In this talk we will discuss the notion of a quiver and its applications to algebraic geometry. In particular we will construct moduli spaces of quiver representations and stability conditions on quivers. We will show how to construct some simple GIT quotients from this data. Along the way Gabriel's theorem and Mumford's criterion will be covered.
I will overview the analytic classification of smooth, simple, 3-fold flops. There are three main aspects: (1) the theoretical advances that reduce the problem to classification of a certain class of noncommutative finite dimensional algebras, (2) a complete understanding of those algebras, then lastly (3) building the associated geometry for each algebra in that class. In the process of proving these results, we also obtain various bonus (and very surprising) geometric corollaries, including to curve-counting invariants, and also to 3-fold crepant divisor-to-curve contractions. Part (1) is joint with Joe Karmazyn and Emma Lepri, the remainder is joint with Gavin Brown.
TBA
Modern cosmological surveys, with their unprecedented precision, have established weak lensing as a powerful probe of the matter distribution in the universe, enabling us to test the nature of dark matter (DM) interactions. The talk will briefly introduce this topic and present our recent investigation of weak lensing constraints on DM interactions with baryons and neutrinos. We utilize cosmological N-body simulations to model the nonlinear evolution of structure formation on weak lensing scales (k ~ 0.1–1 h/Mpc). To efficiently explore the parameter space of cosmological models, we have developed a novel numerical method that facilitates the use of simulation results. Our analysis of DES Year 3 data reveals significant constraints on DM-proton scattering, improving upon previous constraints from cosmic microwave background (CMB) data by up to a factor of five. Furthermore, our analysis of cosmic shear data reveals an intriguing ~3σ hint of a non-vanishing DM-neutrino interaction, which strengthens similar previous findings in high-multipole CMB and Lyman-α forest data. We also discuss how this interaction could simultaneously ameliorate the persisting S8 tension.