Radiotherapy, Carboplatin/Paclitaxel and Nivolumab for High Risk HPV-related Head and Neck Cancer

Inclusion Criteria * Histologically or cytologically proven squamous cell carcinoma of the oropharynx (tonsil, base of tongue, oropharyngeal wall, soft palate) that is p16 positive by immunohistochemistry or HPV positive by in situ hybridization * Clinical stage: stage III AJCC 8th edition staging (cT4 or cN3) OR "matted lymph nodes" (defined as 3 LNs abutting one another with loss of intervening fat plane that is replaced with evidence of extracapsular spread) * Appropriate stage for protocol entry, including no distant metastases, based upon the following minimum diagnostic workup: * History/physical examination, including documentation of weight within 2 weeks prior to registration; * FDG-PET/CT scan for staging and RT plan within 4 weeks prior to registration; Zubrod Performance Status 0-1 within 2 weeks prior to registration; * Age ≥ 18; * CBC/differential obtained within 2 weeks prior to registration on study, with adequate bone marrow function defined as follows: * Absolute neutrophil count (ANC) ≥ 1,000 cells/mm3; Platelets ≥ 75,000 cells/mm3; Hemoglobin ≥ 9.0 g/dL AST/ALT \<3 x ULN * Total Bilirubin \<1.5 x ULN (except subjects with Gilbert Syndrome who must have a total bilirubin level \< 3 x ULN) * Serum creatinine within normal institutional limits or a creatinine clearance ≥ 45 mL/min within 2 weeks prior to registration; * Women of childbearing potential must agree to use a medically effective means of birth control throughout their participation in the treatment phase of the study and for five months after the last treatment. A woman of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) and is not postmenopausal. Menopause is defined as 12 months of amenorrhea in a woman over age 45 years in the absence of other biological or physiological causes. In addition, females under the age of 55 years must have a serum follicle stimulating hormone (FSH) level \> 40 mIU/mL to confirm menopause. Men receiving nivolumab who are sexually active with WOCBP must agree to use effective contraception throughout their participation in the treatment phase of the study and for seven months after the last treatment. * Due to the potential for serious adverse reactions in breastfed infants from carboplatin/paclitaxel and nivolumab, women are advised not to breast-feed during treatment with carboplatin/paclitaxel or nivolumab * The patient must provide study-specific informed consent prior to study entry. Exclusion Criteria * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 3 years (For example, carcinoma in situ of the breast, oral cavity, or cervix are all permissible); * Any prior therapy for the study cancer; note that prior chemotherapy for a different cancer is allowable if \> 3 years prior to study; * Any history of active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields; * Severe, active co-morbidity, defined as follows: * Unstable angina and/or congestive heart failure requiring hospitalization within the last 6 months; * Uncontrolled diarrhea; * Uncontrolled adrenal insufficiency; * Transmural myocardial infarction within the last 3 months; * Acute bacterial or fungal infection requiring systemic antibiotics at the time of registration; * Chronic Obstructive Pulmonary Disease (COPD) exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration; * Acquired Immune Deficiency Syndrome (AIDS) with CD4+ count \< 350 cells per microL; note, however, that HIV testing is not required for entry into this protocol. The need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive. * Pregnancy or women of childbearing potential and men who are sexually active and not willing/able to use contraception; this exclusion is necessary because the treatment involved in this study may be significantly teratogenic. * Women who are breastfeeding and are not willing to discontinue breastfeeding during the trial * Poorly controlled diabetes (defined as fasting glucose level \> 200 mg/dL) despite 2 attempts to improve glucose control by fasting duration and adjustment of medications. Patients with diabetes will preferably be scheduled in the morning and instructions for fasting and use of medications will be provided in consultation with the patients' primary physicians * Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. Short bursts of steroids of 5-7 days (for COPD exacerbation or other similar indication) are allowed. * Known history of, or any evidence of active, non-infectious pneumonitis. * Known history of active TB (Bacillus Tuberculosis). * Hypersensitivity to nivolumab or any of its excipients or known hypersensitivity to carboplatin/paclitaxel. * Known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). * Received a live vaccine within 30 days of planned start of study therapy. * Have any condition that, in the opinion of the investigator, would compromise the well-being of the subject or the study or prevent the subject from meeting or performing study requirements