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Gorsuch, President Donald Trump's first Supreme Court nominee, was sworn in last week, becoming the 113th person to serve on the Supreme Court. He fills the seat left vacant since February 2016, when Justice Antonin Scalia died unexpectedly.
The first case he is scheduled to hear, starting at 10 a.m., along with the eight other justices, is Perry v. Merit Systems Protection Board, a case stemming from an employment issue at the U.S. Census Bureau.
The most anticipated case for this term will come Wednesday, according to ABC News contributor and Cardozo School of Law professor Kate Shaw.
Gorsuch will be there to hear arguments in Trinity Lutheran Church v. Comer, a case about whether a state constitutional provision that prevents state funds from going to religious institutions violates the federal Constitution — both the clause protecting the free exercise of religion and the clause guaranteeing the equal protection of the laws.
In this case, a church that has a playground applied for a state program that helps nonprofits resurface their playgrounds. The church was denied access to the program because of its status as a church, and it argues that this is unconstitutional.
Trump's travel ban
The other big cases to watch are the various challenges to the president's second travel ban executive order.
If the fight over the travel ban reaches the Supreme Court, it's possible that Gorsuch could give the conservative wing the votes needed to rule in favor of the Trump administration.
The 4th Circuit and 9th Circuit courts will hear arguments in May on the constitutionality of the travel ban. Whatever happens, the losing party is virtually certain to seek Supreme Court review, according to Shaw. Although the court doesn't typically hear cases between April and October, it's certainly not unheard of for it to do so.
"It's quite possible here, in particular if the administration loses and asks the court to act quickly. The court could also rule without hearing arguments," said Shaw.
It's likely that the Trump administration took its time appealing district court losses on the ban in Maryland and Hawaii because it wanted Gorsuch in place and thinks it has a good chance of getting his vote, according to Shaw.
There's no guarantee that Gorsuch will vote with the administration, but if the rest of the conservatives on the Supreme Court bench vote to uphold the ban, it's unlikely he will break with them in his first major vote, she added.
Potential cases for SCOTUS
There are also a couple of cases the court has not yet agreed to hear, but eyes will be on Gorsuch as the justices decide whether they will be argued before the Supreme Court.
Perhaps the most watched case is Masterpiece Cakeshop v. Colorado Civil Rights Commission, a religious freedom case stemming from Jack Phillips and his bakery, Masterpiece Cakeshop, refusing to create a wedding cake for a same-sex couple.
Phillips declined because using his talents to promote same-sex marriage would go against his religious belief that marriage is between a man and a woman, according to Alliance Defending Freedom, which is defending the bakery.
"No one — not Jack or anyone else — should be forced by the government to further a message that they cannot in good conscience promote," said ADF senior counsel Jeremy Tedesco in a statement
The bakery asked the Supreme Court to take on the case after the Colorado Civil Rights Commission said that Phillips may not deny services to same-sex couples.
The high court has had the opportunity to take the case since July 2016 but has yet to take action.
The Supreme Court could also agree to hear Peruta v. San Diego, giving Gorsuch the opportunity to weigh in on a controversial Second Amendment case that looks at whether law-abiding citizens may carry concealed guns outside the home for self-defense when openly carrying firearms is forbidden by state law.
San Diego resident Edward Peruta sued the county in 2009 after his application was for a concealed-carry license was denied by the sheriff's office, which is responsible for deciding whether someone has good cause to carry a firearm.
The case made its way to the 9th Circuit Court of Appeals, which said in a divided opinion that there is no constitutional right to carry concealed weapons in public.
Peruta, along with four other California gun owners and the California Rifle and Pistol Association, then petitioned the Supreme Court in January to take the case.
This has the potential to be the biggest gun rights case since District of Columbia v. Heller in 2008, in which the Supreme Court overturned a D.C. law that prevented people from registering and keeping handguns at home.
ABC News' Kate Shaw, Audrey Taylor and Ben Bell contributed to this story.