Mahir Bilen Can

Email: mahirbilencan at gmail dot com Vita

## 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.

## PhD advising.

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)

## I have edited two proceedings.

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.

## Research.

Note: Most of these papers (in some form of their existence) are available on the arXiv, however, only their published versions are final.

- 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.

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

Let G/H be a homogeneous space, and let X be a G-equivariant embedding of G/H. We describe a filtration on the equivariant Borel-Moore homology of X.

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

We study the Levi subgroup actions on Schubert varieties. In particular we focus on the toroidal actions of Levi subgroups and classify the toroidal Schubert varieties in a Grassmann variety. By using Billey-Postnikov decompositions, we find a criterion to see if a Schubert variety in a partial flag variety is a wonderful Schubert variety or not. Here, wonderful means smooth and toroidal.

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

We study the intertwining statistic (of Ehrenborg and Readdy) on set partitions by interpreting it as the function that gives the codimensions of Borel subgroup orbits in the algebraic semigroup of strictly upper triangular matrices.

- 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.

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

Let X_{wQ} be a 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.

- Stirling Posets (with Yonah Cherniavsky). (Last updated: June 8, 2018.)

We transfer the Bruhat-Chevalley-Renner order (BCR order) on the monoid of upper triangular matrices to the set partitions. We define the notion of a "Stirling poset" by restricting the BCR order to the set partitions with k blocks.

- 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.

- 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.

- The rook monoid is lexicographically shellable.

We show that the Bruhat-Chevalley-Renner order on the rook monoid is lexicographically shellable.

- 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.

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

Discrete Mathematics 342 (2019) 201--216.

By finding its recurrence formula, we count the number of Borel orbits in SL_n/ S(SL_pxSL_q). We find a set of lattice paths that is in bijection with the set of Borel orbits in this symmetric variety.

- Classification of Reductive Monoid Spaces Over an Arbitrary Field.

To appear inProceedings of the 2017 Southern Regional Algebra Conference, Mobile.

We review the recent work of Wedhorn on "spherical spaces". We introduce the notion of a reductive monoid space. We classify the normal reductive monoid spaces by using their spherical datum.

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

To appear inColloquium Mathematicum.

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

To appear inAlgebraic Combinatorics.

- 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.

## Some of my recent presentations.

1. AMS Sectional Meeting at Northeastern University 2. AMS Sectional Meeting at UC Riverside

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