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.

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 $(M,\omega)$.

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

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.