Adoptive TReg Cell for Suppression of aGVHD After UCB HSCT for Heme Malignancies

Inclusion Criteria: * Must be ≥18, but \< 70 years of age with no matched 7/8 or 8/8 sibling donor - patients ≥ 70 and ≤ 75 years of age may be eligible if they have a Co-Morbidity score ≤ 2 (http://www.qxmd.com/calculate-online/hematology/hct-ci) * UCB unit(s) composing the graft will be selected according to the current University of Minnesota umbilical cord blood graft selection algorithm and an additional cord blood unit to be used as the source to manufacture the Treg product. This UCB unit must be matched at 4-6/6 to the patient, considering HLA-A, B at the antigen level and DRB1 at the allele level * Acute Leukemias: Must be in remission by morphology. Also a small percentage of blasts that is equivocal between marrow regeneration versus early relapse are acceptable provided there are no associated cytogenetic markers consistent with relapse. Refer to Section 5.2 for complete definitions. * Burkitt's Lymphoma in CR2 or subsequent CR * Natural Killer Cell Malignancies * Chronic Myelogenous Leukemia: all types except refractory blast crisis. Chronic phase patients must have failed at least two different tyrosine-kinase inhibitors (TKIs), or been intolerant to all available TKIs or have T315I mutation. * Myelodysplastic Syndrome: IPSS INT-2 or High Risk; R-IPSS High or Very High; WHO classification: RAEB-1, RAEB-2; Severe Cytopenias: ANC \< 0.8, Anemia or thrombocytopenia requiring transfusion; Poor or very poor risk cytogenetics based on IPSS or R-IPSS definitions; therapy-related MDS. Blasts must be be \< 5%, preferably \< 20% blasts by morphology by bone marrow aspirate morphology.. If ≥ 5% blasts, chemotherapy for cytoreduction to \<5% blasts prior to transplantation may be considered. * Chronic myeloid neoplasms, including but not limited to CMML with blasts must around 5% blasts, preferably \< 20% blasts by morphology by bone marrow aspirate morphology. If ≥5% blasts, chemotherapy for cytoreduction to \<5% blasts prior to transplantation may be considered. * Large-Cell Lymphoma, Hodgkin Lymphoma and Multiple Myeloma with chemotherapy sensitive disease that has failed or patients who are ineligible for an autologous transplant. * Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma (CLL/SLL), Marginal Zone B-Cell Lymphoma, Follicular Lymphoma, which have progressed within 12 months of achieving a partial or complete remission. Patients who had remissions lasting \> 12 months are eligible after at least two prior therapies. Patients with bulky disease should be considered for debulking chemotherapy before transplant. Patients with refractory disease are eligible, unless has bulky disease and an estimated tumor doubling time of less than one month. * Lymphoplasmacytic Lymphoma, Mantle-Cell Lymphoma, Prolymphocytic Leukemia are eligible after initial therapy if chemotherapy sensitive. * Patients must have undergone an autologous transplant ≤ 12 months prior to transplant on this study or have received multi-agent or immunosuppressive chemotherapy within 3 months of the preparative regimen. Performance Status, Organ Function, Contraception Use * Karnofsky score ≥ 70% (Appendix II) * Adequate organ function within 14 days (30 days for cardiac and pulmonary) of registration on-study defined as: * Renal: creatinine ≤ 2.0 mg/dL, for patient with a creatinine \> 1.2 mg/dL or a history of renal dysfunction an estimated glomerular filtration rate ≥ 40 mL/min/1.73 m2 is required * ALT, AST and alkaline phosphatase ≤ 5 x upper limit of normal and total bilirubin ≤ 2.5 mg/dL except for patients with Gilbert's syndrome or hemolysis * Pulmonary function: DLCO, FEV1, FVC ≥ 40% predicted, and absence of O2 requirements. * Cardiac: Absence of decompensated congestive heart failure, or uncontrolled arrhythmia and left ventricular ejection fraction ≥ 40%. * Sexually active females of childbearing potential and males with partners of child-bearing potential must agree to use adequate birth control during study treatment. * Voluntary written consent Exclusion Criteria: * Untreated active infection * History of HIV infection * Pregnant or breast feeding. The agents used in this study may be teratogenic to a fetus and there is no information on the excretion of agents into breast milk. Females of childbearing potential must have a blood test or urine study within 14 days prior to registration to rule out pregnancy * Prior allogeneic transplantation * Less than 3 months from myeloablative conditioning for autologous transplantation (if applicable) * Evidence of progressive disease by imaging modalities or biopsy - persistent PET activity, though possibly related to lymphoma, is not an exclusion criterion in the absence of CT changes indicating progression. * CML in blast crisis * Large cell lymphoma, mantle cell lymphoma and Hodgkin disease that is progressing on salvage therapy. * Active central nervous system malignancy