Study claims that human brain does not produce new brain cells?
The hippocampus does make new brain cells, especially after aerobic exercise; there is an overwhelming amount of evidence for this claim (Erickson, Voss, et al., 2011; Firth, Stubbs, et al., 2017; Patten, Sickmann, et al., 2013). I was unable to find the California study published or peer-reviewed anywhere, and the abstract presented at the conference contained no indication of the sample size used in the study or a justification for why those specific types of neurons were studied. Therefore, it is unclear whether this single study was correctly conducted, and the article discussing the abstract is appropriately cautious about interpreting the results.

There's many different reasons why this study might contradict the scientific literature on hippocampus. One might have to do with a neurogenesis marker they used, doublecortin (DCX). One study notes that DCX levels may be dependent on specific brain regions and can be affected by age, although this is uncertain (Balthazart and Ball, 2014). Another study also warns that DCX reading levels should be interpreted with caution because they can exist at relatively high levels outside of neurogenesis (Kremer, Jagasia, et al., 2013).