# University of Minnesota Combinatorics Seminar

- Seminar meets on Fridays 15:35–16:25 in room 570 of Vincent Hall.
- Seminar announcement list sign-up.

date | speaker | affiliation | title |
---|---|---|---|

Fall 2014 | |||

Sep 12, 2014unusual locationPhys 170 |
William T. Trotter | Georgia Institute of Technology | Duality in the Combinatorics of Posets |

Every 10 years or so, there have been two closely connected theorems in the combinatorics of posets, one for chains and one for antichains. Typically, the statements are exactly the same when roles are reversed, but the proofs are markedly different. And on rare occasions, there has been a case where there is a natural candidate for a dual statement but it turned out not to be true. The classic pair of theorems due to Dilworth and Mirsky were the starting point for this pattern, followed by the more general pair known respectively as the Greene-Kleitman and Greene theorems dealing with saturated partitions. More recently, we have the pair of theorems due to Duffus-Sands and Howard-Trotter dealing with pairwise disjoint families of subsets. And this past spring, a new pair has been discovered dealing with matchings in the comparability and incomparability graphs of a poset. | |||

Sep 19, 2014 | Alexander Woo | University of Idaho | Combinatorics of clans and geometry of B-orbits on GL(n) / (GL(p) x GL(q)) |

Many aspects of the
combinatorics of Weyl groups turn out to be useful in understanding the
geometry of Schubert varieties (which are B-orbit closures) in flag
varieties. For example, Bruhat and weak orders have geometric
interpretations, and the combinatorics of pattern avoidance are known
describe many singularity properties. Kazhdan-Lusztig polynomials,
which capture certain numerical information in complex representation
theory, can be thought of as a singularity invariant of Schubert
varieties and calculated by combinatorial procedures on the Weyl group.
Symmetric spaces play an analogous role in real representation theory through Kazhdan--Lusztig--Vogan polynomials. Symmetric spaces also have finitely many B-orbits, which can be indexed by sets of decorated involutions known as clans, and these indexing sets have analogous combinatorics. At least for the symmetric space GL(n) / (GL(p) x GL(q)), pattern avoidance also seems to describe many singularity properties. I will give an introduction to the combinatorics of clans and talk about some recent work and work in progress using the combinatorics to understand the geometry. This is joint work with Ben Wyser and Alex Yong. | |||

Sep 26, 2014 | Sheila Sundaram | Pierrepont School | Some Open Enumerative and Topological Problems Arising from the Homology of Posets of Partitions |

We describe some questions related to homology representations of subposets of the partition lattice. Although extensively studied, partition posets continue to be a source of interesting open problems, often with tantalising ramifications. The partition lattice and its subposets also serve as a guiding first example of many curious phenomena, both topological and representation-theoretic, that have since been generalised and rediscovered in many other contexts in the last decade. We present some long-standing enumerative conjectures related to the Euler numbers and simsun permutations in the hope of rekindling interest in these problems. | |||

Oct 03, 2014unusual locationPhys 170 |
Jeff Kahn | Rutgers University | On a triangle packing conjecture |

An old conjecture of Zs. Tuza says that in any graph the minimum size of a
triangle cover (set of edges whose removal leaves no triangles) is at most twice
the maximum size of a triangle packing (set of edge-disjoint triangles).
While there are graphs for which equality holds, it's natural to wonder whether
there might typically (whatever that means) be some slack in the inequality.
Here we consider a conjecture of R. Yuster suggesting such an improvement
for dense graphs.
Joint with Jake Baron | |||

Oct 10, 2014 | Damir Yeliussizov | KBTU and IMA | Decomposing digraphs into increasing walks |

Let G be a digraph with distinct edge labels. Consider decompositions
of G into edge-disjoint increasing walks. This setting gives rise to
several algebraic and enumerative applications. For instance, walk
decompositions help to study skew-symmetric polynomials on certain
subspaces of Weyl algebra, similar as Eulerian tours applicable in
Amitsur-Levitski theorem for matrix algebra. We will also discuss some
new types of restricted set partitions and generalized Stirling
numbers. The central idea behind these results is the connection of
walk decompositions and normal ordering of differential operators in
n-th Weyl algebra.
This is a joint work with Askar Dzhumadil'daev. | |||

Oct 17, 2014 | Prasad Tetali | Georgia Institute of Technology | Discrete curvature and what is it good for? |

Inspired by exciting developments in optimal transport and Riemannian geometry (culminating in the work of Lott-Villani and Sturm), several independent groups have formulated a (discrete) notion of curvature in graphs and finite Markov chains. I will describe interesting consequences from a couple of these approaches, including a certain Brunn-Minkowski inequality on the discrete cube and a Buser inequality for graphs. I will also mention some open problems of potential independent interest. | |||

Oct 24, 2014 | Yan Zhang | UC Berkeley | Adinkras: Progress and Problems |

Adinkras are graphical tools created for the study of representations in supersymmetry that turn out to contain many connections with graded posets, the Tutte polynomial, error-correcting codes, and other objects interesting for combinatorialists. In particular, there are many easy-to-state and seemingly accessible open problems. I will present some pleasant results for 1-D supersymmetry and more recent work-in-progress for 2-D supersymmetry, focusing on the combinatorics. In particular, no specialist physics knowledge is required. | |||

Oct 31, 2014 | Emanuele Delucchi | University of Fribourg | Toric arrangements - towards setting up a combinatorial theory |

Recent work of De Concini, Procesi and Vergne on vector partition functions gave a new impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view. In this context, many new discrete structures have appeared in the literature, each describing some aspect of the theory (i.e., either the arithmetic-algebraic one or the topological one) and trying to mirror the combinatorial framework which revolves around arrangements of hyperplanes. I will give a quick overview of the state of the art and, taking inspiration from some recent topological results, I will try to suggest a possible approach towards unifying different objects. | |||

Nov 07, 2014 | Zafeirakis Zafeirakopoulos | University of Athens | Polyhedral Omega: A linear Diophantine system solver |

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. This is joint work with Felix Breuer. | |||

Nov 14, 2014unusual locationPhys 170 |
John Stembridge | University of Michigan | Generalized stability of Kronecker coefficients |

Kronecker coefficients are tensor product multiplicities for symmetric group representations. It is well-known that not much is known about them. In this talk we plan to discuss some new theorems and conjectures about how these coefficients stabilize or de-stabilize in various limiting cases. A typical (easy) case is the classical result of Murnaghan that corresponds to incrementing the first rows of a triple of Young diagrams. | |||

Nov 21, 2014 | Jerrold Griggs | University of South Carolina | Tiling the n-cube Graph with Copies of a Given Graph |

We say that a graph | |||

Nov 28, 2014 | no seminar (Thanksgiving) | ||

Dec 05, 2014 | Tri Lai | IMA | Proof of a Generalization of Aztec Diamond Theorem |

We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a bijection between domino tilings and non-intersecting paths. If time allows, we will go over different proofs of the generalization. | |||

Dec 12, 2014 | no seminar (final exams) | ||

Spring 2015 | |||

Jan 23, 2015 | Jesus De Loera | UC Davis | Affine semigroups and polyhedra with prescribed number of lattice points |

affine semigroup is a semigroup (always containing a zero element) which is finitely generated and can be embedded in
Given an integer matrix
The well-studied semigroup
We also show that, when This is joint work with Iskander Aliev (Cardiff) and Quentin Louveaux (Liege Belgium) | |||

Jan 27, 2015unusual time Tuesday 13:25 |
Jang Soo Kim | SKKU | Multiplex juggling probabilities |

Imagine that a juggler is juggling balls. Let's assume that he catches and throws at most one ball at each discrete beat. At each beat, we can write a 0-1 sequence for which i-th bit is 1 if a ball is going to be landed after i beats, and 0 otherwise. Such a sequence is called a juggling state. Suppose that the juggling can throw a ball at most height h. In other words, when a ball is thrown, it will be landed after at most h beats. Suppose that the juggler throws each ball uniformly randomly such that we still have at most one ball landed at each beat. In 2005, Warrington showed that this random process is a Markov chain with steady state probability and found a formula for the probability. In this talk we will review Warrington's idea and consider multiplex juggling which allows to have more than one ball landed at each beat. | |||

Jan 30, 2015 | Rebecca Patrias | University of Minnesota | Dual filtered graphs |

Using the Hecke insertion algorithm of Thomas and Yong as motivation, we define a K-theoretic analogue of Fomin's dual graded graphs called dual filtered graphs. The key formula in the definition is DU-UD=D+I. We discuss two main constructions of dual graded graphs: the Mobius construction, which corresponds to natural insertion algorithms, and the Pieri construction, which is an algebraic construction. We end with some enumerative results using up-down calculus. | |||

Feb 06, 2015 | Yi Su | University of Michigan | Electrical Networks and Electrical Lie Theory |

Curtis-Ingerman-Morrow studied the space of circular planar electrical networks. Lam and Pylyavskyy introduced an electrical Lie algebra such that the positive part of the corresponding electrical Lie group E_{A_{2n}} acts on circular planar electrical networks via some natural combinatorial operations. Inspired by these operations Lam gave a cell decomposition of the space of electrical networks, which is the analogue of type A Bruhat decomposition. In this talk, I will introduce mirror symmetric circular planar electrical networks, and present some analogous results about this new class of networks and its relation to type B electrical Lie theory. If time permitting, I will talk about the structure of electrical Lie algebras of type C and D. | |||

Feb 13, 2015 | Diana Davis | Northwestern University | Complexity of cutting sequences on translation surfaces |

We can create a surface (such as the square torus) by gluing together edges of polygons, and then study geodesics on the surface. If we write down the polygon edges that a geodesic crosses, it gives us a "cutting sequence." We might ask, for a given surface, which sequences can arise, and what is the complexity of these sequences? I will show how to translate this question about geodesics and polygons into a question about "interval exchange transformations," which are well understood and extensively studied. This helps us to answer the questions about sequences. | |||

Feb 20, 2015 | Olya Mandelshtam | UC Berkeley | Tableaux combinatorics for two-species PASEP probabilities |

The two-species partially
asymmetric exclusion process (PASEP) is a multi-species variation of an
important and well-studied non-equilibrium model from statistical
physics. In the original PASEP, particles hop left and right with on a
finite one-dimensional lattice with open boundaries. New particles can
enter from the left with rate
In the two-species PASEP, there are now two types of particles, one
``heavy'' and one ``light''. Both particles can swap places with
adjacent holes to the right and left at rates 1 and | |||

Feb 27, 2015unusual locationPhys 170 |
Alexander Barvinok | University of Michigan | Complex geometry and computational complexity of the permanent |

The permanent of a square matrix is a polynomial in the matrix entries, which looks similar to the determinant, only simpler: all the monomials are taken with the "+" sign. It turns out that the permanent of a *complex* matrix whose entries are within 0.275 from 1 is never 0 (the constant 0.275 is most certainly non-optimal, it is not clear at the moment by how much it can be improved, though one can show that it cannot be replaced by 0.708, say). This fact allows us to compute the permanent of an nxn real or complex matrix with entries within 0.274 from 1 within a relative error epsilon > 0 in quasi-polynomial n^{O(ln n - ln epsilon)} time. The approach can be readily extended to computing hafnians, multi-dimensional permanents of tensors and other partition functions arising from combinatorial counting problems. | |||

Mar 06, 2015 | Rachel Karpman | University of Michigan | Two families of parametrizations for positroid varieties |

The positroid stratification of the Grassmannian refines the Schubert stratification, and has a rich geometric and combinatorial structure. A parametrization of a positroid variety P is a "nice" birational map from complex d-space to P. There are several remarkable constructions which yield parametrizations of positroid varieties. In this talk, we explore the relationship between two families of such parametrizations. Our first family is defined in terms of Postnikov's theory of planar networks. We focus on a special class of planar networks called bridge graphs, which have applications to particle physics. Our second family arises from Marsh and Rietsch's parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. While the definitions of these two families may appear very different, we show that they encode essentially the same information. | |||

Mar 13, 2015 | Jacob Matherne | Louisiana State University | Combinatorics of exceptional sequences in type A |

Exceptional sequences are certain ordered sequences of quiver representations with applications to noncrossing partitions, factorizations of Coxeter elements, cluster algebras, and the representation theory of algebras. We introduce a class of combinatorial objects called strand diagrams that we use to classify exceptional sequences of representations of type A quivers. We also use variations of the model to classify c-matrices of type A quivers, to interpret exceptional sequences as linear extensions of certain posets, and to give an elementary bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This is part of ongoing work with Alexander Garver, Kiyoshi Igusa, and Jonah Ostroff. | |||

Mar 20, 2015 | no seminar (Spring break) | ||

Mar 27, 2015 | Brendan Pawlowski | University of Minnesota | Cohomology classes of rank varieties and a counterexample to a conjecture of Liu |

Given any diagram (a finite collection of boxes on a grid), one can define an associated symmetric function. In many cases, these symmetric functions contain interesting and nontrivial information related to the diagram: for Young diagrams one obtains Schur functions; for skew diagrams, skew Schur functions; for permutation diagrams, Stanley symmetric functions, which describe reduced words. Liu defined a collection of subvarieties of the Grassmannian indexed by diagrams, and conjectured that their cohomology classes are represented by the corresponding diagram symmetric functions. I will discuss a counterexample to Liu's conjecture. I will also give a connection to Coskun's rank varieties (a special case of Knutson-Lam-Speyer's positroid varieties), and some new results on their cohomology classes. | |||

Apr 03, 2015 | Zachary Hamaker | IMA | Involution Sorting Networks |

The involutions in a Coxeter group | |||

Apr 10, 2015 | Lisa Lamberti | University of Michigan | Geometric model for cluster categories of type E |

Cluster categories provide a categorical framework for studying cluster algebras. In this talk, I will give a geometric description of cluster categories associated to Dynkin diagrams of type E by considering colored oriented diagonals in certain polygons. I will also discuss how this approach helps to understand cluster tilting objects of cluster categories of type E. | |||

Apr 17, 2015 | Steve Butler | Iowa State and IMA | Edge flipping on the complete graph |

Given a graph with vertices colored red and blue we can
consider the following process on the state graph of all possible
colorings: Pick an edge of the graph and with probability This is joint work with Fan Chung, Jay Cummings and Ron Graham. | |||

Apr 24, 2015 | John Wiltshire-Gordon | University of Michigan | Representation theory of combinatorial categories |

Given an interesting sequence of objects X_0, X_1, X_2, ... , it's fun to ponder: is that subscript just a natural number, or is it hiding deeper structure? Maybe these objects ought to be indexed by finite sets, or finite dimensional vector spaces, or finite total orders, or whatever makes sense. The point is, a map relating two of the indexing gadgets should induce a map on the objects themselves. We use this method of argument to give new results on the cohomology of configuration spaces. Next, we give a characterization of indexing categories wherein finitely generated representations are finite length. Finally, we show how computations with these representations can be made effective. Part of this talk is joint work with Jordan Ellenberg. | |||

May 01, 2015 | Gregory Muller | University of Michigan | Twists for positroid cells |

A bipartite graph in a disc determines a subvariety of a Grassmannian called a `positroid cell', as well as two maps.
1) Postnikov's `boundary measurement map' from an algebraic torus to the positroid cell, which is a kind of generating function of perfect matchings. 2) A `cluster': a special collection of Plucker coordinates which collectively define a rational map from the positroid cell to an algebraic torus. Despite closely linked constructions and origins, the exact relation between these two maps has been an open question since Postnikov's original unpublished manuscript featured a missing chapter on the subject. Joint with David Speyer, we introduce a simple `twist' automorphism of each positroid cell, and prove that the two maps are `inverses up to the twist'. This provides simple proofs of several open questions about these maps, as well as a generalization of the `Chamber Ansatz' of Berenstein-Fomin-Zelevinsky. | |||

May 08, 2015 | Nathan Ilten | Simon Fraser University | Counting lines on toric surfaces |

In 1849, Cayley and Salmon proved that any smooth cubic surface contains exactly 27 lines, marking the start of enumerative algebraic geometry. In this talk, I show how to count (with multiplicity) the number of lines on a special class of surfaces, namely, toric surfaces. Such a surface may be encoded combinatorially by a lattice polytope, and the number of lines contained in the surface can be read off in a straightforward fashion from the corresponding polytope. Using degeneration arguments, one may even recover the classic count of 27 lines on a smooth cubic surface. This talk will assume no background in toric or enumerative geometry and will provide a gentle introduction to both subjects. | |||

May 15, 2015 | no seminar (final exams) | ||

May 22, 2015unusual time 13:25 |
Tom Roby | University of Connecticut | Birational Rowmotion: order, homomesy, and cluster connections |

Within dynamical algebraic combinatorics, an action of particular interest on the set of
order ideals of a finite poset rowmotion (aka the "Fon-der-Flaass map" aka
"Panyushev complementation"). Various surprising properties of rowmotion
have been exhibited in work of Brouwer/Schrijver, Cameron/Fon der Flaass, Panyushev,
Armstrong/Stump/Thomas, Striker/Williams, and Propp/R. For example, its order
is Recent work of Einstein/Propp generalizes rowmotion twice: first to the piecewise-linear setting of a poset's "order polytope", defined by Stanley in 1986, and then via detropicalization to the birational setting.
In these latter settings, generalized rowmotion no longer has finite order in the general
case. Results of Grinberg and the speaker, however, show that it still has order | |||

May 22, 2015unusual time 14:30 |
Henri Mühle | Paris Diderot University | Generated Groups, Shellability, and Transitivity of the Hurwitz Action |

Let G be a group generated by a conjugation closed set A. There is a natural action of the braid group on k strands on the set of reduced A-decompositions of any group element of length k, the Hurwitz action. It can informally be described as "shifting letters to the right, and conjugating as you go". Moreover, in the given setting we can naturally define a subword order on G, and it is immediate that the reduced A-decompositions of any group element are in bijection with the maximal chains in the principal order ideal generated by this element. Using this perspective we present a new approach to proving the transitivity of the Hurwitz action, in that we establish a connection between the shellability of this subword order and the Hurwitz transitivity. This work (which is joint work with Vivien Ripoll from the University of Vienna) culminates in the observation that these two properties, whose proofs are in general far from being trivial, follow from a simple local criterion, namely the existence of a well-behaved total order of A. | |||

May 26, 2015unusual time Tuesday 15:35 |
Sebastian Franco | City College of New York | Bipartite Field Theories: Quivers, Graphs and Calabi-Yau Manifolds |

Over the last decade, we have witnessed remarkable progress in our understanding of Quantum Field Theories (QFTs). A common theme in many of the most recent developments has been the definition of QFTs in terms of some underlying geometric or combinatorial objects. In this seminar I will discuss Bipartite Field Theories (BFTs), a new class of QFTs embodying this general approach. BFTs are quiver gauge theories that are defined by bipartite graphs on Riemann surfaces. Remarkably, they underlie a wide spectrum of interesting systems, including: D-branes probing Calabi-Yau manifolds, their mirror configurations, integrable systems and scattering amplitudes. I will introduce new techniques for studying these quiver theories. I will explain how their dynamics is captured graphically and the interesting emergence of concepts such as Calabi-Yau manifolds, the Grassmannian and cluster algebras in their classification. |

- The seminar is organized by Gregg Musiker and Jed Yang; this webpage is maintained by Jed Yang.
- Past seminar archive.
- Combinatorics Problem Session, suspended indefinitely.

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