This is a two-part event. Applicants should ensure they will be able to participate in both parts and collaborate with their groups in between the meetings. Please note: August 5 - 6, 2021 is virtual participation and February 10 - 11, 2022 is in-person at ICERM.
Organizing Committee
Abstract

The Women in Algebraic Combinatorics Research Community will bring together researchers at all stages of their careers in algebraic combinatorics, from both research and teaching-focused institutions, to work in groups of 4-6, each directed by a leading mathematician. The goals of this program are: to advance the frontiers of cutting-edge algebraic combinatorics, including through explicit computations and experimentation, and to strengthen the community of women working in algebraic combinatorics.

Successful applicants will be assigned to a group based on their research interests. The groups will work on open problems in algebraic combinatorics and closely related areas, including representation theory, special functions, and discrete geometry. Several of the proposed projects will extensively involve experimentation and computation, which will increase the likelihood that concrete progress is made over the course of the initial workshop and following 6 months, and provide useful training in computational mathematics.

In their personal statements, applicants should rank in order their top three choices of projects. They should also address their familiarity with the suggested prerequisites. Applicants are expected to attend the opening and closing workshops and meet regularly with their research group for the 6 months in between.

To be considered for this workshop please apply by July 5, 2021.

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Confirmed Speakers & Participants

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Thursday, August 5, 2021
  • 9:45 - 10:00 am EDT
    Welcome
    Virtual
    • Brendan Hassett, ICERM/Brown University
  • 10:00 - 10:20 am EDT
    Opening Remarks
    Virtual
    • Susanna Fishel, Arizona State University
    • Pamela E. Harris, Williams College
    • Rosa Orellana, Dartmouth College
    • Stephanie van Willigenburg, University of British Columbia
  • 10:20 - 10:40 am EDT
    Project # 4 Introduction - Alternating Sign Matrices and Plane Partitions
    Virtual
    • Ilse Fischer, Universität Wien
  • 10:50 - 11:10 am EDT
    Project # 2 Introduction - Row-strict Dual Immaculate Functions
    Virtual
    • Elizabeth Niese, Marshall University
  • 11:10 - 11:30 am EDT
    Break
    Coffee Break - Virtual
  • 11:30 - 11:50 am EDT
    Project # 3 Introduction - Perspectives on Permutations
    Virtual
    • Bridget Tenner, DePaul University
  • 11:50 am - 1:00 pm EDT
    Lunch/Free Time
    Virtual
  • 1:00 - 1:20 pm EDT
    Project # 1 Introduction - Investigating Convex Union Representable Complexes
    Virtual
    • Isabella Novik, University of Washington
  • 1:30 - 1:50 pm EDT
    Project #5 Introduction - Combinatorics of Convex Polytopes
    Virtual
    • Margaret Bayer, University of Kansas
  • 2:00 - 2:20 pm EDT
    Break
    Coffee Break - Virtual
  • 2:20 - 2:40 pm EDT
    Project # 6 Introduction - Enumerative Combinatorics with Filling of Polyominoes
    Virtual
    • Catherine Yan, Texas A&M University
  • 3:00 - 4:00 pm EDT
    Project Group Meetings
    Group Work - Virtual
  • 4:00 - 5:00 pm EDT
    Gathertown Reception
    Reception - Virtual
Friday, August 6, 2021
  • 10:00 - 10:20 am EDT
    Project # 7 Introduction - Dynamical Algebraic Combinatorics
    Virtual
    • Jessica Striker, North Dakota State University
  • 10:30 - 10:50 am EDT
    Project # 8 Introduction - Higher Dimensional Chip-Firing
    Virtual
    • Caroline Klivans, Brown University
  • 11:00 - 11:20 am EDT
    Break
    Coffee Break - Virtual
  • 11:20 - 11:40 am EDT
    Project # 9 Introductions - McKay Matrices for Super Objects and Connections with Characters
    Virtual
    • Georgia Benkart, University of Wisconsin-Madison
  • 11:50 am - 1:00 pm EDT
    Lunch/Free Time
    Virtual
  • 1:00 - 1:20 pm EDT
    Project # 10 Introduction - The A, C, Shifted Berenstein-Kirillov Groups and Cacti
    Virtual
    • Olga Azenhas, University of Coimbra
  • 1:30 - 1:50 pm EDT
    Project # 11 Introduction - Combinatorial dimension formulas for algebraic splines
    Virtual
    • Julianna Tymoczko, Smith College
  • 2:00 - 2:20 pm EDT
    Break
    Coffee Break - Virtual
  • 2:30 - 4:30 pm EDT
    Project Group Meetings
    Group Work - Virtual
  • 4:30 - 5:00 pm EDT
    Closing Remarks
    Virtual
    • Susanna Fishel, Arizona State University
    • Pamela E. Harris, Williams College
    • Rosa Orellana, Dartmouth College
    • Stephanie van Willigenburg, University of British Columbia

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

All event times are listed in .

Application Information

ICERM welcomes applications from faculty, postdocs, graduate students, industry scientists, and other researchers who wish to participate. Some funding may be available for travel and lodging. Graduate students who apply must have their advisor submit a statement of support in order to be considered.

Applications are not currently open. Please check back at a later date.

Your Visit to ICERM

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Project Descriptions

Project 1: Investigating Convex Union Representable Complexes, Isabella Novik, University of Washington

An abstract simplicial complex is called d-representable if it is the nerve of a family of convex sets in the d-dimensional Euclidean space. The research on d-representable complexes has a rich and fascinating history, starting with Helly’s theorem, that to a large degree shaped the modern discrete geometry.

A simplicial complex is d-convex union representable (or d-CUR, for short) if it arises as the nerve of a finite collection of convex open sets in R^d whose union is also convex. This is a fascinating and very natural subclass of representable complexes. In addition, the motivation for investigating such complexes comes from the theory of convex neural codes. While the class of d-representable complexes is well- and long-studied, the notion of d-CUR complexes was introduced only a couple of years ago. Consequently, our present knowledge about this class of complexes is almost null. For instance, while we know that CUR complexes are collapsible and so are their Alexander duals, we do not know the answers to such questions as: is the barycentric subdivision of a CUR complex also CUR? Is every shellable simplicial ball CUR? We are also very short on techniques allowing to construct many such complexes.

The goal of the project will be to further explore the class of CUR complexes and probe some of the open problems.

Recommended background:

  • Some knowledge of discrete geometry and polytopes

References

Aaron Chen, Florian Frick, and Anne Shiu. Neural codes, decidability, and a new local obstruction to convexity, SIAM J. Appl. Algebra Geom. 3 (1), 44−66 (2019)

Amzi Jeffs and Isabella Novik. Convex union representability and convex codes, Int. Math. Res. Not. IMRN, 2021, no. 9, 7132-7158

Martin Tancer. Intersection patterns of convex sets via simplicial complexes: a survey. In Thirty essays on geometric graph theory, pages 521–540. Springer, New York, 2013.

Project 2: Row-strict Dual Immaculate Functions, Elizabeth Niese, Marshall University

There are a number of Schur-like bases for the space of quasisymmetric functions with combinatorial definitions reminiscent of the semi-standard tableaux definition of symmetric Schur functions. One such basis, the dual immaculate functions [2], can be defined as the generating function of a set of composition tableaux of shape α where the first column of α is filled with positive integers so that it is increasing and each row is weakly increasing. This basis expands positively in the Young quasisymmetric Schur basis [1]. Just as the symmetric Schur functions can be defined as the generating function of row-strict tableaux, with weakly increasing columns, there is a row-strict quasisymmetric Schur basis [3]. In this project we will define a row-strict version of the dual immaculate functions and then explore their combinatorics and algebra. Some directions for this research include finding an expansion in terms of the row-strict quasisymmetric Schur functions, identifying Pieri rules, and finding 0-Hecke modules with the row-strict dual immaculate functions as their quasisymmetric characteristic.

References

[1] Edward Allen, Joshua Hallam, and Sarah Mason, “Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions,” J. Combin. Theory Ser. A 157 (2018), 70–108.

[2] Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano, and Mike Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canad. J. Math. 66 (2014), no. 3, 525–565.

[3] Sarah Mason and Jeffrey Remmel, Row-strict quasisymmetric Schur functions, Ann. Comb. 18 (2014), no. 1, 127–148.

Project 3: Perspectives on Permutations, Bridget Tenner, DePaul University

Combinatorialists encounter permutations in many settings, two of which will be the foundation of this group's work. First, the symmetric group is the finite Coxeter group of type A, and this consists of all permutations of a given size. As a Coxeter group, this can be endowed with a natural poset structure (the Bruhat order), and many interesting mathematical questions -- both answered and unanswered -- can be posed about this structure. Second, one can look at patterns occurring within permutations, examining both the characterizing and the enumerative properties that result from such occurrences. A permutation pattern is typically phrased as a binary property: containment or avoidance. However, recent evidence has shown that a higher-order approach can provide an important and influential framework for understanding various phenomena, with the question being a count of how many times a permutation contains a pattern, or in what ways that containment occurs. Connections between the Coxeter-perspective and the pattern-perspective have already been established, and this project aims to extend that relationship. Objects we will focus on may include ideals in the Bruhat order, commutation classes of reduced decompositions, and the RSK correspondence.

Project 4: Alternating Sign Matrices and Plane Partitions, Ilse Fischer, Universität Wien

In this project, we will focus on the mysterious relation between alternating sign matrices and plane partitions. Alternating sign matrices had been introduced in the 1980's and it was soon discovered (conjecturally) that they are counted by the same simple product formula as two classes of plane partitions, namely descending plane partitions and totally symmetric self-complementary plane partitions. Very recently, a fourth class of objects (alternating sign triangles) has been added to the list of objects that are counted by this product formula. There have been many efforts to construct bijective proofs, but so far there only exists a very complicated (recent) construction that proves bijectively an identity that implies the equinumerosity of alternating sign matrices and descending plane partitions.

We have two concrete projects in mind. Alternating sign matrices include permutation (matrices) as a special case, and for this special case bijections have been constructed in the work of Ayyer, of Fulmek and of Striker. The first possibility is to extend this to the next level of complication, by introducing a small number of -1's. Recent work of Höngesberg [3] actually does include -1's for alternating sign triangles, so that this work will be a good starting point. The other project is around unpublished work (at the moment) concerning n+3 pairs of equivalent statistics on two new types of objects that extend alternating sign matrices and descending plane partitions, respectively. That's exciting because for alternating sign matrices and descending plane partitions only four such statistics are currently known.

References

[1] David Bressoud, Proofs and confirmations. The story of the alternating sign matrix conjecture. MAA Spectrum. Mathematical Association of America and Cambridge University Press, Washington, DC and Cambridge, 1999.

[2] Ilse Fischer and Matja{\v z} Konvalinka. The mysterious story of square ice, piles of cubes, and bijections. Proceedings of the National Academy of Sciences, 117(38):23460–23466, 2020.

[3] Hans H{\"o}ngesberg, Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids, arXiv:2102.07555v1.

Project 5: Combinatorics of Convex Polytopes, Margaret Bayer, University of Kansas

The study of the combinatorics of convex polytopes has a long and interesting history, with connections to operations research, algebra and algebraic geometry. There are many open questions in the field; this project will focus on enumeration questions. The face vector counts the number of faces of each dimension; the flag vector counts the numbers of chains of faces (ordered by inclusion) of specified dimensions. Both are characterized for 3-dimensional polytopes, but there is not even a conjectured characterization for general 4-dimensional polytopes. The affine spans of the face vectors and flag vectors of polytopes of dimension d are known. Many inequalities are known. Characterization of both sets of vectors are known for simplicial polytopes, and some results are known for other classes of polytopes. Methods come from combinatorial analysis, but also by applying results in algebra, algebraic geometry, and algebraic topology to polytopes. Depending on the participants' interests and expertise, the project will focus either on a particular class of polytopes or on particular methods.

References

Ron Adin, Daniel Kalmanovich, Eran Nevo, On the cone of f-vectors of cubical polytopes, Sem. Lothar. Combin. 80B (2018), Art. 85, arXiv:1801.00163v2

Louis J. Billera, "Even more intriguing, if rather less plausible..." Face numbers of convex polytopes, in: The Mathematical Legacy of Richard P. Stanley, 65-81, AMS, 2016.

Eran Nevo, Complexity yardsticks for f-vectors of polytopes and spheres, Discrete Comput. Geom. 64 (2020), no. 2, 347-354, arXiv:1908.09628

Isabella Novik, Hailun Zheng, The stresses on centrally symmetric complexes and the lower bound theorems, arXiv:2008.12503

Lei Xue, A proof of Grunbaum's lower bound conjecture for general polytopes, arXiv:2004.08429

Project 6: Enumerative Combinatorics with Filling of Polyominoes, Catherine Yan, Texas A&M University

A polyomino is a finite subset of Z2 , where each element is represented by a square cell. The combinatorial model is obtained by taking a connected polyomino and assigning a non-negative integer to each cell. By considering different shapes, convexity, symmetry, and content of the filling, we get a unified model that contains a hierarchy of combinatorial structures. Explicitly, we would consider the following families of polyominoes, including

  1. square or rectangular shaped polyominoes whose fillings correspond to permutations and words;
  2. Ferrers diagrams which are polyominoes given by an integer partition. Fillings of Ferrers diagrams correspond to matchings, set partitions, and general graphs;
  3. stack polyominoes which are convex, intersection-free, and whose rows are arranged monotonically;
  4. moon polyominoes which are convex and intersection-free;
  5. more general shape of polyominoes with weaker convex properties.

For fillings, we consider two basic cases: the one using 0 and 1, and the one using arbitrary natural numbers. These fillings, together with the aforementioned polyominoes, allow us to use combinatorial operations and transformations to investigate special characteristics of each family. It is particularly convenient to study patterns and combinatorial statistics.

Another approach to fillings of polyominoes is to apply the tools developed in the study of symmetric functions and standard Young tableaux, for example, the Robinson-Schensted-like insertion/deletion processes, growth diagram, and the Knuth equivalence. Such techniques are particularly powerful in dealing with the maximal monotone substructures, such as increasing and decreasing subsequences in integer sequences, and the maximal crossings and nestings in various diagrams.

The objective of this project is to understand the interplay between the combinatorial transformations and the tableaux operations, and to characterize various combinatorial statistics in different families of fillings of polyominoes.

Background needed: Basic techniques in enumerative combinatorics; be familiar with permutation statis- tics. A basic knowledge of symmetric functions, STYs, and the RSK algorithm would be helfpul, (as in Stanley’s EC2.)

Project 7: Dynamical Algebraic Combinatorics, Jessica Striker, North Dakota State University

Do you have a favorite set of combinatorial objects S with a natural bijective action f:S->S? Then come investigate its dynamics with us!

We'll begin the project by discussing everyone's favorite objects and actions in the context of the questions listed below. We will then do some computational exploration, as described below each question. Based on the computations, we will concentrate on studying the most promising actions.

Questions to ask about your favorite objects and actions:

  • Does your action have a predictable order?
    • If so, see if it exhibits the cyclic sieving phenomenon [5] (can predict orbit structure by evaluating a generating function at a certain root of unity).
    • If not, see if it exhibits resonance [2] (projects to another object/action pair with a predictable order).
  • Does your object have important statistics?
    • If so, take a look at how those statistics interact with the action. It may be that they exhibit homomesy [4] (same average value on all orbits).
  • Is your set of objects equivalent to a set of subsets?
    • If so, study its generalized toggle group actions (some examples of this are given in [9, Section 3]).
  • Is your action equivalent to the composition of small local actions?
    • If so, characterize it as a toggle group action [10].

This project will involve computational aspects, so programming experience would be helpful. But it will also be a great way to develop your programming skills in a supportive environment, so prior experience is not necessary. References [8] and [6] are survey articles that give an overview of the area.

References

[1] J. Bernstein, J. Striker, and C. Vorland. P-strict promotion and B-bounded rowmotion, with applications to tableaux of many flavors. https://arxiv.org/abs/2012.12219 (submitted).

[2] K. Dilks, O. Pechenik, and J. Striker. Resonance in orbits of plane partitions and increasing tableaux. J. Combin. Theory Ser. A, 148 (2017), 244-274.

[3] K. Dilks, J. Striker, and C. Vorland. Rowmotion and increasing labeling promotion. J. Combin. Theory Ser. A, 164 (2019), 72-108.

[4] J. Propp and T. Roby. Homomesy in products of two chains. Electron. J. Combin., 22 (2015), no. 3.

[5] V. Reiner, D. Stanton, and D. White. The cyclic sieving phenomenon. J. Combin. Theory Ser. A, 108 (2004), no. 1, 17-50.

[6] T. Roby. Dynamical Algebraic Combinatorics and the Homomesy Phenomenon. A. Beveridge, J. Griggs, L. Hogben, G. Musiker, P. Tetali, eds., Recent Trends in Combinatorics (IMA Volume in Mathematics and its Applications), Springer, 2015.

[7] J. Striker and N. Williams. Promotion and rowmotion. Eur. J. Combin., 33 (2012), no.8, 1919-1942.

[8] J. Striker. Dynamical algebraic combinatorics: promotion, rowmotion, and resonance. Notices of the American Mathematical Society, 64 (2017), no. 6, 543-549.

[9] J. Striker. Rowmotion and generalized toggle groups. Discret. Math. Theor. Comput. Sci., 20 (2018), no. 1.

[10] J. Striker. The toggle group, homomesy, and the Razumov-Stroganov correspondence. Electron. J. Combin., 22 (2015), no. 2.

Project 8: Higher Dimensional Chip-Firing, Caroline Klivans, Brown University

Chip-firing processes are combinatorial dynamical systems. Traditionally the dynamics are envisioned on a graph or network. A commodity (chips, sand, dollars) is exchanged between sites of a network according to simple local rules. Although governed by local rules, chip-firing systems result in compelling global behavior.

More recent work has sought to broaden the domain of chip-firing to other settings such as root systems, M-matrices and matroids. This project will focus on chip-firing in higher-dimensions. Higher dimensional chip-firing can be thought of as a flow-rerouting process on cellular complexes. A commodity (now thought of as a flow) is diffused along the faces of a cell complex. The diffusion is dictated by the structure of the complex or equivalently by a combinatorial Laplacian. We will investigate questions about the behavior of such systems, related for example to stabilization, confluence and expansion.

Preferred background: Comfort with basic programming in a language such as matlab or python will be helpful but is not necessary. Someone(s) with skills in visualization would be excellent.

References

A. Duval, C. Klivans, and J. Martin, Critical groups for simplicial complexes. Annals of Combinatorics vol. 17, Issue 1, 2013.

P. Felzenszwalb and C. Klivans, Flow-firing processes. Journal of Combinatorial Theory A, vol. 177, 2021.

C. Klivans, The Mathematics of Chip-firing, Publisher: CRC Press. 2018.

Project 9: McKay Matrices for Super Objects and Connections with Characters, Georgia Benkart, University of Wisconsin-Madison

The McKay matrix records the result of tensoring the finite-dimensional simple modules with a fixed finite-dimensional module V. In the case of a finite group G and its complex modules, the columns of the character table of G give a complete set of right eigenvectors for the McKay matrix determined by any finite-dimensional G-module V. The corresponding eigenvalues depend on V and are the values of the character of V on a set of conjugacy class representatives of G. This project involves investigating the analogous situation for supergroups, for small Lie superalgebras such as gl(1,1) and osp(1,2), and for their quantum analogues defined using roots of unity.

The centralizer algebra of transformations that commute with the action of the (super)group or (super)algebra on tensor powers of the module V often have significant connections with diagram algebras and braid groups. Another goal is to investigate the centralizer algebras that arise from studying these super objects and their representations.

Project 10: The A, C, Shifted Berenstein-Kirillov Groups and Cacti, Olga Azenhas, University of Coimbra

This project studies a symplectic version of the Berenstein-Kirillov group, BKC, and its relationship with the combinatorial internal action of the cactus group Jsp2n, constructed by Halacheva [HKRW20] using the Dynkin diagram I = [n] of the root system in type Cn, on a normal crystal of type Cn via partial Sch ̈utzenberger-Lusztig involutions.

We focus on the type Cn crystals of Kashiwara-Nakashima tableaux [KN94, L07]. They are compatible with a plactic monoid, bumping algorithm and symplectic jeu de taquin [L07]. Considering KN = KN(λ, n) to be the crystal of Kashiwara-Nakashima tableaux of straight shape λ on the alphabet [±n], a full evacuation algorithm [S21], whose construction depends on the long element of the Weyl group W = Bn (the hyperoctahedral group), also exists and agrees with the Lusztig involution on the set KN. For 1 ≤ q < n, the crystal KNq, the restriction of the crystal KN to the connected subdiagram [1, q] of the Dynkin diagram I, is of type Aq with Weyl group Sq+1. Partial evacuation algorithms agreeing with the Lusztig involution on the set KNq are not known. The action of the cactus group Jsp2n on KN makes manifest the action of the Weyl group Bn on that crystal but symplectic Bender-Knuth involutions for Kashiwara-Nakashima tableaux are not known. However, the action of certain elements of the cactus group Jn = Jgln on a straight shape semistandard Young tableau (or Gillespie-Levinson-Purbhoo shifted tableau) crystal is (shifted) Bender- Knuth like [BK95, R21]. We hope that a similar behaviour of the action of certain elements in the cactus Jsp2n on KN gives an indication on what a symplectic Bender Knuth involution for the Kashiwara-Nakashima tableaux should be.

The Berenstein-Kirillov group or Gelfand-Tsetlin group, BK, studied by Berenstein and Kirillov [BK95], is the free group generated by the Bender-Knuth involutions, modulo the relations they satisfy on semistandard Young tableaux of any shape. Fixing the subgroup BKn, it has an alternative set of generators whose action on the straight shaped semistandard Young tableau crystal SSYT(λ, n) agrees with the partial Sch ̈utzenberger-Lusztig involutions, and, therefore, coincides with the internal action of the cactus group Jn on SSYT(λ, n), studied by Halacheva [H20]. Chmutov, Glick and Pylyavskii [CGP20] also related the Berenstein-Kirillov group and the cactus group Jn using semistandard growth diagrams, and determined implications between subsets of relations in the two groups, which yields a presentation for cactus groups Jn in terms of Bender-Knuth generators. In the vein of Halacheva and Chmutov-Glick-Pylyavskii, a shifted version of the Berenstein-Kirillov group has been studied in [R21] considering shifted Bender-Knuth involutions and an action of the cactus group Jn on a Gillespie-Levinson-Purbhoo shifted tableau crystal [GLP17]. A similar presentation of the cactus group Jsp2n in terms of symplectic Bender-Knuth generators also makes sense to be explored.

Suggested background: Combinatorics of crystal graphs and tableau combinatorics along the lines of D. Bump and A. Schilling’s book ”Crystal Bases. Representations and Combinatorics”.

References

[BK95] A. D. Berenstein and A. N. Kirillov. Groups generated by involutions, Gelfand–Tsetlin patterns, and combinatorics of Young tableaux, Algebra i Analiz, 7, 1995, 1, 92–152.

[CGP20] M. Chmutov and M. Glick and P. Pylyavskyy. The Berenstein-Kirillov group and cactus groups, J. Combinatorial Algebra, 2020,4,2, 111–140, arXiv:1609.02046v2.

[GLP17] M. Gillespie, J. Levinson. Kevin Purbhoo. A crystal-like structure on shifted tableaux, J. Algebraic Combinatorics, 3, 2020, 693–725, arXiv:1706.09969.

[H20] I. Halacheva. Skew Howe duality for crystals and the cactus group, arxiv 2001.02262v1.

[HKRW20] I. Halacheva and J. Kamnitzer and L. Rybnikov and A. Weeks. Crystals and monodromy of Bethe vectors, 169, Duke Math. Journal, 12, 2337 – 2419, 2020.

[KN94] M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra, 165, 2 (1994), pp. 295–345.

[L07] C. Lecouvey. Combinatorics of crystal graphs for the root systems of types An, Bn, Cn, Dn, G2, in Combinatorial Aspects of Integrable Systems, MSJ Memoirs vol 17, 11–41, 2007.

[R21] I. Rodrigues. A shifted Berenstein-Kirillov group and the cactus group, arXiv:2104.11799.

[S21] J. M. Santos. Symplectic keys and Demazure atoms in type C, to appear in Electron. J. Comb, 2021, arXiv:1910.14115v3.

Project 11: Combinatorial dimension formulas for algebraic splines, Julianna Tymoczko, Smith College

Splines are a fundamental tool in applied mathematics and analysis, classically described as piecewise polynomials on a combinatorial decomposition of a geometric object (a triangulation of a region in the plane, say, or a convex polytope) that agree up to a specified differentiability on faces of codimension one. Generalized splines extend this idea algebraically and combinatorially: instead of certain classes of geometric objects, we start with an arbitrary combinatorial graph; instead of labeling faces with polynomials, we label vertices with elements of an arbitrary ring; and instead of applying degree and differentiability constraints, we require that the difference between ring elements associated to adjacent vertices be a multiple of a fixed ring element labeling the edge. Billera showed that these two characterizations coincide in most cases of real-world interest.

In this project, we’ll try to make headway on the so-called upper bound conjecture from analysis, which asks for the dimension of the collection of splines on certain triangulations in the plane. We’ll think about generalized splines

  1. on planar trivalent graphs
  2. whose edges are labeled by two-variable polynomials of the form (ax+by)^2
  3. and whose vertices are labeled by polynomials of degree at most two

Our goal is to conjecture and/or prove formulas for the dimension of the space of splines in as special or general context as we can, whether in terms of combinatorial features of the graph, or for certain families of trivalent graphs, or by developing algorithms that crawl through the graphs in particular ways or using other strategies that we come up with.

References

Gilbert, Tymoczko, Viel. Generalized splines on arbitrary graphs. Pacific Journal of Math, Vol. 281 (2016), No. 2, 333-364.

Anders, Crans, Foster-Greenwood, Mellor, Tymoczko. Graphs admitting only constant splines. Pacific Journal of Math, Vol. 304 (2020), No. 2, 385-400.