Combinatorics seminar
Combinatorics Seminar  Fall 2017
Tuesday 1pm2pm in Goldsmith 300.
Organizers: Olivier Bernardi and Yan Zhuang
The Combinatorics Seminar is an introductory seminar for combinatorics. The talk should be accesible to first year graduate students.

September 12
Speaker: Kate Moore (Dartmouth)
Title: Permutations in Time Series and Dynamical Systems
Abstract:
Given a time series, we can extract a collection of permutations of a fixed length by sliding a window across our data and recording the relative order of the values in the window. It is known that the distribution of patterns that appear conveys information about the complexity of the time series; one commonly studied measure is permutation entropy. Interestingly, in the specific case that the time series is defined by iterating a piecewise monotone map of the interval, there are forbidden permutations. Moreover, the number of allowed permutations encodes the system’s topological entropy, a measure of complexity that is itself closely related to the permutation structure of periodic points.
Using permutationbased measures of entropy in time series analysis as motivation, I will talk about some interesting problems that arise at this intersection of combinatorics, dynamics and time series analysis.

September 19
Speaker: Melody Chan (Brown)
Title: A moduli stack of tropical curves; or, Stacks for combinatorialists
Abstract:
The tropical moduli space of curves is a combinatorial moduli space
parametrizing weighted metric graphs, with close connections to many
interesting spaces in geometry: moduli spaces of Riemann surfaces, the
space of phylogenetic trees, CullerVogtmann Outer Space, Harvey's
curve complex...
Recent joint work with Renzo Cavalieri, Martin Ulirsch, and Jonathan
Wise provides a construction of a moduli _stack_ of tropical curves. I
will use this as an excuse to give, from scratch, a combinatorialist's
introduction to stacks. No prior knowledge of stacks will be assumed.

September 26
Speaker: Ira Gessel (Brandeis)
Title: An introduction to rook theory
Abstract:
Rook theory deals with placements of nonattacking rooks on a subset of a chessboard, and has interesting applications to permutation enumeration and other enumerative problems. I’ll talk about the basic formulas of rook theory, the analogous theory for matchings and partitions, and the reciprocity theorem for factorial rook polynomials.

October 10
Speaker: Oliver Knill (Harvard)
Title: The energy of a simplicial complex
Abstract:
A finite abstract simplicial complex G defines
a matrix L, where L(x,y)=1 if two simplicies x,y
in G intersect and L(x,y)=0 else. This matrix L
always has an inverse which is integer valued.
The sum of the matrix values of the inverse is the
energy of the complex. We explain why this energy
is equal to the Euler characteristic of the complex.

October 17
Speaker: Duncan Levear (Brandeis)
Title: The representation theory of symmetric groups via OkounkovVershik (1/2)
Abstract:
For any group G, a natural goal is to enumerate the vector spaces on which G acts (in the sense of L[G] modules). For finite groups, many amazing results are known, such as the number of these `irreducible' vector spaces is equal to the number of conjugacy classes. The symmetric group is a poignant example, and its representation theory is closely tied to other combinatorial subjects. This case was essentially solved by the work of Frobenius and Young 100 years ago, but recently (1995) Okounkov and Vershik established an alternative approach by constructing an ``inductive representation theory'' for the chain of symmetric groups, working from the abstract claim that S_{n1} is a multiplicityfree subgroup of S_{n}. In this talk, I will prove that claim, explain the resulting work of Okounkov and Vershik, and demonstrate some of its corollaries. In particular, the Branching Rule and MurnaghanNakayama Rule for an ncycle are rendered to trivial observations in this light.

October 24
Speaker: Duncan Levear (Brandeis)
Title: The representation theory of symmetric groups via OkounkovVershik (2/2)
Abstract:
For any group G, a natural goal is to enumerate the vector spaces on which G acts (in the sense of L[G] modules). For finite groups, many amazing results are known, such as the number of these `irreducible' vector spaces is equal to the number of conjugacy classes. The symmetric group is a poignant example, and its representation theory is closely tied to other combinatorial subjects. This case was essentially solved by the work of Frobenius and Young 100 years ago, but recently (1995) Okounkov and Vershik established an alternative approach by constructing an ``inductive representation theory'' for the chain of symmetric groups, working from the abstract claim that S_{n1} is a multiplicityfree subgroup of S_{n}. In this talk, I will prove that claim, explain the resulting work of Okounkov and Vershik, and demonstrate some of its corollaries. In particular, the Branching Rule and MurnaghanNakayama Rule for an ncycle are rendered to trivial observations in this light.

October 31
Speaker: Joshua Heike (Brandeis)
Title: TBA
Abstract:
TBA

November 7
Speaker: Jay Pantone (Dartmouth)
Title: TBA
Abstract:
TBA

November 14
Speaker: Brittney Ellzey (University of Miami)
Title: TBA
Abstract:
TBA

November 21
Speaker: Thao Do (MIT)
Title: Zarankiewicz's problem for semialgebraic hypergraphs
Abstract:
Zarankiewicz’s problem asks for the largest possible number of edges in a graph with $n$ vertices that does not contain K_{s,t} for some fixed integers $s, t$. Recently, Fox, Pach, Sheffer, Sulk and Zahl considered this problem for semialgebraic graphs, the ones whose vertices are points in Euclidean spaces and edges are defined by some semialgebraic relations. In this talk, I will explain their results and how to extend to semialgebraic hypergraphs. As an application, we find an upper bound for the number of unit $d \times d$ minors in a $d\times n$ matrix.

November 28
Speaker: Konstantin Matveev (Brandeis)
Title: TBA
Abstract:
TBA

December 5
Speaker: Konstantin Matveev (Brandeis)
Title: TBA
Abstract:
TBA
Here are some indications for reaching Brandeis, and the math department.
Previous semesters:
Spring 2017,
Fall 2016,
Spring 2016,
Fall 2015,
Spring 2015,
Fall 2014,
Spring 2014,
Fall 2013.