Yes, you can build complex apps in Appery.io. Appery.io creates HTML/JS apps and hybrid apps (PhoneGap). Anything these technologies support -- you can build such app. Here is a good post on what kind of apps you can build with Appery.io: https://blog.appery.io/2015/04/what-k...
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[X] A calendar to schedule events, tasks, projects with the ability to receive reminders/notifications.
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Yes. You can enable push notifications for your app as well: https://devcenter.appery.io/documenta...
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[X] Subject specific research resources (Pulled out from Google search)
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As long as there is a REST API, you can do this.
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[X] Login system
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Yes, use the Login Registration example to see how this works: https://devcenter.appery.io/documenta...
With Appery.io you get out-of-the-box backend services: database, push and server code to write any business logic.
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[X] Ability to create user profile with avatar or picture
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Yes
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[X] Social share (Facebook and/or Twitter)
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Yes, you can connect to any social network via REST API. We also provide plugins to help you with that: https://appery.io/plugins/
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[X] Ability to add photos, videos and share class notes with other students
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Yes, you can implement such logic
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[X] Admin portal to control content
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You can use the Customer Console feature: https://devcenter.appery.io/documenta... or you can build your own app that will act as an admin portal.
If you are new to Appery.io, I recommend to start here: http://devcenter.appery.io/getting-st... and check out the videos from our YouTube channel: http://youtube.com/apperyio