The purpose of this study is to see if Isatuximab can alter the immune system in patients with multiple myeloma or lymphoma upon recovery from the autologous stem cell transplantation. The investigators will see if Isatuximab makes changes to the immune system so that upon recovery from the transplant, the immune system can fight the cancer. This study will have two arms. On one arm (control arm), participants will receive standard transplant procedures and on the other arm (experimental arm), participants will receive Isatuximab in addition to the standard transplant procedures. The assignment to these arms is done randomly (determined by chance, like flipping a coin) by a computer. Each participant will have about 66% chance of getting on the experimental arm and about 33% chance of getting on the control arm.

Relapse post-autologous stem cell transplantation (ASCT) remains a major challenge in the treatment of multiple myeloma (MM) and Lymphoma. The immune reconstitution post-ASCT has a major impact on the outcomes of ASCT, however effective methods to improve upon immune reconstitution have not been developed and the use of novel immunomodulators remains relatively unexplored. In addition, numerous studies have demonstrated the profound impact of graft composition on transplant outcomes, but not a single prospective study has addressed this issue successfully. In this study, the investigators intend to test a novel double pronged method of changing the immune repertoire post ASCT by modifying graft composition and improving effector T cell recovery and function post ASCT. In this study, the investigators intend to generate new information on immune modulation post-ASCT. In addition, the CD38 antibodies have not been evaluated as therapy for B-cell non-Hodgkin Lymphoma (NHL). If this study shows significant immunomodulator activity of this approach, cluster of differentiation 38 (CD38) antibodies could be further evaluated in combination with ASCT in NHL.

Inclusion Criteria:

1. Following diagnoses are eligible for inclusion in the study:

   A) Multiple Myeloma with ASCT used as consolidation after first line induction therapy or at first relapse.

   B) Relapsed/Refractory Hodgkin's disease

   C) Non-Hodgkin's Lymphomas as follows

   • Relapsed/Refractory Diffuse large B cell lymphoma
   • Relapsed/Refractory indolent or relapsed/refractory transformed indolent B cell lymphomas as consolidation after second line therapy
   • Mantle Cell lymphoma as consolidation after first-line therapy
   • Peripheral T cell lymphoma as consolidation after first-line therapy or at relapse or primary refractory disease

2. Patients undergoing first ASCT will be eligible for the study.

3. Any prior therapy for the malignancy except CD38 antibody within the last 12 months is allowed.

4. Age ≥18 years

5. Life expectancy of greater than 6 months.

Exclusion Criteria:

1. Previously exposure to a CD38 antibody during the last 12 months.
2. Participants who are receiving any other investigational agents concurrently or received any investigational agent within the last 8 weeks.
3. History of severe allergic reactions or anaphylaxis attributed to compounds of similar chemical or biologic composition to Isatuximab.
4. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements.
5. Pregnant and Lactating women
6. HIV-positive status due to increased risk of infection when treated with immunosuppressive therapy