Randomized experiments have long been considered to be a gold standard for causal inference. The classical analysis of randomized experiments was developed under simplifying assumptions such as homogeneous treatment effects and no treatment interference leading to theoretical guarantees about the estimators of causal effects. In modern settings where experiments are commonly run on online networks (such as Facebook) or when studying naturally networked phenomena (such as vaccine efficacy) standard randomization schemes do not exhibit the same theoretical properties. To address these issues we develop a randomization scheme that is able to take into account violations of the no interference and no homophily assumptions. Under this scheme, we demonstrate the existence of unbiased estimators with bounded variance. We also provide a simplified and much more computationally tractable randomized design which leads to asymptotically consistent estimators of direct treatment effects under both dense and sparse network regimes.

Title and Abstract: TBA.

Let M be a finitely generated graded module over a polynomial ring of a field. Then we can define a function corresponding to that module which is called "Hilbert Function". Now in my lecture I will tell you how to compute that function for such a module, with examples of course. I will also introduce "Betti numbers" and will show how these things are connected to each other.

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Let I be a complete m-primary ideal in a two dimensional regular local ring R. The Hoskin-Deligne (HD) formula expresses the length of the Artin local ring R/I in terms of the m-adic order of the transforms of I in various local quadratic transforms of R. It uses a structure theorem of Zariski and Abhyankar about two dimensional regular local rings which birationally dominate R. The HD formula implies several fundamental theorems about complete ideals proved by Zariski, Lipman, Rees and Huneke-Sally. We shall prove these in a series of talks.

In the first part of my talk, some preliminaries on microlocal analysis will be discussed. In the second part of the talk, some techniques to study the controllability of a system using microlocal analysis will be explained. As an example of this approach, the lack of null controllability of viscoelastic flows will be discussed.

We establish a Gysin formula for Kempf-Laksov flag bundles and prove a duality formula for Grassmann bundles. We then combine them to study Schubert bundles, their push-forwards and fundamental classes. This is a joint work with Lionel Darondeau.

We will start by giving some broad overview of the general area of inverse spectral theory and the associated area of so-called shape optimization. Then, we will make an attempt to give a proof of the classical Faber-Krahn theorem. MSc students are encouraged to attend. Please note tea and snack will be available in Room 215 at 5:00 PM

Many of the historical motivations for symplectic geometry come from classical mechanics. Conversely, classical mechanics can be treated elegantly with the tools of symplectic geometry. We will present a "dictionary" between classical mechanics and symplectic geometry that is well-known to practitioners in both fields, but is often not explicitly taught in beginning courses in either subject, but left to the mathematical maturity of the student to pick up. The dictionary may be of pedagogic value to students embarking on a study of symplectic geometry, allowing them to ground their understanding of notions like Lagrangian submanifolds in concrete physical situations. Our presentation will be broadly based on notes by John Baez which can be found here: http://www.math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf . We will assume familiarity with basic notions of differential geometry, and some previous exposure to both classical mechanics and symplectic geometry.