Based on the press release (2009 Nobel Peace Prize press release) and historical records from his first year in office (2009), here are specific actions President Obama took for each of the categories mentioned:
Nuclear Disarmament
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Prague Speech (April 2009): He delivered a landmark speech in Prague where he committed the U.S. to the long-term goal of a "world without nuclear weapons," shifting the focus of U.S. national security strategy.
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UN Security Council Resolution 1887: In September 2009, he became the first U.S. president to chair a UN Security Council summit, which unanimously passed a resolution calling for an end to the spread of nuclear weapons.
Multilateral Diplomacy
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Rejoining the UN Human Rights Council: Reversing a previous policy of boycotting the group, the U.S. successfully ran for and won a seat on the council in 2009 to engage more directly in international human rights monitoring.
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G-20 Elevation: He worked to elevate the G-20 as the premier forum for international economic cooperation in response to the global financial crisis.
Conflict Resolution
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Cairo Speech (June 2009): In his "A New Beginning" speech, he called for improved relations and "sustained effort to listen" between the United States and Muslim communities worldwide.
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Middle East Peace Envoys: On his second full day in office, he appointed veteran diplomat George Mitchell as a Special Envoy for Middle East Peace to immediately jumpstart negotiations.
Climate Change
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Copenhagen Accord (December 2009): At the COP15 summit, he personally brokered a last-minute political agreement between major economies (including China and India) to limit global temperature increases to below 2°C.
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U.S.-China Strategic Dialogue: He established a new strategic and economic dialogue with China in July 2009, specifically identifying climate change as a top priority for bilateral cooperation.
Human Rights and Democracy
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Banning Torture: Within his first days, he signed executive orders to unequivocally ban torture and "enhanced interrogation" techniques and ordered the closure of CIA "black sites."
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Signing Disability Rights Convention: In July 2009, he signed the UN Convention on the Rights of Persons with Disabilities, signaling a renewed U.S. commitment to international human rights treaties.