MSCS Seminar Calendar

The study of decision making has been dominated by economic perspectives, which model people as rational agents who carefully weigh costs and benefits and try to maximize the utility of every choice, without consideration of issues such as cultural norms, religious beliefs and moral rules which exist outside the market. However, psychological findings indicate that in many situations people are not optimal nor rational decision makers as defined by the economic theories. One of the domains in which the rational actor perspective fails to explain human behavior is that of moral decision making. A body of research illustrates that in the presence of moral values, such as those outlined in religious texts or folk stories, people tend to focus on the obligations and duties outlined in their culture and as a result are less concerned about the outcome utility of their choice. In this talk, I present a computational model of recognition-based moral decision making, MoralDM, which integrates several AI techniques in order to model recent psychological findings on moral decision making. MoralDM uses a natural language system to produce formal representations from psychological stimuli, reducing tailorability. The impacts of secular versus sacred values are modeled via qualitative reasoning, using an order of magnitude representation. MoralDM uses a combination of first-principles reasoning and analogical reasoning to determine consequences and utilities when making moral judgments. I describe how MoralDM works and show that it can model psychological results and capture the impact of cultural narratives on decision making.

If X is a compact set in the plane then, by a classical theorem of Marstrand, almost every projection onto a line maps X to a set of the maximal possible Hausdorff dimension, i.e. the smaller of dim(X) and 1. While in general the set of exceptional direction can be large, in certain situations arising from dynamical, arithmetic or combinatorial contexts, it is predicted that there should be either no exceptions, or some small explicit set of exceptions. One example of this is an old conjecture of Furstenberg's, predicting that, if X=A\times B, and A,B are, respectively, subsets of the unit interval invariant under times-2 mod 1 and times-3 mod 1, then the image of X under projection should behave in this manner for every (not just almost every) projection, the only exceptions being the coordinate projections. I will explain the background of this problems and my recent work with Pablo Shmerkin in which we resolve this conjecture positively. If time permits I will describe some other applications of our methods.

A classical theorem of S.Bernstein states that, for any increasing sequence of finite dimensional subspaces $E_1 \hookrightarrow E_2 \hookrightarrow \ldots \hookrightarrow X$ ($X$ is a Banach space), and for any sequence $\alpha_i \searrow 0$, there exists $x \in X$ s.t. $dist(x,E_i) = \alpha_i$ for every $i$. In this talk, we present several related results. (1) Establishing a non-commutative version of Bernstein's result, we prove that, for any pair of infinite dimensional Banach spaces $X$ and $Y$, and for any sequence $\alpha_i \searrow 0$, there exists $T \in B(X,Y)$ whose sequences of approximation, Gelfand, and Kolmogorov numbers ``behave like'' $(\alpha_i)$. Other $s$-scales are also considered. (2) We show that, for many dictionaries in Banach spaces (including Markushevich bases, and also certain highly redundant dictionaries), the error of the best $n$-term approximation may decay arbitrarily slowly. Part of this work was carried out in collaboration with J.Almira.

The theory of valued D-fields provides an interesting setting for the study of difference and differential Galois groups. The extra structure provided by the valuation can be used to relate groups arising from these two types of equations; however, these groups are not always what one might expect. I will talk about the some of the advantages and limitations of working in this particular theory, and look at a few specific equations.