Georgia Institute of Technology College of Sciences

The Witt group of a scheme is a globalization to schemes of the classical Witt group of a field. It is a part of a cohomology theory for schemes called the derived Witt groups. In this talk, we introduce two problems about the derived Witt groups, the Gersten conjecture and a finite generation question for arithmetic schemes, and explain recent progress on them.

Mechanism design for distributed systems is fundamentally concerned with aligning individual incentives with social welfare to avoid socially inefficient outcomes that can arise from agents acting autonomously. One simple and natural approach is to centrally broadcast non-binding advice intended to guide the system to a socially near-optimal state while still harnessing the incentives of individual agents. The analytical challenge is proving fast convergence to near optimal states, and we present the first results that carefully constructed advice vectors yield stronger guarantees. We apply this approach to a broad family of potential games modeling vertex cover and set cover optimization problems in a distributed setting. This class of problems is interesting because finding exact solutions to their optimization problems is NP-hard yet highly inefficient equilibria exist, so a solution in which agents simply locally optimize is not satisfactory. We show that with an arbitrary advice vector, a set cover game quickly converges to an equilibrium with cost of the same order as the square of the social cost of the advice vector. More interestingly, we show how to efficiently construct an advice vector with a particular structure with cost $O(\log n)$ times the optimal social cost, and we prove that the system quickly converges to an equilibrium with social cost of this same order.

In this era of "big data", Mathematics as it applies to human behavior is becoming a much more relevant and penetrable topic of research. This holds true even for some of the less desirable forms of human behavior, such as crime. In this talk, I will discuss the mathematical modeling of crime on various "scales" and using many different mathematical techniques, as well as the results of experiments that are being performed to test the usefulness and accuracy of these models. This will include: models of crime hotspots at the scale of neighborhoods -- in the form of systems of PDEs and also statistical models adapted from literature on earthquake predictions -- along with the results of the model's application within the LAPD; a model for gang retaliatory violence on the scale of social networks, and its use in the solution of an inverse problem to help solve gang crimes; and a game-theoretic model of crime and punishment at the scale of a society, with comparisons of the model to results of lab-based economic experiments performed by myself and collaborators.

We show how to construct frames for square integrable functionsout of modulated Gaussians. Using the frame representation of the Cauchydata, we show that we can build a suitable approximation to the solutionfor low regularity, time dependent wave equations. The talk will highlightthe relationship of the construction to harmonic analysis and will explorethe differences of the new construction to the standard Gaussian beamansatz.