# Faculty Activities

*Abstract:*

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

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By Quantum Matrix algebras one usually means the algebras defined via braidings,i.e. solutions to the Quantum Yang-Baxter equation. I plan to discuss the problemof classification of braidings. Also, I plan to introduce some Quantum Matrixalgebras and exhibit their properties. In particular, I plan to definequantum analogs of basic symmetric polynomials (elementary, full, Schur...)and to present a quantum version of the Cayley-Hamilton identity.The talk is supposed to be introductory.

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Under the assumption of the GRH(Generalized Riemann Hypothesis), we show that there is a real quadratic field K such that the étale fundamental group of the spectrum of the ring of integers of K is isomorphic to A5. To the best of the author's knowledge, this is the first example of a nonabelian simple étale fundamental group in the literature under the assumption of the GRH. (The talk will be basic and tha above notions will be defined).

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I intend to sketch well-known facts about ellipsoids, viewed as a particular case of symmetric convex sets, giving some background on the latter. The ambient spaces will be (finite or infinite dimensional) real linear spaces (some notions not depending on specifying a topology there).

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In tame geometry, a cell (or cylinder) is defined as follows. A onedimensional cell is an interval; a two-dimensional cell is the domainbounded between the graphs of two functions on a one-dimensional cell;and so on. Cellular decomposition (covering or subdiving a set intocells) and preparation theorems (decomposing the domain of a functioninto cells where the function has a simple form) are two of the keytechnical tools in semialgebraic, subanalytic and o-minimal geometry.

Cells are normally seen as intrinsically real objects, defined interms of the order relation on $\mathbb R$. We (joint with Novikov)introduce the notion of \emph{complex cells}, a complexification ofreal cells where real intervals are replaced by complexannuli. Complex cells are naturally endowed with a notion of analyticextension to a neighborhood, called $\delta$-extension. It turns outthat complex cells carry a rich hyperbolic-geometric structure, andthe geometry of a complex cell embedded in its $\delta$-extensionoffers powerful new tools from geometric function theory that areinaccessible in the real setting. Using these tools we show that thereal cells of the subanalytic cellular decomposition and preparationtheorems can be analytically continued to complex cells.

Complex cells are closely related to uniformization and resolution ofsingularities constructions in local complex analytic geometry. Inparticular we will see that using complex cells, these constructionscan be carried out uniformly over families (which is impossible in theclassical setting). If time permits I will also discuss how thisrelates to the Yomdin-Gromov theorem on $C^k$-smooth resolutions andsome modern variations.

*Abstract:*

(This is is the second of two lectures on this subject)

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

Dirichlet's Theorem states that for a real mxn matrix A, ||Aq+p||^m ≤ t, ||q||^n < t has nontrivial integer solutions for all t > 1. Davenport and Schmidt have observed that if 1/t is replaced with c/t, c<1, almost no A has the property that there exist solutions for all sufficiently large t. Replacing c/t with an arbitrary function, it's natural to ask when precisely does the set of such A drop to a null set. In the case m=1=n, we give an answer using dynamics of continued fractions. We then discuss an approach to higher dimensions based on dynamics on the space of lattices. Where this approach meets an obstruction, a similar approach to the analogous inhomogeneous approximation problem will succeed. Joint work with Dmitry Kleinbock.

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We will survey recent developments in the symplectictopology that lead to various notions of distance on the category ofLagrangian submanifolds of a symplectic manifold. We will explain boththe algebraic as well as geometric sides of the story and outline someapplications.

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The Landau-de Gennes model is a widely used continuum description of nematic liquid crystals, in which liquid crystal configurations are described by fields taking values in the space of real, symmetric traceless $3\times 3$ matrices (called $Q$-tensors in this context). The model is an extension of the simpler $S^2$- or $RP^2$-valued Oseen-Frank theory, and provides a relaxation of an ${\mathbb R}P^2-$, $S^2-$ or $S^3$-valued harmonic map problem on two- and three-dimensional domains. There are similarities as well as differences with the $\mathbb{C}$-valued Ginzburg-Landau model.There is current interest in understanding the structure and disposition of defects in the Landau-de Gennes model. After introducing and motivating the model, I will discuss some recent and current work on defects in two-dimensional domains, in the harmonic-map limit as well as perturbations therefrom This is joint work with G di Fratta, V Slastikov and A Zarnescu.

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