Events
Department of Mathematics and Statistics
Texas Tech University
We give a gentle introduction to supermanifolds and the splitting problem. Supermanifolds are a mildly noncommutative geometry where the coordinate functions either commute or anticommute. We recount Batchelor's theorem: every supermanifold in the smooth category is noncanonically isomorphic to a vector bundle. Such an isomorphism is called a splitting. Splittings may not exist at all in the holomorphic category, which turns out to be deeply significant to string theory in a way we will sketch. We close with basic examples of splittings and nonsplittings.For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^n, n = 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter. This property distinguishes hyperbolic spaces from Euclidean and spherical ones, where the quantity is bounded below by $3 \pi^2$. We finish by talking about horoconvex domains.
Watch online via this Zoom link.The regulation and interpretation of transcription factor levels is critical in spatiotemporal regulation of gene expression in development biology. However, concentration-dependent transcriptional regulation, and the spatial regulation of transcription factor levels are poorly studied in plants. WUSCHEL, a stem cell-promoting homeodomain transcription factor was found to activate and repress transcription at lower and higher levels respectively. The differential accumulation of WUSCHEL in adjacent cells is critical for spatial regulation on the level of CLAVATA3, a negative regulator of WUSCHEL transcription, to establish the overall gradient. However, the roles of extrinsic spatial cues in maintaining differential accumulation of WUSCHEL are not well understood. We have developed a 3D cell-based computational model which integrates sub-cellular partition with cellular concentration across the spatial domain to analyze the regulation of WUS. By using this model, we investigate the machinery of the maintenance of WUS gradient within the tissue. We also developed a stochastic model to study the binding and unbinding of WUS to cis-elements regulating CLV3 expression to understand the concentration dependent manner mechanistically. The robustness mechanism and the concentration-dependent machinery discovered by the modeling analysis can be general principles for stem cell homeostasis in different biological systems.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 20th at 4 PM -- meeting ID: 937 2431 1192
Differential modules are modules equipped with the additional data of
a square-zero endomorphism. As such, they form a natural
generalization of complexes and have recently seen a more focused
study for the novel perspective they provide on older problems in
commutative algebra, algebraic geometry, and representation theory. In
full generality, the theory of differential modules (and their
resolutions) can stray wildly from the classical case, but recent work
of Brown and Erman has shown that the classical theory of minimal free
resolutions still plays an important role in understanding the
homological properties of differential modules. In this talk, we will
explore differential modules whose homology is generated by a regular
sequence, comparing and contrasting the classical theory of minimal
free resolutions of complete intersections (which is quite simple)
with the analogous generalization to differential modules (which turns
out to be much more subtle). This work is joint with Maya Banks.