Title: Tangents of Uniformly Distributed Measures - II

Date: 11/21/2016

Time: 1:40 PM - 3:00 PM

Place: C304 Wells Hall

We will prove Theorem 3.8 of Preiss's paper: roughly
speaking, given a uniformly distributed measure and a point in its
support, it is possible to find a nearby point for which an
appropriate scaling of the measure looks like Lebesgue measure on a
hyperplane. The proof will involve a Taylor expansion of the Gaussian
moments studied a few weeks ago.

Title: Risk sharing and risk aggregation via risk measures

Date: 11/21/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk, we discuss two problems in risk management using the tools of risk
measures.
In the first part of the talk, we address the problem of risk sharing among agents using
a two-parameter class of quantile-based risk measures, the so-called RangeValue-at-Risk
(RVaR), as their preferences. We first establish an inequality for RVaRbased
risk aggregation, showing that RVaR satisfies a special form of subadditivity.
Then, the risk sharing problem is solved through explicit construction.
Comonotonicty and robustness of the optimal allocations are investigated. We
show that, in general, a robust optimal allocation exists if and only if none of the risk
measures is a VaR. Practical implications of our main results for risk management
and policy makers will be discussed.
In the second part of the talk, we study the aggregation of inhomogeneous risks with
a special type of model uncertainty, called dependence uncertainty, evaluated by a
generic risk measure. We establish general asymptotic equivalence results for the
classes of distortion risk measures and convex risk measures under different mild
conditions. The results implicitly suggest that it is only reasonable to implement a
coherent risk measure for the aggregation of a large number of risks with uncertainty
in the dependence structure, a relevant situation for risk management practice.

This talk will be a follow up for the previous talk during the the cluster algebra seminar. We will explore specific cases of the amplituhedron when m=4. The goal will be to work through few examples of this specific case, and see if we can state some of the main results in the area.

Title: Restricted Stirling and Lah numbers, and their inverses

Date: 11/22/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The matrix of Stirling numbers of the second kind (counting partitions of a set into non-empty blocks) is lower triangular with integer entries and 1’s down the diagonal, so its inverse shares the same properties. It’s well-known that the entries in the inverse matrix have a nice combinatorial meaning — they are Stirling numbers of the first kind (counting permutations by number of cycles).
We explore restricted Stirling numbers of the second kind, in which the block sizes are required to lie in some specified set. As long as this set contains 1, the matrix of these restricted Stirling numbers has an inverse with integer entries, so it is natural to ask: do these integers count things?
In many cases, we find that they do. This is joint work with John Engbers (Marquette) and Cliff Smyth (UNC Greensboro).