Study of Ipilimumab After Stem Cell Transplantation in People With Relapsed/Refractory Multiple Myeloma

Inclusion Criteria: Inclusion Criteria prior to Allogeneic Hematopoietic Stem Cell Transplantation (alloHSCT): * Willing and able to participate as a research subject and provide informed consent (Note: an LAR may sign the consent form on the partipant's behalf) * Diagnosis of relapsed refractory multiple myeloma defined as more than 2 lines of prior therapy with at least a very good partial remission to most recent salvage therapy. * Patients should have R-ISS stage II or III disease at diagnosis or high risk cytogenetics by IMWG criteria (t(4;14), del(17/17p), t(14;16), t(14;20), nonhyperdiploidy, and gain(1q)) at any time since diagnosis Note:. A line of therapy is treatment between diagnosis and progression or between two progressions * Eligible for CD34-selected HSCT according to MSKCC adult BMT guidelines. * Have a 10/10 matched donor * Age ≥ 21, \< 73 years. * Karnofsky (adult) Performance Status ≥ 70%. * Patients must have adequate organ function measured by: 1. Cardiac: LVEF at rest must be ≥ 50% 2. Hepatic: * \< 3x ULN ALT * \< 1.5 ULN total serum bilirubin, unless there is congenital benign hyperbilirubinemia. 3. Renal: serum creatinine \<1.2 mg/dl or if serum creatinine is outside the normal range, then CrCl \> 40 ml/min (measured or calculated/estimated) with dose adjustment of Fludarabine for \<70ml/min 4. Pulmonary: DLCO \> 50% of predicted (corrected for hemoglobin). Inclusion Criteria prior to Ipilimumab: * Non progressive myeloma (partial response or better) as defined by International Myeloma Working Group (IMWG) criteria * Engraftment of all cell lines without transfusion dependence, defined as: * absolute neutrophil count \> 1.0K/mcL x 3 consecutive days * platelets \> 50K/mcLx 7 consecutive days without platelet transfusion * no platelet or RBC transfusions within the preceding 7 days * ≥ 80% donor chimerism in the bone marrow Exclusion Criteria: Exclusion Criteria prior to Allogeneic Hematopoietic Stem Cell Transplantation (alloHSCT): * Patients ineligible for therapy with ipilimumab, for example:. 1. Active autoimmune disease or any condition requiring systemic treatment with either corticosteroids (\>10 mg daily of prednisone equivalents) or other immunosuppressive medications at enrollment. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. 2. History of motor neuropathy considered to be of autoimmune origin (e.g., Guillain-Barre Syndrome, Myasthenia Gravis). * Female patients who are pregnant or breast-feeding. * Patients with plasma cell leukemia at the time of diagnosis. * Patients who have undergone prior allogeneic hematopoietic stem cell transplantation. * Patients who have had a previous malignancy that is not in remission. Exclusion Criteria prior to Ipilimumab: * Active infection or treatment for infection (patients on Cytomegalovirus (CMV) therapy will be considered eligible; patients with CMV viremia by PCR or disease with end-organ involvement will not be eligible) * Active GVHD of any grade or prior grade 3-4 GVHD * Active immune suppression, defined as: * active use of calcineurin inhibitors, mycophenolate mofetil, or other immunomodulators * steroid dosing exceeding 10 mg/d prednisone or equivalent * Receiving immunomodulatory agents (ex. thalidomide, lenalidomide, pomalidomide)