Correlation coefficient and non linear association
Yes, Dr. Sullivan is right. For example, take a perfect quadratic relationship between X and Y. 

Here is some Python code to show nine sample points and calculate their Pearson correlation coefficient. You can skip the code and just look at the results below if you trust me.

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn')
plt.rcParams["figure.figsize"] = [5, 8]

x = np.arange(-4, 5)
y = x**2  # This is x squared. 

plt.scatter(x, y)

plt.axis('equal')
plt.title('$X = Y^2$', fontsize=20)
plt.xlabel('X', fontsize=14)
plt.ylabel('Y', fontsize=14, rotation=0)
plt.xticks([-4, -2, 0, 2, 4])
plt.yticks([0, 1, 4, 9, 16])

# Calculate the Pearson correlation coefficient:
print('r =', np.corrcoef(x, y)[0, 1])


r = 0.0



Surely there is a relation between X and Y in this example, and yet the Pearson correlation coefficient is zero.