Monday, October 12, 2020
Time  Items 

All day 

2pm 
10/12/2020  2:30pm Abstract: Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural. We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial differential equations. We discuss the theoretical motivation of this method and show its benefits in practice on a number of PDEs. We also discuss the limitations of the methodology and present ideas for improvement and future works. Location:
https://yale.zoom.us/j/98739390999

4pm 
10/12/2020  4:00pm This is joint work with Ian Alevy and Ren Yi. We construct minimal “domain exchange maps “ Location:
Zoom
10/12/2020  4:30pm We associate to a projective $n$dimensional toric variety $X_{\Delta}$ a pair of cocommutative (but generally noncommutative) Hopf algebras $H^{\alpha}_X, H^{T}_X$. These arise as Hall algebras of certain categories $\Coh^{\alpha}(X), \Coh^T(X)$ of coherent sheaves on $X_{\Delta}$ viewed as a monoid scheme  i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When $X$ is smooth, the category $\Coh^T(X)$ has an explicit combinatorial description as sheaves whose restriction to each $\mathbb{A}^n$ corresponding to a maximal cone is determined by an $n$dimensional generalized skew shape. The (nonadditive) categories $\Coh^{\alpha}(X), \Coh^T(X)$ are treated via the formalism of protoexact/protoabelian categories developed by DyckerhoffKapranov.
The Hall algebras $H^{\alpha}_X, H^{T}_X$ are graded and connected, and so enveloping algebras $H^{\alpha}_X \simeq U(\n^{\alpha}_X)$, $H^{T}_X \simeq U(\n^{T}_X)$, where the Lie algebras $\n^{\alpha}_X, \n^{T}_X$ are spanned by the indecomposable coherent sheaves in their respective categories.
We explicitly work out several examples, and in some cases are able to relate $\n^T_X$ to known Lie algebras. In particular, when $X = \mathbb{P}^1$, $\n^T_X$ is isomorphic to a nonstandard Borel in $\mathfrak{gl}_2 [t,t^{1}]$. When $X$ is the second infinitesimal neighborhood of the origin inside $\mathbb{A}^2$, $n^T_X$ is isomorphic to a subalgebra of $\mathfrak{gl}_2[t]$. We also consider the case $X=\mathbb{P}^2$, where we give a basis for $\n^T_X$ by describing all indecomposable sheaves in $\Coh^T(X)$.
This is joint work with Jaiung Jun.
Location:
https://yale.zoom.us/j/99433355937 (password was emailed by Ivan on 9/11, also available from Ivan by email)
