Some title

Some Person

USomething

Wednesday

Sept. 3, 2014

8:00-8:50am

Vincent Hall 313

Some Abstract. $e^x$

TBA

No seminar- 1st week of class

Friday

Sept. 7, 2018

3:35-4:25pm

Vincent Hall 570

No seminar- 1st week of class

Friday

Sept. 6, 2019

3:35-4:25pm

Vincent Hall 570

No seminar - AMS Meeting in Madison

Friday

Sept. 13, 2019

3:35-4:25pm

Vincent Hall 570

Cyclic sieving for plane partitions and symmetry

Sam Hopkins

UMN

Friday

Sept. 20, 2019

3:35-4:25pm

Vincent Hall 570

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count fixed points for a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by studying the interaction of promotion with symmetries of plane partitions. We obtain cyclic sieving-like formulas in this context where the relevant polynomial is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews. We then go on to consider the way the symmetries interact with rowmotion, another operator acting on plane partitions which is closely related to promotion. We end by explaining the connection of our work to some earlier conjectures we made concerning rowmotion acting on the P-partitions of various $``$triangular$''$ posets P.

A positivity phenomenon in Elser's Gaussian-cluster percolation model

Galen Dorpalen-Barry

UMN

Friday

Sept. 27, 2019

3:35-4:25pm

Vincent Hall 570

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call $\textit{ Elser numbers}$ $\text{els}_k(G)$, where $G$ is a connected graph and $k$ a nonnegative integer. Elser had proven that $\text{els}_1(G)=0$ for all $G$. By interpreting the Elser numbers as Euler characteristics of appropriate simplicial complexes called $\textit{nucleus complexes}$, we prove that for all graphs $G$, they are nonpositive when $k=0$ and nonnegative for $k\geq2$. The last result confirms a conjecture of Elser. At the end, we will present an open problem naturally arising from our proof of Elser's conjecture.

Higher Cluster Categories and QFT Dualities

Gregg Musiker

UMN

Friday

Oct. 4, 2019

3:35-4:25pm

Vincent Hall 570

We present a unified mathematical framework that elegantly describes minimally supersymmetric gauge theories in various dimensions, and their dualities. Though this approach utilizes higher Ginzburg algebras and higher cluster categories (also known as m-cluster categories), we show that the constructions can be given explicitly and combinatorially. We emphasize the connections to cluster algebras and two classes of examples: one class related to toric geometry and giving rise to brane bricks, which are 3-dimensional analogues of certain bipartite graphs on surfaces, and one class corresponding to higher partial triangulations of surfaces. This is based on joint work with Sebastian Franco of City College of New York. No prior knowledge of Quantum Field Theories or Cluster Algebras will be assumed.

Kazhdan-Lusztig Immanants for $k$-Positive Matrices

Sunita Chepuri

UMN

Friday

Oct. 11, 2019

3:35-4:25pm

Vincent Hall 570

Immanants are matrix functionals that generalize the determinant. One notable family of immanants are the Kazhdan-Lusztig immanants. These immanants are indexed by permutations and are defined as sums involving Kazhdan-Lusztig polynomials. One notable property of Kazhdan-Lustzig immanants is that they are nonnegative on totally positive matrices. We give a condition on permutations that allows us to extend this theorem to the setting of $k$-positive matrices. We also conjecture a larger class of permutations for which our theorem holds true.

Mobius functions for real hyperplane arrangements

Marcelo Aguiar

Cornell

Friday

Oct. 18, 2019

3:35-4:25pm

Vincent Hall 570

We discuss a number of algebraic structures attached to a
real hyperplane arrangement,
leading to the beginnings of a theory of noncommutative Mobius functions.
Background on hyperplane arrangement and Mobius functions will be reviewed.
The talk will contain geometric, combinatorial and algebraic aspects
and there will be many pictures. All based on joint work with Swapneel
Mahajan.

Combinatorial methods for Integrable Systems

Nick Ovenhouse

UMN

Friday

Oct. 25, 2019

3:35-4:25pm

Vincent Hall 570

An integrable Hamiltonian system is a dynamical system with "enough conserved quantities" to guarantee that it can, in principle, be solved, or "integrated". I will give some basic definitions of Poisson algebras and what it means to be integrable in this context. I will then show, by way of an example (namely the "pentagram map"), how some combinatorial techniques using weighted directed graphs can be used to model the system and demonstrate its integrability. This method also hints at connections with cluster algebras and Postnikov's constructions related to stratifications of the positive Grassmannian. Time permitting, I will also discuss recent work generalizing this example.

Two-row W-graphs in affine type A

Dongkwan Kim

UMN

Friday

Nov. 1, 2019

3:35-4:25pm

Vincent Hall 570

For a Coxeter group $W$, a $W$-graph is a graph which produces a nice basis of the corresponding representation of $W$ and also describes the action of $W$ on the basis elements. Even when $W$ is finite and its irreducible characters are known, $W$-graphs are still useful for understanding representations of $W$. In this talk, I will talk about $W$-graphs when $W$ is an (extended) affine symmetric group, especially when these graphs are associated with "two-row partitions." Also I will discuss the connection between them and Lusztig's periodic $W$-graph. This work is joint with Pavlo Pylyavskyy.

Combinatorics via Deligne Categories

Chris Ryba

MIT

Friday

Nov. 8, 2019

3:35-4:25pm

Vincent Hall 570

The Deligne category Rep$(S_t)$ can be thought of as "interpolating" the representation categories of symmetric groups. After describing this category, I will explain how a calculation in the Deligne category can be used to prove stability properties of permutation patterns within conjugacy classes (joint with Christian Gaetz).

Simplicial generation of Chow rings of matroids

Chris Eur

UC Berkeley

Friday

Nov. 15, 2019

3:35-4:25pm

Vincent Hall 570

Matroids are combinatorial objects that capture the essence of linear independence. We first give a gentle introduction to the recent breakthrough in matroid theory, the Hodge theory of matroids, developed by Adiprasito, Huh, and Katz. By combining two prominent recent approaches to matroids, tropical geometric and Lie/Coxeter theoretic, we give a new presentation for the Chow ring of a matroid that further tightens the interaction between combinatorics and geometry of matroids. We discuss various applications, including a simplified proof of the main portion of the Hodge theory of matroids. This is joint work with Spencer Backman and Connor Simpson.

A Pieri rule for key polynomials

Danjoseph Quijada

USC

Friday

Nov. 22, 2019

3:35-4:25pm

Vincent Hall 570

The Pieri rule for the product of a Schur function and a single row Schur function is notable for having an elegant bijective proof that can be intuited by the rule's concise diagrammatic interpretation, to wit, by appending cells to a Young diagram. Now, key polynomials generalize Schur polynomials to a basis of the full polynomial ring, in which they also refine the Schubert basis via a nice formula. In this talk, I will describe a Pieri rule for the product of a key polynomial and a single row key polynomial that can be analogously interpreted as appending cells to a key diagram, albeit potentially dropping some cells in between each cell addition. I will also outline the main points of the rule's bijective proof, and in the process hopefully illustrate the utility of understanding the rule from a diagrammatic perspective. Joint work with Sami Assaf.

No seminar- Thanksgiving Break

Friday

Nov. 29, 2019

3:35-4:25pm

Vincent Hall 570

A non-iterative formula for straightening fillings of Young diagrams

Reuven Hodges

UIUC

Friday

Dec. 6, 2019

3:35-4:25pm

Vincent Hall 570

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. It has been a long-standing open problem to give a non-iterative formula for this straightening process. In this talk I will give such a formula. I will then use this non-iterative formula to give a proof that the coefficient of the leading term in the straightening is either 1 or $-1$, generalizing a theorem of Gonciulea and Lakshmibai.

No seminar- final exams

Friday

Dec. 13, 2019

3:35-4:25pm

Vincent Hall 570

- Seminar meets on Fridays 3:35-4:25 in room 570 of Vincent Hall. The Spring 2020 seminar site is here.
- Seminar announcement list sign-up.
- Organizers: Chris Fraser and Vic Reiner.
- Past seminar archive.
- Student Combinatorics and Algebra Seminar; meets on Thursdays.