A Study of Carboplatin and Paclitaxel Chemotherapy With Nivolumab With or Without Ipilimumab in Patients With Ovarian Cancer

Inclusion Criteria: * Patients must have clinically and radiographically suspected International Federation of Gynecology and Obstetrics (FIGO) stage (Appendix 2) 3 or 4 high grade serous ovarian, primary peritoneal or fallopian tube cancer, high grade, for whom the plan of management will include neoadjuvant chemotherapy (NACT) with interval tumor reductive surgery (TRS). Patients will be selected for NACT according to established criteria (Society of Gynecologic Oncology and the American Society of Clinical Oncology Guideline, https://www.ncbi.nlm.nih.gov/pubmed/27650684). Patients must have undergone core needle biopsy for histologic confirmation prior to start of treatment. See Appendix 3 for guidelines to aid in histologic diagnosis. * Institutional confirmation of pathology on or laparoscopic biopsy performed at MSKCC with sufficient tissue for analysis or willingness to undergo repeat laproscopic biopsy if diagnostic biopsy was performed at an outside hospital. * Appropriate stage for study entry based on the following diagnostic workup: * History/physical examination within 28 days prior to registration; * CT imaging of the chest, abdomen and pelvis within 28 days prior to registration documenting disease consistent with FIGO stage III or IV disease (patients who cannot receive contrast may instead undergo MRI of abdomen and pelvis along with non-contrast chest CT); * Further protocol-specific assessments as detailed below. * Age ≥18 * ECOG Status of 0, 1, or 2 within 28 days prior to registration. * Adequate hematologic function within 14 days prior to registration defined as follows: * ANC ≥1,000/mcl. This ANC cannot have been induced by granulocyte colony stimulating factors. * Platelets ≥75,000/mc * Hemoglobin ≥8.0 mg/dl (transfusions are permitted to achieve baseline hemoglobin level) * Adequate renal function within 14 days prior to registration defined as follows: °Creatinine \<=1.5 x ULN. * Adequate hepatic function within 14 days prior to registration defined as follows: * Bilirubin ≤ 1.5 x ULN * ALT and AST ≤ 3 x ULN * Alkaline phosphatase ≤ 2.5 x ULN * For patients with Gilbert's syndrome, Bilirubin ≤ 3 xULN is acceptable. * Neurologic function: Neuropathy (sensory and motor) less than or equal to CTCAE Grade 1. * Adequate thyroid function within 14 days prior to registration defined as serum TSH within the normal range. * The patient or legally authorized representative must provide study-specific informed consent prior to study entry. Exclusion Criteria: * Subjects with a history of other invasive malignancies, with the exception of non-melanoma skin cancer, are excluded if there is any evidence of other malignancy being present within the last 3 years. Subjects are also excluded if their previous cancer treatment contraindicates this protocol's therapy. * Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis or thoracic cavity within the last three years are excluded. Prior radiation for localized cancer of the breast, head and neck or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease. * Patients who have received any prior treatment for management of their epithelial ovarian, fallopian tube or peritoneal primary cancer. * Patients with any histology other than high grade serous evident on pretreatment biopsy. * Patients with synchronous primary endometrial cancer, or a past history of primary endometrial cancer, unless all of the following conditions are met: Stage not greater than I-A, grade 1 or 2, no more than superficial myometrial invasion, without vascular or lymphatic invasion; no poorly differentiated subtypes, including serous, clear cell or other FIGO grade 3 lesions. * Severe, active co-morbidity defined as follows: * Chronic or current active infectious disease requiring systemic antibiotics, antifungal or antiviral treatment (with the exception of uncomplicated UTI) * Clinically significant cardiac disease including unstable angina, acute myocardial infarction within 6 months from enrollment, New York Heart Association Class III or IV congestive heart failure, and serious arrhythmia requiring medication (this does not include asymptomatic atrial fibrillation with controlled ventricular rate). * Partial or complete gastrointestinal obstruction * Patients with anticipated contraindications to interval tumor reductive surgery. * Patients with any condition that in the judgment of the investigator would jeopardize safety or patient compliance with the protocol. * Patients who are unwilling to be transfused with blood components. * Concurrent anticancer non protocol directed therapy (e.g. chemotherapy, radiation therapy, biologic therapy, immunotherapy, hormonal therapy, investigational therapy). * Receipt of an investigational study drug for any indication within 28 days or 5 half-lives (whichever is longer) prior to Day 1 of protocol therapy. * Patients who, in the opinion of the investigator, are unable or unlikely to comply with the dosing schedule and study evaluations. * Patients who are pregnant or nursing. The effects of nivolumab on the developing human fetus are unknown. For this reason, women of child-bearing potential (WOCBP) must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. WOCBP must have a negative serum or urine pregnancy test within 7 days of study enrollment, prior to dosing nivolumab, then every 6 weeks. After discontinuation from nivolumab these should be repeated at approximately 30 days and approximately 70 days. Women must not be breastfeeding. 1. Women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile) do not require contraception. 2. Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy and/or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 month amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level greater than 40mIU/mL. * Patients are excluded if they have active brain metastases or leptomeningeal metastases. Subjects with brain metastases are eligible if metastases have been treated and there is no magnetic resonance imaging (MRI) evidence of progression for 4 weeks or more after treatment is complete and within 28 days prior to the first dose of nivolumab administration. There must also be no requirement for immunosuppressive doses of systemic corticosteroids (\> 10 mg/day prednisone equivalents) for at least 2 weeks prior to study drug administration. * In order for patients with known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) to be eligible, they must be on a stable highly active antiretroviral therapy (HAART) regimen, have CD4 counts \> 350, with no detectable viral load on quantitative PCR. * Patients with treated hepatitis virus infections (Hepatitis B or Hepatitis C) are eligible if they have been definitively treated for 6 months, have no detectable viral load on quantitative PCR, and LFTs meet eligibility requirements. * Patients with active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including systemic corticosteroids, should be excluded. These include but are not limited to patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as SLE, connective tissue diseases, scleroderma, inflammatory bowel disease, Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease. Patient with vitiligo, endocrine deficiencies including thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible. Patients with rheumatoid arthritis and other arthropathies, Sjogren's syndrome and psoriasis controlled with topical medication and patients with positive serology, such as antinuclear antibodies (ANA), anti-thyroid antibodies should be evaluated for the presence of target organ involvement and potential need for systemic treatment but should otherwise be eligible. * Patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement (such as Hashimoto's thyroiditis), psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (precipitating event). * Patients should be excluded if they have a condition requiring systemic treatment with either corticosteroids (\>10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \<10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. Patients are permitted to use topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption). A brief course of corticosteroids for prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions (e.g., delayed-type hypersensitivity reaction caused by contact allergen) is permitted. * Any of the following within 2 months of registration: active peptic ulcer disease, diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, malabsorption syndrome. Any of the following within 6 months of registration: Intra-abdominal abscess, gastrointestinal obstruction requiring parenteral hydration and/or nutrition, gastrointestinal perforation. Note: complete resolution of an intra-abdominal abscess must be confirmed prior to registration even if the abscess occurred more than 6 months prior to registration.