Mathematical Research Seminar - Archive

Motion planning is among the core problems in robotics, see for example LaValle, Planning Algorithms for a nice overview of the subject.
Mathematically, one can consider all possible positions of a robot, view it as a topological space and try to find a continuous path ('motion plan') in that space from a given initial position to a required final position of the robot. A further refinement arises if we require that the motion plan is predictable in the sense that the chosen path depends continuously on the input and output data (think of motion plans for self-driving cars).
It turns out that predictable motion planning has inherent instabilities, which are caused by the global topology of the space of all robot positions.
Topological complexity was introduced some twenty years ago by Michael Farber as a measure for these instabilities. We will present some basic results of the theory and describe some applications and open problems.

Mark your calendar and join us for what promises to be an unforgettable experience.

Feel free to invite any friends or colleagues who may be interested.

We look forward to sharing the passion for math with you!

The Erdős-Ko-Rado (EKR) theorem is a fundamental result in extremal set theory which asserts that if $\mathcal{F}$ is a collection of pairwise intersecting $k$-subsets of $\{1,2,\ldots,n\}$, for $n\geq 2k$, then $|\mathcal{F}| \leq \binom{n-1}{k-1}$. Moreover, if $n\geq 2k+1$ then equality holds if and only if $\mathcal{F}$ is an orbit of a conjugate of a stabilizer of a point of the symmetric group $\sym(n)$ in its natural action.
    
 The EKR theorem has been extended to various combinatorial objects throughout the years. In this talk, I will present some powerful algebraic combinatorics tools,  such as association schemes and representation theory, to prove EKR type results on the symmetric group.

It will also be held virtually via Zoom.

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We look forward to (virtually) sharing the passion for math with you!

Altanisation originated in the chemical literature as a formal device for constructing generalised coronenes from smaller structures. The altan graph of $G$, $\mathfrak{a}(G, H)$, is constructed from graph $G$ by choosing an attachment set $H$ from the vertices of $G$ and attaching vertices of $H$ to alternate vertices of a new perimeter cycle of length $2|H|$. We prove sharp bounds for the nullity of altan and iterated altan graphs based on a general parent graph: the nullity of the altan exceeds the nullity of the parent graph by at most $2$. The case of excess nullity $2$ had not been noticed before; for benzenoids it occurs first for a parent structure with merely $5$ hexagons. We also exhibit an infinite family of convex benzenoids with $3$-fold dihedral symmetry (point group $D_{3h}$), where nullity increases from $2$ to $3$ under altanisation. This family accounts for all known examples with the excess nullity of $1$ where the parent graph is a singular convex benzenoid.

This is joint work with Patrick W. Fowler.

Join the Zoom Meeting HERE!

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We look forward to (virtually) sharing the passion for math with you!