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August 2018
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Events on Friday 02 March 2018

A Hecke module structure on the KK-theory of arithmetic groups
13:10 Fri 2 Mar, 2018 :: Barr Smith South Polygon Lecture theatre :: Bram Mesland :: University of Bonn

Media...
Let $G$ be a locally compact group, $\Gamma$ a discrete subgroup and $C_{G}(\Gamma)$ the commensurator of $\Gamma$ in $G$. The cohomology of $\Gamma$ is a module over the Shimura Hecke ring of the pair $(\Gamma,C_G(\Gamma))$. This construction recovers the action of the Hecke operators on modular forms for $SL(2,\mathbb{Z})$ as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair $(\Gamma, C_{G}(\Gamma))$ maps into the $KK$-ring associated to an arbitrary $\Gamma$-C*-algebra. From this we obtain a variety of $K$-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of $\Gamma$-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of $SL(2,\mathbb{Z})$. This is joint work with M.H. Sengun (Sheffield).
Calculating optimal limits for transacting credit card customers
15:10 Fri 2 Mar, 2018 :: Horace Lamb 1022 :: Prof Peter Taylor :: University of Melbourne

Credit card users can roughly be divided into `transactors', who pay off their balance each month, and `revolvers', who maintain an outstanding balance, on which they pay substantial interest. In this talk, we focus on modelling the behaviour of an individual transactor customer. Our motivation is to calculate an optimal credit limit from the bank's point of view. This requires an expression for the expected outstanding balance at the end of a payment period. We establish a connection with the classical newsvendor model. Furthermore, we derive the Laplace transform of the outstanding balance, assuming that purchases are made according to a marked point process and that there is a simplified balance control policy which prevents all purchases in the rest of the payment period when the credit limit is exceeded. We then use the newsvendor model and our modified model to calculate bounds on the optimal credit limit for the more realistic balance control policy that accepts all purchases that do not exceed the limit. We illustrate our analysis using a compound Poisson process example and show that the optimal limit scales with the distribution of the purchasing process, while the probability of exceeding the optimal limit remains constant. Finally, we apply our model to some real credit card purchase data.
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