Mahir Bilen Can

Email: mahirbilencan at gmail dot com Vita

Research.

Note: While some of the unpublished papers listed below are under review in a journal, I am currently working on improving the main results of the others. If you would like to learn the state of a particular article from this list, please ask me.

- On the Dual Canonical Monoids.

We investigate the conjugacy decomposition, nilpotent variety, the Putcha monoid, as well as the two-sided weak order on the dual canonical monoids.

- The Betti Numbers of a Determinantal Variety.

We determine the Poincare polynomial of the determinantal variety det=0 in the projective space associated with the monoid of n×n matrices.

- Sects (with Aram Bingham).

By explicitly describing a cellular decomposition we determine the Borel invariant cycles that generate the Chow groups of the quotient of a reductive group by a Levi subgroup. For illustrations we consider the variety of polarizations $\mbf{SL}_n / \mbf{S}(\mbf{GL}_p\times \mbf{GL}_q)$, and we introduce the notion of a sect for describing its cellular decomposition. In particular, for p=q, we show that the Bruhat order on the sect corresponding to the dense cell is isomorphic, as a poset, to the rook monoid with the Bruhat-Chevalley-Renner order.

- Diagonal Orbits in a Double Flag Variety of Complexity One, Type A (with Tien Le).

We investigate the structure of the inclusion poset of diagonal SL(n)-orbit closures in a double flag variety of complexity one. We find that there are exactly 28 nonisomorphic such posets. In particular we show that the number of SL(n)-orbits in any of these posets is bounded by 10.

- Ansatz for $(-1)^{n-1}\nabla p_n$ (with Soumya D Banerjee and Adriano Garsia).

We construct an S_n equivariant vector bundle on the Hilbert scheme of n points in the plane whose bigraded Euler characteristic gives the symmetric function (-1)^{n-1}\nabla p_n.

- Adjoint Representations of the Symmetric Group (with Miles Jones).

We compute the Frobenius character of the S_n representation that is obtained from the restriction of adjoint representation of GL_n.

- Complexity $c$ Pairs in Simple Algebraic Groups.

We introduce the notion of a complexity $c$ pair in reductive groups. We classify the pairs of closed subgroups (G_1,G_2) from a simple algebraic group such that G_1 is reductive and (G_1,G_2) is a complexity one pair. Also, we classify the triplets (G,H,P) such that the diagonal action of G on G/H x G/P is spherical. Here, G/H is an affine homogeneous space, and G/P is a partial flag variety.

- Combinatorial models for the variety of complete quadrics (with Soumya D. Banerjee and Michael Joyce).

We show that the action of SL_n on the variety of complete quadrics, denoted by Q_n, is cancellative. Let T denote the maximal torus of diagonal matrices in SL_n. We determine the T-invariant surfaces and curves in Q_n. Let B denote a Borel subgroup in SL_n. We describe the inclusion order (Bruhat-Chevalley order) on the B-orbit closures in Q_n. Our description of the inclusion order applies to other wonderful compactifications.

- Equivariant K-theory of smooth projective spherical varieties (with Soumya D. Banerjee).

We give a concrete description of the equivariant K-rings of smooth complete spherical varieties.

- Sphericity of Smooth Schubert and Richardson Varieties (with Reuven Hodges).

Let X_{wQ} be a smooth Schubert variety in a (partial) flag variety G/Q, P denote the stabilizer subgroup of X_{wQ} in G, and let L denote the standard Levi factor of P. We show that the action of L on X_{wQ} is spherical.

- On the cohomology rings of Grassmann varieties and Hilbert schemes (with Jeff Remmel).

We review the works of Akyildiz, Carrell, and Brion on the regular varieties. We compute the equivariant cohomology rings of Hilbert scheme of n points in the plane and the Grassmann variety. We study the relationship between these two cohomology rings.

- Remarks on Enveloping Semigroups.

We revisit the works of Rittatore and Vinberg on the asymptotic and enveloping semigroups. The local structures of enveloping semigroups of simple groups are investigated.

- Toroidal Schubert Varieties(with Reuven Hodges and Venkatramani Lakshmibai).

To appear in Algebras and Representation Theory.

- Stirling Posets (with Yonah Cherniavsky).

To appear in Israel Journal of Mathematics.

- A Filtration on Equivariant Borel-Moore Homology (with Aram Bingham and Yıldıray Ozan).

Forum Math. Sigma 7 (2019), e18.

- The rook monoid is lexicographically shellable.

European Journal of Combinatorics, Volume 81, October 2019, 265--275.

- A Geometric Interpretation of the Intertwining Number (with Yonah Cherniavsky and Martin Rubey).

Electronic Journal of Combinatorics, Volume 26, Issue 2 (2019) Article Number P2.7.

- The genesis of involutions, polarizations and lattice paths (with Özlem Uğurlu).

Discrete Mathematics 342 (2019) 201--216.

- Classification of Reductive Monoid Spaces Over an Arbitrary Field.

Advances in Algebra, 67--94, Springer Proc. Math. Stat., 277, Springer, Cham, 2019.

- Monoid embeddings of symmetric spaces (with Roger Howe and Lex Renner).

Colloquium Mathematicum, Volume 157, no. 1, 2019.

- Loop-augmented forests and a variant of the Foulkes' conjecture (with Jeff Remmel).

Algebraic Combinatorics, Volume 1 (2018) no. 5 p. 573--601

- Counting Borel Orbits in Symmetric Varieties of Types BI and CII (with Özlem Uğurlu).

Arnold Mathematical Journal, 4 (2018), no. 2, 213--250.

- The cross section of a spherical double cone.

Advances in Applied Mathematics. Volume 101, October 2018, Pages 215--231

- Wonderful symmetric varieties and Schubert polynomials (joint with Michael Joyce and Ben Wyser).

Ars Mathematica Contemporanea. Vol 15, No 2 (2018)

- A representation on labeled rooted forests.

Comm. Algebra 46 (2018), no. 10, 4273--4291.

- Complex G2 and associative grassmannian (with Selman Akbulut).

Journal of Gökova Geometry Topology Volume 11 (2017) 56--79.

- Monodromy of torus fibrations and decomposability problem (with Mustafa Topkara).

Topology and Its Applications 222 (2017), 165--176.

- An analogue of Springer fibers in certain wonderful compactifications (joint with Roger Howe and Michael Joyce).

Journal of Algebra and Its Applications 16 (2017), no. 9.

- Corrigendum to ''Generators of the Hecke algebra of (S_2n,B_n)'' (with Şafak Özden).

Adv. Math. 308 (2017), 1337--1339.

- Toroidal affine Nash groups.

Journal of Lie Theory 26 (2016), no. 4, 1069--1077.

- Chains in weak order posets associated with involutions (with Michael Joyce and Ben Wyser).

Journal of Combinatorial Theory Series A 137 (2016), 207--225.

- Divisors and specializations of Lucas polynomials (with Tewodros Amdeberhan and Melanie Jensen).

Journal of Combinatorics 6 (2015) no 1-2, 69--89

- Lexicographic shellability of the Bruhat-Chevalley order on fixed point free involutions (with Yonah Cherniavsky and Tim Twelbeck)

Israel Journal of Mathematics 207 (2015) no. 1.281--299.

- Omitting Parentheses from the Cyclic Notation (with Yonah Cherniavsky)

Mediterranean Journal of Mathematics 12 (2015), no. 4, 1199--1214.

- Some plethystic identities and Kostka-Foulkes polynomials.

Ars Combinatoria 122 (2015), 411--421.

- Bruhat order on partial fixed point free involutions. (with Yonah Cherniavsky and Tim Twelbeck)

Electronic Journal of Combinatorics 21 (2014), no. 4, Paper 4.34, 23 pp.

- Branching through G2 (with Roger Howe)

Proceedings of the Gökova Geometry-Topology Conference 2013, 41--75, Gökova Geometry/Topology Conference (GGT), Gökova, 2014.

- Calculating Heegaard-Floer homology by counting lattice points in tetrahedra (with Çağrı Karakurt)

Acta Mathematica Hungarica 144 (2014), no. 1, 43--75.

- Lexicographic shellability of partial involutions (with Tim Twelbeck)

Discrete Math. 335 (2014), 66--80.

- SL_2-regular subvarieties of complete quadrics (with Michael Joyce)

Algebraic monoids, group embeddings, and algebraic combinatorics, 271--284, Fields Inst. Commun., 71, Springer, New York, 2014.

- Unipotent invariant matrices (with Roger Howe and Michael Joyce).

Linear Algebra and its Applications, 439 (2013), no.1, 196--210.

- Weak order on complete quadrics (with Michael Joyce)

Transactions of the American Mathematical Society 365 (2013), no. 12, 6269--6282.

- Ordered Bell Numbers, Hermite Polynomials, Skew Young Tableaux, and Borel Orbits (with Michael Joyce).

Journal of Combinatorial Theory, Series A 119 (2012) 1798--1810.

- Bruhat-Chevalley Order on the Rook Monoid (with Lex Renner).

Turkish Journal of Math. no. 36 (2012), 499--519.

- Irreducible representations of semisimple algebraic groups and supersolvable lattices.

Journal of Algebra. 351 (2012), no. 1, 235--250.

- From Parking functions to Gelfand pairs (with K. Aker).

Proceedings of AMS. 140 (2012), 1113--1124.

- Generators of the Hecke algebra of (S_2n,B_n) (with K. Aker).

Advances in Math. 231 (2012), no. 5, 2465--2483.

- Broken Bracelets, Molien Series, Paraffin Wax and Elliptic Curve of Conductor 48 (with T. Amdeberhan and V.Moll).

SIAM J. on Discrete Math. 25(4) (2011), pp. 1843--1859.

- Partitions, rooks, and symmetric functions in noncommuting variables (with Bruce Sagan).

Electronic J. Combinatorics 18 (2011), no. 2, Paper 3.

- R-polynomials of finite monoids of Lie type (with K. Aker and M. Taskin).

Internat. J. of Algebra and Computation, 20 (2010) no. 6, 793--805.

- H-polynomials and Rook Polynomials (with Lex Renner).

Internat. J. of Algebra and Computation 18 (2008), no. 5, 935--949.

- Nested Hilbert schemes and the nested q,t-Catalan series.

Discrete Mathematics and Theoretical Computer Science Proceedings (2008) 61--70.

- A proof of the q,t-square conjecture (with N. Loehr).

Journal of Combinatorial Theory, Series A, 113 (2006), no. 7, 1419--1434.

Books that I edited:

1. Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, Vol 71 (jointly edited with Zhenheng Li, Benjamin Steinberg, Qiang Wang).

2. Algebraic Groups: Structure and Actions. Proceedings of Symposia in Pure Mathematics series of the AMS. Volume 94.

PhD advising (current and past)

I am currently working with

Aram Bingham (expected to graduate in 2020)

Tien Li (expected to graduate in 2019)

Previously I had the fortune of working with

Özlem Uğurlu (graduated - 2018)

Tim Twelbeck (graduated - 2013)

Editorial

I am on the editorial board of Journal of Gökova Geometry Topology .

The Journal of GGT is a fully refereed electronic journal, specializing in articles in Geometry and Topology.

Current Teaching

Algebraic Geometry (G)

Linear Algebra (UG)