The Arukh haShulchan (OC 56:1,3) discusses the meaning of the opening words of Qaddish “Yisgadel veyisqadeish shemeih rabba“. Praying that Hashem’s Great Name be made great and holy. And since a name is how we refer to something, we are talking about how humans perceive Hashem’s presence.

The Arukh haShulchan mentions a tradition that “שמיה – His name” is an allusion to “שם י-ה — the name Y-H”.

As we read on parashas Zakhor (Shemos 17:16):

וַיֹּ֗אמֶר כִּֽי־יָד֙ עַל־כֵּ֣ס יָ֔הּ מִלְחָמָ֥ה לַיהוָ֖ה בַּֽעֲמָלֵ֑ק מִדֹּ֖ר דֹּֽר׃

Moshe said, “Because there is a hand upon the throne of Y-H, Hashem is at war with Amaleiq from gemeration to generation.”

Chazal (Tanchuma ad loc) famously (quoted by Rashi ad loc) ask why “throne” here is rendered in the contraction “כס” rather than the full “כסא” and Hashem’s full name is shortened to just the first two letters “י-ה”. Until evil is vanquished, Hashem’s rule is incomplete, and His name is incomplete.

Thus in Qaddish, we are praying for an end of evil, when שמיה — Hashem’s name י-ה — is enlarged to the full tetragrammaton and made holy in a world consecrated to do His work.

Similarly (se’if 3), when we respond, “יהא שמיה רבא מברך — May His great name be blessed”. (That’s where the Rama says the comma belongs.) Which is why the Zohar refers to the utterance of this response as “one of the pillars of creation.”

As Yechezqeil quotes Hashem’s promise, in the pasuq (38:23) from which we get the idiom with which Qaddish opens:

וְהִתְגַּדִּלְתִּי֙ וְהִתְקַדִּשְׁתִּ֔י וְנ֣וֹדַעְתִּ֔י לְעֵינֵ֖י גּוֹיִ֣ם רַבִּ֑ים וְיָדְע֖וּ כִּֽי־אֲנִ֥י ה’׃

Thus will I manifest My Greatness and My Holiness, and make Myself known in the sight of the many nations; and they shall know that I am Hashem.

… בְּחַיֵּיכון וּבְיומֵיכון וּבְחַיֵּי דְכָל בֵּית יִשרָאֵל, בַּעֲגָלָא וּבִזְמַן קָרִיב, וְאִמְרוּ אָמֵן!

… in our lifetimes and in our days, and in the lifetimes of all of the Jewish Home,  quickly, and soon. And let us say, Amein!