Per Investopedia, Portfolio Management -
The art and science of making decisions about investment mix and policy, matching investments to objectives, asset allocation for individuals and institutions, and balancing risk against performance. Portfolio management is activity that could be done through a financial advisor or an individual investor.
Portfolio management is all about strengths, weaknesses, opportunities and threats in the choice of debt vs. equity, domestic vs. international, growth vs. safety, and many other tradeoffs encountered in the attempt to maximize return at a given appetite for risk.
Per Wikipedia, Private Equity:
Private equity is an asset class consisting of equity securities and debt in operating companies that are not publicly traded on a stock exchange.
Equity research is the compilation of information about a particular common or preferred stock. It can involve the review of share issued, common financial ratios, dividend history, past stock price performance, insider trading, institutional ownership, industry comparisons, debt structure, professional analysts' projections............
Equity capital markets are where equity stock issues are traded. Trading involves matching buyers with sellers and vice versa. This is historically rooted in auction environment where a market maker for any given equity kept a "book" of orders and did the matching between buyers and sellers.