Patients are not eligible if they have had or are planned for solid organ transplant; patients who have received allogeneic hematopoietic stem cell transplant are eligible if:\r\n* The transplant occurred at least days prior to registration, \r\n* Patient has no prior acute graft versus host disease (GVHD), and \r\n* Within hours of registration, patient demonstrates at least % engraftment, defined as: absolute neutrophil count (ANC) >= mcl, measured over consecutive days or day with an ANC >= , mcl, or platelets >= , mcl measured, wherein the patient did not receive any platelet transfusions within days prior to laboratory assessment Patients may have received prior allogeneic transplant or autologous transplant; however, patients with prior allogeneic bone marrow transplant will be eligible only if both of the following conditions are met:\r\n* The transplant must have been performed >= days prior to registration\r\n* The patient must not have >= grade acute graft versus host disease (GvHD) or either moderate or severe limited chronic GvHD within days prior to registration ELIGIBILITY CRITERIA - PHASE I (ARMS A, B, C): Patient may be enrolled with a prior allogeneic hematopoietic stem cell transplant (HSCT) but the transplant date must be at least days before date of enrollment; patient must be off immunosuppression and without active graft versus host disease (GVHD) prior to enrollment if previous HSCT COHORT : Not be appropriate candidate for intensive salvage chemotherapy due to co-morbidities or other disease- or treatment-related factors\r\n* NOTE: Subjects who received prior treatment with hypomethylating agents either for myelodysplastic syndrome (MDS), myeloproliferative neoplasm (MPN), or AML will be eligible\r\n* NOTE: Subjects who had prior allogeneic stem cell transplant (alloHSCT) will be eligible as long as they have been at least months after allogeneic HSCT and are off of all immune suppression for at least weeks (> days) and have no evidence of active graft versus host disease (GVHD); subjects with prior alloHSCT will NOT be eligible for enrollment during the safety run in phase Allogenic or autologous transplant for hematological malignancy with infusion of stem cells within days before Cycle Day , or on active immunosuppressive therapy for graft-versus-host disease (GVHD) or GVHD prophylaxis within weeks of Cycle Day . No prior allogeneic transplant unless all of the following apply:\r\n* At least years from time of transplant\r\n* Absence of clinically significant graft-versus-host disease (GVHD)\r\n* Not on immune suppression\r\n* Approval of overall PI Patients with history of allogeneic stem cell transplantation are eligible if at least days post-transplant, if there is no evidence of active graft versus host disease (GVHD) and no longer taking immunosuppressive agents for at least days prior to enrollment Receipt of an allogeneic transplant within months or an autologous transplant within the preceding months; evidence of ongoing graft-versus-host disease (GVHD) For subjects with prior allogeneic stem cell transplant, no evidence of active graft-versus host disease (GVHD), and must be >= weeks off immunosuppressive therapy Prior allogeneic transplant performed ? months prior to first dose of AMV is allowed provided there is no evidence of active graft-versus-host disease (GVHD) and the patient has been off immunosuppressive therapy for ? weeks. Prior allogeneic transplant if performed < months prior to first dose of AMV, if patient has active GVHD, or if patient has not been off immunosuppressive If prior allogeneic stem cell transplant, history of moderate to severe chronic graft versus host disease (GVHD) Prior first allogeneic stem cell transplant, with any graft source, donor type, and GVHD prophylaxis Patients with active graft-versus-host-disease (GVHD) status post stem cell transplant, i.e. patients requiring therapy more than chronic steroid immunosuppression and/or phototherapy for chronic skin GVHD will be excluded. Cohort #: patients with MCL with prior allogeneic hematopoietic stem cell transplant, minimum months after transplant, not on immunosuppression, and without prior or active graft versus host disease (GVHD), are allowed Patients within year of allogeneic stem cell transplant, patients with active graft versus host disease (GVHD) or requiring immunosuppression are excluded No evidence of active graft-versus-host disease (GVHD) and at least days must have elapsed after allogeneic bone marrow transplant or stem cell infusion prior to study drug administration Patients who have undergone allo-stem cell transplant (SCT) are eligible if they are at least months post SCT, have relapsed AML, are not on treatment or prophylaxis for graft versus host disease (GVHD), and have no active GVHD. Patients with a prior allogeneic transplant ARE eligible UNLESS previously or currently experienced graft versus host disease (GvHD) that required systemic steroids or other systemic lymphotoxic therapy Patients with hematologic malignancies status post allogeneic SCT without evidence of disease relapse, active graft versus host disease (GVHD) or history of more than stage I skin acute GVHD; and off immunosuppression for at least weeks; at least days after allo-SCT Patients who are < days post allogeneic stem cell transplant will be excluded; patients beyond days post-allogeneic stem cell transplant with active uncontrolled graft versus host disease (GVHD) > grade will be excluded; patients who are on a stable dose of immunosuppressive therapy (tacrolimus, cyclosporine, or other) for > weeks will be eligible but those with recent increase in the immunosuppressive medication dose within last weeks to control GVHD will not be included; Note: subjects may be using systemic corticosteroids or topical or inhaled corticosteroids post allogeneic stem cell transplant; patients requiring >= mg/kg prednisone for GVHD management at the time of screening will not be eligible until the prednisone can be weaned to < mg/kg; such patients should be monitored for at least days and if no flare of GVHD requiring re-escalation of steroids or additional interventions for the GVHD they will be eligible Patients with previous allogeneic stem cell transplant (SCT) if they meet either of the following criteria:\r\n* =< days from allogeneic SCT \r\n* Active acute or chronic graft-versus-host-disease (GvHD) or receiving immunosuppressive therapy as treatment for GvHD Patient participants who have undergone allogeneic SCT (alloSCT) are eligible if they are >= days from stem cell infusion, have no evidence of graft versus host disease (GVHD) > grade , and are >= weeks off all immunosuppressive therapy Patients with previous allogeneic stem cell transplant (SCT) are excluded within months or with active acute or chronic graft-versus host disease are excluded; patients must be off immunosuppression for graft versus host disease (GVHD) for at least days before cycle day Patients with relapsed disease after prior allogeneic SCT (myeloablative or nonmyeloablative) will be eligible if they meet all other inclusion criteria and:\r\n* Have no active graft versus host disease (GVHD) and require no immunosuppression\r\n* Are more than months from transplant Have undergone autologous or allogeneic stem cell transplant < days prior to receiving the first dose of ponatinib; have any evidence of ongoing graft-versus-host disease (GVHD) or GVHD requiring immunosuppressive therapy or are being considered for stem cell transplant within - months of enrollment (note: ponatinib is not to be used as a bridge to stem cell transplant in this trial) Participants who are less than days post-transplant, or greater than days post-transplant with active graft versus host disease (GVHD), or are still continuing post-transplant immunosuppressant therapy within days prior to the first dose of study drug. Prior allogeneic transplant with graft-versus-host disease (GVHD) requiring ongoing immunosuppressive therapy Patients with previous allogeneic stem cell transplant (SCT) within months or with active acute or chronic graft-versus host disease are excluded; patients must be off immunosuppression for graft-versus host disease (GVHD) for at least days before cycle day Patients with previous allogeneic stem cell transplant (SCT) if they meet either of the following criteria:\r\n* < days from allogeneic SCT\r\n* Active acute or chronic graft-versus-host disease (GvHD), or\r\n* Receiving immunosuppressive therapy as treatment for GvHD within the last days Prior allogeneic transplant with graft-versus-host disease (GVHD) requiring immunosuppressive therapy If a research participant has undergone prior allogeneic stem cell transplant, he/she must be off all immunosuppressants for graft versus host disease (GVHD) for at least weeks before undergoing leukapheresis\r\n* Note: the above is not applicable if the research participant's done is undergoing leukapheresis If a research participant has undergone prior allogeneic stem cell transplant (alloSCT), and has documented =< grade graft versus host disease (GVHD) but the donor is undergoing leukapheresis, the research participant may be considered eligible for enrollment (at the discretion of the study principal investigator [PI]) provided that immunosupressants can be tapered off completely prior to lymphodepletion Patients with relapsed disease after prior allogeneic SCT (myeloablative or non-myeloablative) will be eligible if they meet all other inclusion criteria and:\r\n* Have experienced graft rejection (no evidence of donor cells by short tandem repeat (STR) analysis on occasions separated by at least month)\r\n* Have no active graft-vs-host disease (GVHD) and require no immunosuppression\r\n* Are more than months from transplant Patients may have had a prior stem cell transplant (autologous or allogeneic), however they may not have active graft-vs-host disease (GvHD), nor be on any immunosuppression No active graft versus host disease (GVHD): patients with a history of stem cell transplant are eligible but cannot have evidence of active GVHD as determined by the investigator Previous allogeneic stem cell transplant within months prior to enrolment, active graft vs host disease (GVHD), or requiring transplant-related immunosuppression Subjects after prior allogeneic SCT are allowed if the allogeneic transplant was performed >= years prior to study treatment, and if subject has no active Graft-versus- host disease (GVHD) requiring treatment, and meet the remaining eligibility criteria. Evidence of ongoing graft-versus-host disease (GvHD) if prior stem cell transplant (SCT). Prior allogeneic stem cell transplant with day < months prior and/or with chronic graft versus host disease (GVHD) requiring current use of immunosuppression; patients with prior allogeneic stem cell transplant with day > months prior who do not require immunosuppression for GVHD will be eligible Recipients of previous allogeneic transplants who have rash involving more than % body surface area attributed to graft versus host disease (GVHD) (> grade GVHD of skin); stem cell transplant recipients will be excluded if they are still receiving immunosuppression including steroids for GVHD or have active GVHD in any organ (except for grade only of skin, not requiring treatment) Active, uncontrolled graft vs. host disease (GVHD) following allogeneic transplant for non-AML condition (e.g. MDS, lymphoid malignancy, aplastic anemia); patients with GVHD controlled on stable doses of immunosuppressants are eligible At least one prior therapy for CLL/SLL; prior autologous or allogeneic stem cell transplant is allowed; patients may not be on chronic immunosuppressive therapy for graft-versus-host disease (GVHD); patients who are on only oral steroids must be on an oral dose of mg or less of prednisone (or equivalent) daily Patients who have undergone allogeneic SCT (alloSCT) are eligible if they are >= days post stem cell infusion, have no evidence of graft-versus-host disease (GVHD) > grade , and are >= weeks off all immunosuppressive therapy Patients with an allogeneic transplant must meet the following conditions: the transplant must have been performed more than days before registration to this study, the patient must not have >= grade acute graft versus host disease (GvHD) or either moderate or severe limited chronic GvHD, or extensive chronic GvHD of any severity; the patient must be off all immunosuppression for at least weeks Patients with history of allogeneic stem cell transplantation are eligible if at least days post-transplant, if there is no evidence of active graft-versus-host disease (GVHD) and no longer taking immunosuppressive agents for at least days prior to enrollment Prior allogeneic bone marrow progenitor cell transplant within days or on active immunosuppression for graft versus host disease (GVHD) treatment or prophylaxis within days prior to enrollment If participant had prior allogeneic stem cell transplant (SCT), participant has evidence of ongoing graft-versus-host disease (GvHD). Patients with relapsed disease after prior allogeneic SCT (myeloablative or non-myeloablative) will be eligible if they meet all other inclusion criteria and\r\n* Have no active graft-versus-host disease (GVHD) and require no immunosuppression\r\n* Are more than months from transplant Patients with previous allogeneic stem cell transplant (SCT) within months or with active acute or chronic graft-versus host disease are excluded; patients must be off immunosuppression for GVHD for at least days before cycle day Subjects can be enrolled and treated under this protocol regardless of their CLL treatment history or number of previous treatments; in addition, subjects with history of allogeneic stem cell transplant can be enrolled and treated unless they have active manifestations of graft versus (vs.) host disease (GVHD) or chronic illness or infections that will prevent them from completing the study Active graft-versus-host disease (GVHD) following allogeneic stem cell transplant for non-AML condition (ex. MDS, MPN, lymphoid malignancy, aplastic anemia) requiring ongoing use of immunosuppressants Patients within year of allogeneic stem cell transplant, patients with active graft versus host disease (GVHD) or requiring immunosuppression are excluded Patients with relapsed disease after prior allogeneic SCT (myeloablative or nonmyeloablative) will be eligible if they meet all other inclusion criteria and:\r\n* Experienced graft rejection (no evidence of donor cells by short tandem repeat [STR] analysis on occasions separated by at least month)\r\n* Have no active graft-versus-host disease (GVHD) and require no immunosuppression\r\n* Are more than months from transplant Patients who underwent stem cell transplant (SCT) in first complete remission are eligible, as long as all of the following criteria are met:\r\n* At least days have elapsed since stem cell infusion\r\n* At least days off of all medications for graft-versus-host-disease (GVHD) prophylaxis or treatment\r\n* No evidence of acute GVHD Patients previously treated with allogeneic stem cell transplant (SCT) that is currently complicated by active graft versus host disease (GVHD) requiring T cell suppressive therapy Hematopoietic Stem Cell Transplant: Patients who have had previous allogeneic HSCT must have grade I or less of Graft-versus-Host Disease (GVHD) and have not received immunosuppressive medication for at least days. Patients with previous allogeneic stem cell transplant (SCT) if they meet either of the following criteria: < days from allogeneic SCT, acute or chronic graft-versus-host disease (GvHD), or receiving immunosuppressive therapy as treatment for or prophylaxis against GvHD within the last days Allogeneic stem cell transplant within days, active acute or chronic graft-versus-host disease (GvHD), or receiving immunosuppressive therapy as treatment for GvHD Participants who are less than days post-transplant, or greater than or equal to days post-transplant with active graft versus host disease (GVHD), or are receiving immunosuppressant therapy within days prior to first dose of study drug. Patients must have recovered from the acute side effects of all prior anticancer therapy\r\n* At least week from prior cytotoxic chemotherapy\r\n* At least weeks from craniospinal irradiation\r\n* At least months since hematopoietic stem cell transplant (HSCT) with no evidence of active graft-versus-host disease (GVHD) Patients with previous allogeneic stem cell transplant (SCT) within months or with active acute or chronic graft-versus host disease are excluded; patients must be off immunosuppression for graft-versus host disease (GVHD) for at least days before cycle day Hematopoietic stem cell transplant recipients with chronic thrombocytopenia due to chronic graft-versus-host disease (GVHD) or other causes Previous allogeneic hematopoietic stem cell transplant within months prior to enrollment, active graft versus host disease (GVHD), or requiring transplant-related immunosuppression. Prior allogeneic stem cell transplant with active graft-versus-host- disease (GVHD). Prior allogeneic stem cell transplant, no evidence of active graft-versus host disease (GVHD) and must be ? weeks off immunosuppressive therapy. The patient underwent autologous or allogeneic stem cell transplant within days prior to receiving the first dose of omacetaxine and has any evidence of ongoing graft versus host disease (GVHD), or GVHD requiring immunosuppressive therapy.