on an n-dimensional complete manifold m, consider an h-almost gradient ricci soliton, which is a generalization of gradient ricci solitons and $(\lambda , n + m)$-einstein manifolds. in this talk, we show that if the manifold is bach-flat and $dh/du > 0$, then the manifold m is either einstein or rigid. in particular, such a manifold has harmonic weyl curvature. when the dimension of m is four, the metric is locally conformally flat.

optimal power flow (opf) problems are non-convex and large-scale optimization problems which appear in operation analysis, scheduling, and energy management of power systems. various algorithms have been developed to solve the opf problems, while in many cases, only local optimal solutions are available. recent studies show that semidefinite programming (sdp) can either provide an exact or near global optima for the opf problems. in this regard, there are enormous potential for sdp in solving the opf problems. however, sdp-based approaches are far from real-world implementations. this talk will cover our recent results that partially address the limitations of sdp-based approaches for the opf, including scalability, incorporation of binary variables, and distributed formulation.

motivated by local-global compatibility in the $p$-adic langlands program, emerton and helm (and others) studied how the local langlands correspondence for $gl(n)$ can be interpolated in zariski families. in this talk i will report on joint work with c. johansson and j. newton on the interpolation in rigid families. we take our rigid space to be an eigenvariety $y$ for some definite unitary group $u(n)$ which parametrizes hecke eigensystems appearing in certain spaces of $p$-adic modular forms. the space $y$ comes endowed with a natural coherent sheaf $\mathcal{m}$. our main result is that the dual fibers $\mathcal{m}_y'$ essentially interpolate the local langlands correspondence at all points $y \in y$. this make use of certain bernstein center elements which appear in scholze's proof of the local langlands correspondence (and also in work of chenevier). in the pre-talk i will talk about the local langlands correspondence, primarily for $gl(2)$.

the class number formula is a precise relationship between two
cornerstones of number theory -- the zeta function of a number field and
its class group. a mysterious transcendental quantity, the regulator,
serves as a bridge that links these objects. a central open question in
number theory is beilinson's vast conjectural generalization of this
formula.

after explaining and motivating beilinson's conjecture, we state some
new cases obtained jointly with aaron pollack. the key behind them is
to find a setting where one can simultaneously realize beilinson's
regulator pairing in representation theory and algebraic geometry. we
mention some diverse ingredients involved in the proof, such as the
relative fourier-mukai transform and explicit solutions of systems of
vector-valued pdes, as well as some next steps.

calabi-yau varieties and fano varieties are building blocks of varieties in the sense of birational geometry. birkar recently proved that fano varieties with bounded singularities belong to just finitely many algebraic families. one can then ask if an analogous result holds for calabi-yau varieties. if one only considers rationally connected calabi-yau varieties with klt singularities - those calabi-yau varieties behaving most like fano - shokurov actually conjectured that also these varieties should be bounded in any fixed dimension. we show that rationally connected klt calabi-yau 3-folds form a birationally bounded family. in many cases, we can actually give more precise statements and we are able to relate the boundedness problem to the study of a quite mysterious birational invariant: the minimal log discrepancy. this is a joint work in progress with w. chen, g. di cerbo, j. han, and c. jiang.