Schack showed indirectly that the Hochschild cohomology of such a functor is also a Gerstenhaber algebra. I will describe my efforts to construct operations on the complex that induce this structure. My motivation comes from algebraic quantum field theory. My methods rely on operad theory. |
12 PhD Thesis Defence
PhD Thesis Defence Speaker: Baran Serajelahi (Western) "Quantization of two types of multisymplectic manifolds" Time: 13:00 Room: MC 107 We will be interested in quantization in a setting where the algebraic structure on $C^{\infty}(M)$ is given by an m-ary bracket $\{.,\dots,.\}:\otimes^m C^{\infty}(M)\rightarrow C^{\infty}(M)$. Quantization in this context is the same as in the symplectic case, where we have a bracket of just two functions except that now we are interested in a correspondence $\{.,\dots,.\}\rightarrow [.,\dots,.]$, between an m-ary bracket and a generalizeation of the commutator. In particular we will be interested in two situations where the m-ary bracket comes from an $(m-1)$-plectic form defined on M (i.e. a closed non-degenerate $m$-form), $\Omega$, for $m\ge 1$. The case $m=1$ is when $\Omega$ is symplectic. Let $(M,\omega)$ be a compact connected integral K\"ahler manifold of complex dimension $n$. In both of the cases that we will be looking into, the $(m-1)$-plectic form $\Omega$ on $(M,\omega)$ is constructed from a K\"ahler form (or forms): (I) $m=2n$, $\Omega = \frac{\omega^n}{n!}$ (II) $M$ is, moreover, hyperk\"ahler, $m=4$, $$ \Omega = \omega_1\wedge \omega_1 + \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3 $$ where $\omega_1, \omega_2, \omega_3$ are the three K\"ahler forms on $M$ given by the hyperk\"ahler structure. It is well-known (and easy to prove) that a volume form on an oriented $N$-dimensional manifold is an $(N-1)$-plectic form, and that the $4$-form above is a $3$-plectic form on a hyperk\"ahler manifold. It is intuitively clear that in these two cases the classical multisymplectic system is essentially built from Hamiltonian system(s) and it should be possible to quantize $(M,\Omega)$ using the (Berezin-Toeplitz) quantization of $(M,\omega)$. |
13 |
14 PhD Thesis Defence
PhD Thesis Defence Speaker: Mayada Shahada (Western) "Combinatorial Polynomial Identity Theory" Time: 10:30 Room: MC 107 Algebras with polynomial identities generalize commutative and finite-dimentional algebras. This talk will consist of two parts. Part I examines certain Burnside-type conditions on the multiplicative and the adjoint semigroups associated with an associative algebra $A$. A semigroup $S$ is called $n$-collapsing if, for every $s_1,\ldots,s_n$ in $S$, there exist functions $f \neq g$, such that
$$
s_{f(1)} \cdots s_{f(n)} = s_{g(1)} \cdots s_{g(n)}.
$$
More specifically, $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations. Semple and Shalev extended Zelmanov's solution of the Restricted Burnside Problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. In Part I of this talk, we will consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we prove the following conditions are equivalent, for all unital algebras $A$ over an infinite field: the multiplicative semigroup of $A$ is collapsing, $A$ satisfies a multiplicative semigroup identity, and $A$ satisfies an Engel identity. Furthermore, we will see that, if the multiplicative semigroup of $A$ is rewritable, then $A$ must be commutative. In Part II of this talk, we will consider algebraic analogues to well-known problems of Philip Hall on the verbal and marginal subgroups of a group. Consider the canonicl descending and ascending central series of ideals of an associative algebra $A$:
$$A=A^{(1)}\supseteq A^{(2)}\supseteq \cdots \supseteq A^{(n)}\supseteq \cdots\supseteq 0\quad\text {and}$$ $$0=F^{(0)}(A)\subseteq F^{(1)}(A)\subseteq\cdots \subseteq F^{(n)}(A) \subseteq \cdots \subseteq A.$$
Jennings proved that $A^{(n+1)}=0$ precisely when $A=F^{(n)}(A)$.
First we will prove that, if $A/F^{(n)}(A)$ is finite-dimensional, then so is $A^{(n+1)}$. This result is an analogue of a group-theoretic result of Baer, which was proved first by Schur in the case when $n=1$. We also will see that the converse holds whenever $A$ is finitely generated. While this is not true for arbitrary algebras $A$, we do show that, if $A^{(n+1)}$ is finite-dimensional, then at least the quotient $A/F^{(3n-1)}(A)$ is finite-dimensional. These two partial converses are analogues of group-theoretic results due to Hall, albeit with a different bound in the second result.
Our main technique is to first describe the ideals $A^{(n+1)}$ and $F^{(n)}(A)$ as the verbal and marginal subspaces of $A$ corresponding to a certain polynomial $g_n$ and then apply Setwart's result in the algebraic analogue of Hall's First Problem. Moreover, we will consider algebraic analogues to the other Hall's Problems and will support them positively in some cases. Noncommutative Geometry
Noncommutative Geometry Speaker: Mitsuru Wilson (Western University) "Hilbert C*-Modules" Time: 11:00 Room: to be determined Lecture series based on the book: Hilbert $C^*$-Modules: A Toolkit for Operator Algebraists, Cambridge University Press. |
15 |
16 |
17 |
18 Noncommutative Geometry
Noncommutative Geometry Speaker: Shahab Azarfar (Western) "Selberg Trace Formula" Time: 11:00 Room: TBA Consider a closed smooth hyperbolic surface ${\Sigma = \Gamma \backslash \mathbb{H}}$. Let ${k(x,y)}$ be a continuous function which depends only on the hyperbolic distance between ${x,y \in \mathbb{H}}$, and has some ``nice'' decay properties. Using ${k(x,y)}$, we construct a trace-class integral operator ${T_k}$ on
${L^2 (\Sigma)}$. The trace of ${T_k}$ is computed in two different ways using the Lidski's trace formula. The resulting Selberg's trace formula gives a relation between the length of closed geodesics and the eigenvalues of the hyperbolic Laplacian on $\Sigma$. PhD Thesis Defence
PhD Thesis Defence Speaker: Javad Rastegari Koopaei (Western) "Fourier inequalities in Lorentz and Lebesgue spaces" Time: 13:30 Room: MC 107 This talk is on the mapping properties of the Fourier transform between
Banach function spaces. These are generalizations of Hausdorff-Young and
Pitt's inequalities. We provide several relations between weight functions, that guarantee the
boundedness of the Fourier series coefficients, viewed as a map between
weighted Lorentz spaces. As a useful machinery, we briefly introduce the
quasi concave functions and generalize a number of known inequalities.
Finally, we apply our results to Fourier inequalities in weighted Lebesgue
spaces and Lorentz-Zygmund spaces |
19 PhD Thesis Defence
PhD Thesis Defence Speaker: Allen O'Hara (Western) "A study of Green's relations on algebraic semigroups" Time: 11:00 Room: MC 107 |
20 Colloquium
Colloquium Speaker: Mieczysław Mastyło (Adam Mickiewicz University) "On the convergence of Fourier series" Time: 15:30 Room: MC 107 |
21 |
22 |
30 |
31 |
1 |
2 |
3 |
4 Noncommutative Geometry
Noncommutative Geometry Speaker: TBA (Western) "Learning Seminar" Time: 11:00 Room: MC 108 The NCG learning seminar's topic this year will be Geometric Analysis. One of our goals is
to go through a heat equation proof of the celebrated Atiyah-Singer index theorem. The following topics will be covered:
1. Operators of Dirac type and its main examples,
2. Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula,
3. Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula,
4. The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory,
5. Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.
6. Approach via path integrals and quantum mechanics,
7. Atiyah-Bott-Lefschetz fixed point formula. |
5 |
Mathematics Calendar 
August 2015
