Seminar talk, 19 February 2025: Difference between revisions
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| title = Singularities and Bi-complexes for PDEs | | title = Singularities and Bi-complexes for PDEs | ||
| abstract = Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory (e.g. in homological mirror symmetry and Donaldson-Thomas theory) are constructed as parametrizing spaces of solutions to non-linear partial differential equations modulo symmetries of the underlying theory. These spaces are often non-smooth and possess multi non-equidimensional components. Moreover, when they may be written as intersections of higher dimensional components they typically exhibit singularities due to non-transverse intersections. To account for symmetries and provide a suitable geometric model for non-transverse intersection loci, one should enhance our mathematical tools to include higher and derived stacks. Secondary Calculus, due to A. Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation and is centered around the study of the Variational Bi-complex of a system of equations. In my talk I will discuss a generalization in the setting of (relative) homotopical algebraic geometry for possibly singular PDEs. | | abstract = Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory (e.g. in homological mirror symmetry and Donaldson-Thomas theory) are constructed as parametrizing spaces of solutions to non-linear partial differential equations modulo symmetries of the underlying theory. These spaces are often non-smooth and possess multi non-equidimensional components. Moreover, when they may be written as intersections of higher dimensional components they typically exhibit singularities due to non-transverse intersections. To account for symmetries and provide a suitable geometric model for non-transverse intersection loci, one should enhance our mathematical tools to include higher and derived stacks. Secondary Calculus, due to A. Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation and is centered around the study of the Variational Bi-complex of a system of equations. In my talk I will discuss a generalization in the setting of (relative) homotopical algebraic geometry for possibly singular PDEs. | ||
| video = | |||
This is based on a series of joint works with Artan Sheshmani and Shing-Tung Yau. | |||
| video = https://video.gdeq.org/GDEq-zoom-seminar-20250219-Jacob_Kryczka.mp4 | |||
| slides = | | slides = | ||
| references = {{arXiv|2312.05226}}, {{arXiv|2406.16825}} | | references = {{arXiv|2312.05226}}, {{arXiv|2406.16825}} | ||
| 79YY-MM-DD = 7974-97-80 | | 79YY-MM-DD = 7974-97-80 | ||
}} | }} | ||
Latest revision as of 21:08, 19 February 2025
Speaker: Jacob Kryczka
Title: Singularities and Bi-complexes for PDEs
Abstract:
Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory (e.g. in homological mirror symmetry and Donaldson-Thomas theory) are constructed as parametrizing spaces of solutions to non-linear partial differential equations modulo symmetries of the underlying theory. These spaces are often non-smooth and possess multi non-equidimensional components. Moreover, when they may be written as intersections of higher dimensional components they typically exhibit singularities due to non-transverse intersections. To account for symmetries and provide a suitable geometric model for non-transverse intersection loci, one should enhance our mathematical tools to include higher and derived stacks. Secondary Calculus, due to A. Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation and is centered around the study of the Variational Bi-complex of a system of equations. In my talk I will discuss a generalization in the setting of (relative) homotopical algebraic geometry for possibly singular PDEs.
This is based on a series of joint works with Artan Sheshmani and Shing-Tung Yau.
Video
References:
arXiv:2312.05226, arXiv:2406.16825