Will glycerol and citric acid react? If yes, under what conditions?
As @Mithoron said, it will form esters. Actual name is Citric acid esters of tri-glycerides or citroglycerides.


  
  
  (source)
  
  Where at least one of R1, R2 or R3 represents a citric acid moiety,
  one represents a fatty acid moiety and the remainder may represent
  citric acid, fatty acid or hydrogen.
  
  Obtained by esterification of glycerol with citric acid [...] acid;
  consists of mixed esters of citric acid and edible fatty acids with
  glycerol; may contain minor parts of free fatty acids, free glycerol,
  free citric acid and mono- and diglycerides; may be wholly or
  partially neutralized with sodium hydroxide or potassium hydroxide.
  
  They are white to ivory coloured, oily to waxy material. They are used
  as stabilizer, emulsifier, dough conditioner, antioxidant synergist.


According to this site:


  The influence of various heating methods without catalysis to prepare
  polyesters from citric acid : glycerol blends were studied. In the
  presence of short-term microwave treatments, i.e., 60 s at 1200 W,
  blends of glycerol and citric acid invariably formed solid amorphous polyesters. Fourier transform infrared spectroscopy showed that
  citric acid and glycerol blends can form highly stable polymers
  composed of ester bonds. The glycerol citrate polyester polymers
  exhibited the least degradation in water, more in acid solutions
  (0.1–1.0M HCl), and the most deterioration in strong alkaline
  solutions (0.1–1.0M NaOH) after 72 h soakings. Polyesters of glycerol
  and citric acid were studied with differential scanning calorimetry
  and thermal gravimetric analysis. The polyesters were found to be
  thermally stable (up to 313°C).


A noble application of glycerol + citric acid esters or citroglycerides is in the making of a biodegradable thermoset polymers.

1. Source 1


  A new biodegradable thermoset polymer, was made from simple,
  economical, biocompatable monomers without the need for a catalyst.
  Glycerol and citric acid, non-toxic and renewable reagents, were
  crosslinked by a melt polymerization reaction at temperatures from
  90-150°C. Consistent with a condensation reaction, water was
  determined to be the primary byproduct. The amount of crosslinking was
  controlled by the reaction conditions, including temperature, reaction
  time, and ratio between glycerol and citric acid. Also, the amount of
  crosslinking was inversely proportional to the rate of degradation. [...]


2. Source 2


  The notable increase in global consumption of plastics and their long
  residence time in the atmosphere show the great need for biodegradable
  products. In this project, we developed biodegradable polymers based
  on citric acid and glycerol. Their synthesis was carried out at
  different conditions of constant temperature and concentration and
  they were synthesized using an economically viable method.
  Characterization was undertaken using: Acid number, FTIR spectroscopy,
  refractive index, viscosity, impact analysis, tensile test, hardness,
  calorimetry, the % humidity (oven method with recirculating air), and
  density determination, along with qualitative evidence to corroborate
  their biodegradability. These elaborated polymers were mixed with
  medical grade PVC formulation, obtaining a hybrid polymer, showing
  their mechanical properties were changed.