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Quantized lattices, or q-lattices, appear naturally through categorification constructions (for example from zigzag-algebras), but they haven't been studied from a lattice theory point of view. After establishing the necessary background, we'll explain the q-versions of various lattice theory concepts illustrated with examples from combinatorics and graph theory, and we'll list a number of open problems.

In this talk we will discuss three (intimately related) examples of 4-manifold invariants based on higher structures:
   - VOA[M4] from transgression of EFTs;
   - SW and Donaldson invariants as chiral algebra correlators;
   - Massey triple products from trisections.
These topics are based, respectively, on work with A.Gadde, P.Putrov (and work to appear with B.Feigin); recent paper with M.Dedushenko and P.Putrov; and a solo paper of the speaker.

Skew Howe duality is based on a very simple observation: the set of n by m matrices has commuting action of SL_n and SL_m.  We can use this observation to study morphisms of GL_m representations using GL_n.  This perspective has proven very useful in recent years for studying quantum knot invariants and their categorifications.  I will survey work in this direction from the last 10 years, including more recent developments concerning annular skew Howe duality and annular knot invariants.

In this talk, intended for a broader audience, I promise to use techniques only at the level of university calculus. While staying at this level, our goal will be to learn conceptual lessons for categorification of quantum group invariants of knots and 3-manifolds, also known as the Witten-Reshetikhin-Turaev (or WRT) invariants. In particular, we will introduce new q-series invariants of 3-manifolds that have integer powers and integer coefficients and, if time permits, discuss their various constructions and properties. The talk is based on several papers with D. Pei and/or P. Putrov, C. Vafa, as well as ongoing work with C. Manolescu.