MSCS Seminar Calendar

Nonlinear Schrödinger equations can be used as a mean-field description of Bose-Einstein condensates. Recent experimental breakthroughs require extensions of the classical Gross-Pitaevskii model to include new physical phenomena. A particularly interesting field concerns condensates in so-called dipolar quantum gases. We review several recent mathematical results on nonlinear Schrödinger equations arising in this context, including existence of solutions, the possibility of finite time-blow-up, and dimension reduction through scaling limits.

This talk continues, from last week, an introduction to spectral sequences.

In the 1970s S. Shelah gave the following characterization of stable theories: a theory is stable if and only if any indiscernible sequence in a model of the theory is an indiscernible set. I will present a similar characterization of NIP theories, as theories in which any random ordered graph-indiscernible in a model of the theory remains indiscernible strictly with respect to the order. In this talk I will explain what I mean by a random ordered graph-indiscernible and I will indicate how the result is proved using the Nesetril-Rodl theorem. If time permits, I will discuss an additional example of a characterization of stable theories by generalized indiscernibles that generalizes more faithfully on Shelah's.

With no doubt, Thurston's stability theorem is still the most striking rigidity result for group actions. In this talk I will concentrate on its 1-dimensional version, which establishes that the group of C^1 diffeomorphisms of the interval is locally indicable (i.e. every finitely generated subgroup surjects onto Z). I will show by an example that the converse statement does not hold. More precisely, the semidirect product SL(2,Z) \rtimes Z^2, though locally indicable (and finitely generated) does not act faithfully by C^1 diffeomorphisms of neither the real line nor the circle. Several open questions will be addresed.

I'll explain joint work with Greg Arone that decomposes the Goodwillie tower of a functor from spaces to spaces. We construct an approximation to the tower built from a bimodule structure on the derivatives of the functor, and show that the fibre of the map from the real tower to the approximation can be described in terms of Tate spectra. In particular, in cases where the Tate spectra vanish, such as rationally, we obtain models for the Goodwillie tower explicitly in terms of this bimodule. I'll also mention our plans to apply a result of Nick Kuhn on vanishing Tate spectra to this setting.

A new statistical methodology is developed for analysis of spontaneous adverse event reports from post-marketing drug surveillance data. The method involves both empirical Bayes and fully-Bayes estimation of rate multipliers for each drug within a class of drugs, for a particular adverse event, based on a mixed-effects Poisson regression model. Both parametric and semi-parametric models for the random effect distribution are examined. The method is applied to data from FDA`s Adverse Event Reporting System (AERS) on the relationship between antidepressants and suicide. We obtain point estimates and 95% confidence intervals for the rate multiplier for each drug (e.g., antidepressants), which can be used to determine if a particular drug has an increased risk of association with a particular adverse event (e.g., suicide). Confidence intervals that do not include 1.0 provide evidence for either significant protective or harmful associations of the drug and the adverse effect. We also examine empirical Bayes, parametric Bayes and semi-parametric Bayes estimators of the rate multipliers and associated confidence intervals. Results of our analysis of the FDA AERS data revealed that newer antidepressants are associated with lower rates of suicide. This finding contradicts previous findings of FDA that newer antidepressants are causally related to increased suicidal thinking in children and young adults. Finally, we suggest changes in the AERS system to improve our ability to discover these adverse events.

The majority of imaging technologies are sensitive to the linear properties of the imaged medium, for example Computed Tomography is sensitive to X-ray absorption and seismic imaging is sensitive to changes in wave velocity. When differentiating between healthy and cancerous tissue in medical imaging or oil and water in Earth imaging, there is growing evidence that contrasts in nonlinear material properties can also provide important information. A particular example is Ultrasound Vibro-Acoustography, in which the nonlinear interaction of ultrasound waves at multiple frequencies is used to generate images sensitive to both linear and nonlinear material parameters. Creating and testing a mathematical model of this experiment presents many computational challenges, arising from the range of scales and the intrinsic nonlinearity of the problem. To address these challenges, we have developed a set of integral equation methods specifically tailored to this problem, allowing the rapid, accurate, simulation of the experiment.

This talk will discuss the following paper: arXiv:0710.4300 Odd Khovanov homology by Peter Ozsvath, Jacob Rasmussen, Zoltan Szabo.

For families of intermediate Jacobians over a curve, there are two constructions of an analytic Neron model: one introduced by Green-Griffiths-Kerr, using admissible normal functions (ANF); the other introduced by Kato-Nakayama-Usui, using log mixed Hodge theory. In this talk, we will talk about the two constructions, and state our main result: the existence of a map between these Neron models.

A classification of the dynamics of polynomials in one complex variable has remained elusive, even when considering only the simpler "structurally stable" polynomials. In this talk, I will describe the basics of polynomial iteration, leading up to recent results in the direction of a complete classification. In particular, I will describe a (singular) metric on the complex plane induced by the iteration of a polynomial. I will explain how this geometric structure relates to topological conjugacy classes within the moduli space of polynomials.

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.