C5

TBA

I will present my recent results on global regularity for a canonical shell model of the Navier-Stokes equations, covering the unforced equation and certain classes of forcing. I will discuss the distinction between the forced and unforced regularity problems in this model, with the corresponding Navier-Stokes questions as motivation.

We consider a cross diffusion system of two populations, often called the Busenberg-Travis system. The two species are transported by the same pressure gradient with Darcy’s law, modeling overcrowding effect (populations tend to move away from regions of high pressure). However, their mobility is different: the first species moves with mobility 1, whereas the second moves with mobility \nu. The difficulty to prove existence is to prove strong compactness of each densities, which we achieve with a variant of the div-curl lemma applied to evolution PDEs.

Compactifying topological actions using only de Rham forms fails to capture torsion sectors encoded in integral cohomology. Differential cohomology remedies this by combining integral characteristic classes, differential-form curvatures, and holonomy data into a single framework. In the context of deriving SymTFTs from M-theory, such a refinement is crucial for capturing background gauge fields for discrete 1-form global symmetries in the physical theory. In this talk, we will review the construction of differential cohomology and, time permitting, show how a refined Kaluza-Klein compactification leads to background gauge fields that encode these higher-form symmetries.

In this seminar, we consider an isoperimetric problem for planar tilings with possibly unequal repeating cells. We present general existence and regularity results, and we study the classification of planar isoperimetric double tilings, namely tilings with two repeating cells of minimal perimeter. In this case, we explicitly determine the associated energy profile and provide a complete description of the phase transitions. We also comment on possible extensions and discuss some open problems. This is based on joint work with M. Novaga and E. Paolini.

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Any two 1 by 1 real matrices commute.  This is in general not the case for 2 by 2 real matrices.  However, if A, B, C, and D are any 2 by 2 real matrices, then ABCD - ABDC - ACBD + ACDB + ADBC - ADCB - BACD + BADC + BCAD - BCDA - BDAC + BDCA + CABD - CADB - CBAD + CBDA + CDAB - CDBA - DABC + DACB + DBAC - DBCA - DCAB + DCBA = 0.  This identity is the first instance of a general result of Amitsur and Levitski; I will explain a simple graph-theoretic proof due to Swan.