Upcoming Seminars
Title: Can computational math help settle down Morrey's and Iwaniec's conjectures? |
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Seminar: Analysis and Differential Geometry |
Speaker: Wilfrid Gangbo, PhD of UCLA |
Contact: Dr. Levon Nurbekyan, lnurbek@emory.edu |
Date: 2025-02-14 at 11:00AM |
Venue: MSC W303 |
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Abstract: In 1987, D. L. Burkholder proposed a very simple looking and explicit energy functionals $I_p$ defined on $\mathbb{S}$, the set of smooth functions on the complex plane. A question of great interest is to know whether or not $\sup_{\mathbb{S}} I_p \geq 0$. Since the function $I_p$ is homogeneous of degree $p$, it is very surprising that it remains a challenge to prove or disprove that $\sup_{\mathcal{S}} I_p \geq 0$. Would $\sup_{\mathbb{S}} I_p \geq 0$, the so-called Iwaniec's conjecture on the Beurling--Ahlfors Transform in harmonic analysis would hold. Would $\sup_{\mathcal{S}} I_p > 0$, the so-called Morrey's conjecture in elasticity theory would hold. Therefore, proving or disproving that $\sup_{\mathbb{S}} I_p \geq 0$ is equally important. Since the computational capacity of computers has increased exponentially over the past decades, it is natural to hope that computational mathematics could help settle these two conjectures at once. |
Title: Independent transversals in multipartite graphs |
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Seminar: Discrete Math |
Speaker: Yi Zhao, PhD of Georgia State University |
Contact: Dr. Cosmin Pohoata, cosmin.pohoata@emory.edu |
Date: 2025-02-19 at 4:00PM |
Venue: MSC E408 |
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Abstract: An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that given any positive even integer r, every r-partite graph with parts of size n and maximum degree r n / (2r-2) - t (t>0) contains c t n^{r-1}) independent transversals. This is best possible up to the constant c=c_r, confirming a conjecture of Haxell and Szabo from 2006 and partially answering a question Erdos from 1972 and a question of Bollobas, Erdos and Szemeredi from 1975. We also show that for every s\ge 2, even r\ge 2 and sufficiently large n, every r-partite graph with parts of size n and maximum degree \Delta |