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for events the day of Friday, April 22, 2016.

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Friday, April 22, 2016

2:00 pm in 445 Altgeld Hall,Friday, April 22, 2016

Weyl’s Law and the Heat Kernel on a Compact Manifold

Hadrian Quan (UIUC Math)

Abstract: For a compact oriented n-dimensional manifold $M$, the Laplace-Beltrami operator $\Delta$ acting on $L^2(M)$ has discrete spectrum, with eigenvalues limiting to infinity. One may hope for a more refined growth estimate, by asking how many eigenvalues of $\Delta$ lie below a fixed value of $\lambda$. In 1911, Hermann Weyl proved an asymptotic statement of the form $$ \#(\lambda) \sim \frac{\text{Vol}(M)}{(4\pi)^{n/2}\Gamma((n/2)+1)} \lambda^{n/2} $$ as $\lambda\rightarrow \infty$, where $\#(\lambda)=\max\{j:\lambda_j\leq \lambda \}$ is the counting function. That eigenvalue growth is determined by the volume of the manifold was one of the early results indicating that spectral invariants are closely tied to geometry. Here we shall present a proof of this statement based on the short time behavior of the heat kernel $e^{-t\Delta}$ of $M$, and a Tauberian theorem of Karamata.

4:00 pm in 241 Altgeld Hall,Friday, April 22, 2016

Noncommutative Topology

Chris Gartland (UIUC Math)

Abstract: We'll review the classical Gelfand-Naimark theory establishing a duality between the category of compact Hausdorff spaces and the category of commutative unital C*-algebras. A noncommutative space can then be rightly viewed as a noncommutative unital C*-algebra. We show how to naturally extend the cohomology functors $K^0$, $H^0$, $H^1$, and $H^2$ from the category of classical spaces to the category of noncommutative spaces.

4:00 pm in 345 Altgeld Hall,Friday, April 22, 2016

Algebraically closed fields with a generic multiplicative character

Chieu Minh Tran (UIUC)

Abstract: We consider the 2-sort structure $(\mathbb{F}_p^a, \mathbb{Q}^a; \chi)$ where $\chi: \mathbb{F}_p^a \to \mathbb{Q}^a$ is injective and multiplicative preserving. I will briefly summarize some earlier results on the model theory of this structure: axiomatization, categoricity, regular model completeness, quantifier reduction, $\omega$-stability. Then I will discuss some attempts to geometrically understand definable sets in connection to proving that the theory has the definability of multiplicity property. (This is a joint work with T. Hakobyan and the talk is a part of my preliminary exam).

4:00 pm in 347 Altgeld Hall,Friday, April 22, 2016

Hook Formulas for Skew Shapes

Alejandro Morales   [email] (UCLA)

Abstract: The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for skew shapes. In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using the excited diagrams of Ikeda-Naruse, Kreiman and Knutson-Miller-Yong. We prove Naruse's formula algebraically and combinatorially. We give two q-analogues of this formula involving semistandard tableaux and reverse plane partitions of skew shape. We end by applying our results to border strips to obtain relations between Euler numbers and Dyck paths. Joint work with Igor Pak and Greta Panova.