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Everyone knows that aftershocks follow large earthquakes. But why? Embarrassingly, the mechanism by which earthquakes trigger other earthquakes is still an unresolved question in seismology. We have evidence that the ground shaking from seismic waves plays a key role in this process. 
For distant earthquakes, seismic waves clearly generate local seismicity (Brodsky et al., 2000; Brodsky and Prejean, 2005; Brodsky et al. 2006; Harrington and Brodsky, 2006; Miyazawa and Brodsky, 2008). There is some indication from the spatial distribution of aftershocks that the same process occurs near mainshocks (Felzer and Brodsky, 2006). Using a large dataset of distant and local triggering, we can show that triggering at all distances can be well predicted based on the amplitude of the seismic waves from previous earthquakes (Van der Elst and Brodsky, submitted). Solving the mystery of earthquake triggering provides constraints on the conditions for generally initiating earthquakes.
Seismic waves also have other effects on distant systems. Large permeability increases sometimes occur when the ground shakes (Brodsky et al., 2003). The larger the shaking, the larger the permeability enhancement (Elkhoury et al., 2006). These observations suggest that increased flow in the hydrological systems of faults may play a role in earthquake generation.
As a fault slips, a complex series of processes occur. 
The rocks around the fault fracture (Savage and Brodsky, submitted). Rocks on the fault are ground to a powder. Shear is focussed into narrow slip surfaces and the shape of these surfaces is controlled by the undulations of the powder layer. Those undulations in turn accumulate stress that controls the location of future slip. Each of these parts of an earthquake leave some imprint on the geological record. We are trying to unravel the feedbacks involved in slip on natural faults by measuring the geometry and internal architecture of exposed fault zones (Sagy et al., 2007; Sagy and Brodsky, 2009; Brodsky et al., 2009).
Earthquakes occur when tectonic stresses overcome friction. Tectonic stresses are reasonably well-understood, but friction is not. The resisting stress of complex, multiphase faults moving at the high slip rates of an earthquakeis the largest single unknown in physical models of earthquake rupture. My group is trying to constrain the processes and values of friction through a combination of laboratory and observational approaches. In the lab, we study the rheology of granular flows as analogs to fault gouge (Lu et al., 2007). In the field, we are involved in efforts to quickly measure the heat dissipated by friction immediately after a major earthquake (www.pmc.ucsc.edu/~rapid/).
Volcanoes produce distinct seismic signatures. Some of these can be understood as ordinary earthquakes travelling through extraordinarily messy rocks (Harrington and Brodsky, 2007). Other signals are related to the actual eruptive process (Brodsky et al., 1999). The waves from eruptions can be used to infer the dynamics of the eruption. In collaboration with Stephanie Prejean of the US Geological Survey, I have found that the amplitudes of seismic waves from recent Alaskan eruptions are surprisingly well-predicted by thermal buoyancy models of volcanic plumes combined with observations of plume heights. This work points the way for using seismology as a remote sensing technology for issuing aircraft alerts during explosive eruptions.