Arcane adepts are a specialized school of wizardry who not only study magic, but are tapped into the direct source of magic itself. This source may be called the Weave, Arcana, the Fade, the Source, or ley lines depending on the campaign, or even just some primal ancient force that no one fully understands. (Ask your DM.) Sometimes bursts of extra power come to an arcane adept due to their inclusion in various powerful magical families or other arcane lineages, or by being members of exceptional mage guilds, but other times the adept just possesses a stronger connection to raw magic than others for unknown reasons (this can be expanded upon in the character's background). The research of arcane adepts focuses on influencing raw magic directly and on their minimizing their own expenditure of magical energy by drawing power back into themselves from other sources. Dedicated to the power of magic itself, they gain a deeper understanding of magic as a whole at the cost of specialized knowledge involving a specific magical school instead.

Arcane Intuition

Your profound understanding of the laws of magic allows you to easily sense magic in your surroundings. You gain proficiency in the Arcana skill. When using the Arcana skill, you add double your proficiency bonus to those skill checks. 

Whenever you observe a spell of 1st level or higher being cast within 60 feet of you, you automatically know the school of magic of the spell as well as the level of the spell slot expended. 

Intensify Spell

Starting at 2nd level, you can use your keen understanding of the Weave to increase the potency of your spells. Whenever you cast a spell of first level or higher, you can use a bonus action to briefly increase the spell's DC by 2. This increase lasts until you start your next turn.

You can use this feature a number of times equal to your proficiency bonus, and you regain all expended uses of it when you finish a long rest.

Magic Absorption

Starting at sixth level, your control over the Weave (or whatever is the source of arcane magic in your campaign) allows you to repurpose or absorb magic from other spellcasters. When a creature other than you expends a spell slot to cast a spell within 10 feet of you, you can use your reaction to regain an expended spell slot whose level is no greater than half the level of the spell slot the other creature expended (rounded down, minimum of 1).

You can use this ability a number of times equal to your proficiency bonus. This number resets after a short rest.

At level 10, the range of this feature increases to 30 feet.

Shielding Weave

At 10th level, you can manipulate the Weave instinctually to shield yourself and others around you from certain spells. Whenever a spell is cast on you or another target within 10 feet of you, you can use your reaction to grant all creatures of your choice within a 10-foot radius sphere centered around you advantage on saving throws against that spell. 

Synchronized Casting

At 14th level, you've learned to harness the expended power of spells to simultaneously cast spells of your own. When a creature other than you expends a spell slot to cast a spell within 30 feet of you, you can use your reaction to cast a spell you have prepared whose casting time is no longer than one action. You may not expend a spell slot higher than the spell slot of the mirrored spell you are harnessing to cast your own prepared spell. 

You can use this feature once, and regain use of it once you have completed a short rest.