Lecture 1: Motivation, definition of Koszul algebras

Lecture 2: Methods for proving Koszulity, and more examples

Lecture 3: Bar complex, topology and inequalities

Lecture 4: Group actions

Exercises for Session 1

Exercises for Session 2

- A very old one from MSRI 2003.
- A recent one from Oberwolfach, December 2023.
- A recent one from Brenti Fest 2023.

**Koszulity original sources, books, and surveys**- Priddy, Koszul resolutions
- Fröberg, Determination of a class of Poincare series
- Fröberg, Koszul algebras (survey)
- Backelin and Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions
- Backelin, Low degrees in a Groebner basis may force the Koszul property
- Polishchuk and Positselski's book, "Quadratic algebras" (AMS University Lecture Series)
- McCullough and Peeva, Infinite graded free resolutions
- Avramov, Infinite free resolutions
- Jöllenbeck and Welker, Resolution of the residue class field via algebraic discrete Morse theory
- Ufnarovskii, V.A., Combinatorial and asymptotic methods in algebra

**Recent and not-so-recent papers**- Almousa, Reiner and Sundaram, Koszulity, supersolvability, and Stirling representation
- Angarone, Nathanson and Reiner, Chow rings of matroids as permutation representations
- Backman and Liu, A regular unimodular triangulation of the matroid base polytope
- Bruns, Herzog and Vetter, Syzygies and walks
- Conca, De Negri and Rossi, Koszul algebras and regularity
- Conca, Trung and Valla, Koszul property for points in projective spaces
- Coron, Supersolvability of built lattices and Koszulness of generalized Chow rings
- Dorpalen-Barry, The Varchenko-Gelfand ring of a cone
- Dotsenko, Homotopy invariants for M_{0,n} via Koszul duality
- Eisenbud, Reeves and Totaro, Initial ideals, Veronese rings, and rates of algebras
- Faber, Juhnke-Kubitzke, Lindo, Miller, R. G., Seceleanu, Canonical resolutions over Koszul algebras
- Jambu, Koszul algebras and hyperplane arrangements
- Josuat-Verges and Nadeau, Koszulity of dual braid monoid algebras via cluster complexes
- Maestroni and McCullough, Chow rings of matroids are Koszul
- Peeva, Hyperplane arrangements and linear strands in resolutions
- Peeva, Reiner and Sturmfels, How to shell a monoid
- Reiner and Stamate, Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
- Reiner and Welker, On the Charney-Davis and Neggers-Stanley Conjectues
- Roos, Some non-Koszul algebras
- Sam and VandeBogert, From total positivity to pure free resolutions
- Shelton and Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements
- VandeBogert, Ribbon Schur functors

- Is Neil White's
*matroid basis ring*a quadratic algebra, and is it Koszul? See White, Blasiak, Kashiwabara, Lason-Michalek

What about for polymatroids? See Herzog-Hibi.

- The
*Charney-Davis-Gal Conjecture*: gamma-nonnegativity for flag simplicial homology spheres, or equivalently, Koszul Gorenstein Stanley-Reisner rings.

See this paper and note by Gal, and Athanasiadis.

See also a stronger conjecture by Nevo-Petersen with results by Aisbett-Volodin.

Lastly, note the cautionary tale in D'Ali-Venturello (Cor. 7.4).

- For the
*Orlik-Solomon algebra OS(M)*of a matroid M, does OS(M) Koszul imply M is*supersolvable*? See Yuzvinsky, Peeva.

Same for the (graded)*Varchenko-Gelfand ring VG(M)*of an oriented matroid M: Does VG(M) Koszul imply M is supersolvable? See Dorpalen-Barry. - For which matroids M is the
*quadratic closure*of their Orlik-Solomon algebra O(M) a Koszul algebra? See Peeva

Does this hold for the matroid of the reflection arrangement of type D_n? -
*Bøgvad's Conjecture*: the*toric ring*of a*smooth*projectively normal toric variety is Koszul; see Bøgvad - For which
*quadratic monomial*quotients of the free associative algebra is the Hilbert series a*Polya-Frequency*sequence? See Chapter 7 of Brenti's thesis. -
*Thrall's Problem*: give a combinatorial formula for the irreducible expansion of the higher Lie characters of S_{n},

essentially describing the type A reflection arrangement's graded Varchenko-Gelfand ring and its Orlik-Solomon algebra. See Schocker.

- Is the S
_{n}-representation on the cohomology of M_{0,n}a*permutation representation*?

See Ramadas-Silversmith and Dotsenko. - Let A=S/I where S=k[x
_{1},...,x_{n}] is the commutative polynomial algebra, and assume A is a Koszul algebra.

Do the Betti numbers b_{i}=b_{i}^{S}(A) in the*finite*minimal free resolution of A as an S-module satisfy b_{i}≤ ( b_{1}choose i )?

See Conca, De Negri, Rossi. - Conjectures 4.7, 4.11, 4.23 on
*matroid Chow rings*

from our paper with Angarone and Nathanson - Problems and conjectures on
*Koszul duals of type A*Orlik-Solomon algebras and graded Varchenko-Gelfand rings

from Sections 10, 11, 12 of our paper with Almousa and Sundaram - For a poset P, is the
*ideal of P-partitions*within the ring of weak P-partitions always a*nongraded Koszul*module?

See Sections 8,9,12 of our paper with Féray - For which finite subgroups of GL
_{n}(k) acting on a polynomial ring S=k[x_{1},...,x_{n}] does one have ...- ... that the
*invariant ring*S^{G}is a*nongraded Koszul algebra*? - ... the polynomial ring S is a
*nongraded Koszul module*over the invariant ring S^{G}?

- ... that the