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<title>Cohomology in algebra, geometry, physics and statistics</title>
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Wednesday, 20 March 2024 - 13:30 to 14:30 <br />
Place: ZOOM meeting
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Speaker: K V Subrahmanyam, Chennai Mathematical Institute<br />
Title: Stabilizer limits and Orbit closures with applications to Geometric Complexity Theory
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Abstract <br />
<p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">Let $G\subseteq GL(X)$ be a reductive group acting on a finite dimensional vector space $V$ over $\C$. A central problem in Geometric Complexity Theory is the study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$.</span></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$ for which $z$ is observed as a limit of $y$.</span></p><p><br></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">Using $\lambda$, we develop a leading term analysis which applies to $V$ as well as to ${\mathcal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively.</span></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. Here $\hat{\cal K} is the leading term Lie algebra obtained from ${\cal K}$ by the adjoint action of $\lambda(t)$. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$.</span></p><p><br></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">We examine the case when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$. We call this a {\em alignment}. We describe some consequences of alignment and relate it to existing work on lower bounds in the case of the determinant and permanent.</span></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">We also connect alignment to {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.</span></p><p><br></p><p><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);" class="ql-font-monospace">This is joint work with Bharat Adsul and Milind Sohoni.</span></p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);">-----------------------------------------------------------------------------------------------------------------------</span></p><p class="ql-align-justify"><span style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);" class="ql-font-monospace">We shall open ZOOM at 13.15 for virtual coffee</span></p><p class="ql-align-justify"><span style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);" class="ql-font-monospace">Join Zoom Meeting</span></p><p class="ql-align-justify"><a href="https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09" rel="noopener noreferrer" target="_blank" style="background-color: rgb(255, 255, 255); color: rgb(0, 0, 204);" class="ql-font-monospace">https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09</a></p><p class="ql-align-justify"><span style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);" class="ql-font-monospace">MeetingID:99598413922</span></p><p class="ql-align-justify"><span style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);" class="ql-font-monospace">Passcode:Galois</span></p><p><br></p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/16
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Cohomology in algebra, geometry, physics and statistics mailing list <br />
phk_seminar@math.cas.cz <br />
https://list.math.cas.cz/listinfo/phk_seminar@math.cas.cz
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