L4

Transmission problems model phenomena in domains composed of several adjacent phases. While a variational ''divergence-form'' theory is by now classical, a non-variational framework has only emerged more recently. This talk concerns the regularity of viscosity solutions to such transmission problems in non-divergence form.

I will present new regularity results in the case of flat interfaces, where the transmission condition may depend on both the normal and tangential derivatives of the solution. This condition can be viewed as a nonlinear coupling of oblique derivatives across the interface. Our main result establishes optimal piecewise $C^{1,\alpha}$ regularity for viscosity solutions.

As a scheme undergoes a simple normal crossing degeneration its coherent sheaves - and their derived category - behave unpredictably. In this talk I will describe a proposal for a category of logarithmic (quasi-)coherent sheaves associated to such a degeneration. The corresponding derived category exhibits Grothendieck-Neeman duality with the expected dualizing complex, and has a perfect diagonal (the categorical signature of smoothness).  Structures on the logarithmic derived category of the central fibre most closely resemble the smooth situation when considered relative a tropical derived category. This category is compatible with Maulik and Ranganathan’s logarithmic Donaldson—Thomas theory and has an associated (logarithmic) moduli stack of logarithmic coherent sheaves.

 

Based on joint work with De Deyn, Dell, Hu, Manali-Rahul, and Schimpf, and on separate joint work with Poiret, and Song for the moduli stack.

In algebraic geometry, two useful ways of classifying cycles are up to rational equivalence, or up to a weaker equivalence relation known as algebraic equivalence. Using the dictionary of mirror symmetry, these correspond to classifications of Lagrangians in the mirror symplectic manifold: Sheridan—Smith showed that rational equivalence is related to a version of Lagrangian cobordism, and in previous work I introduced an equivalence relation called alg-cobordism as a Lagrangian counterpart to algebraic equivalence.

In the early 90’s, Voevodsky and Voisin independently proved the following surprising Theorem: if an algebraic cycle Z in an algebraic variety Y is algebraically trivial, then for k sufficiently large, Z^k is rationally trivial in Y^k. I will present joint work with Danil Kozevnikov, in which we translate their Theorem into the symplectic realm: if a Lagrangian L is alg-nullcobordant, then for k sufficiently large, L^k is null-cobordant.