Combinatorics seminar
Combinatorics Seminar  Fall 2016
Tuesday 1pm2pm in Goldsmith 226.
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 13  SPECIAL SEMINAR  In special room: 317
Speaker: Jonathan Touboul
Title: Complex oscillations and synchrony in models of largescale random neural networks
Abstract:
Unprecedented biological advances in neuroscience and fine descriptions of the brain based on novel data acquisition techniques have brought to light the extraordinary complex functioning of the nervous system whose dynamical behavior emerges from structured networks of interacting cells in random interactions and subject to noise.
I will present two or three recent progress in the mathematical characterization of such systems allowing finer understanding of brain’s dynamics. First, using largedeviations techniques and probability theories, I will show that depending on the level of fluctuations of the interaction amplitudes between cells (synaptic weights), two distinct and somewhat universal limits describe the dynamics of randomly connected networks as the network size diverges. These are complex implicit equations in the space of stochastic processes. I will show how dynamical systems and spectral theory of operators allow describing their complex dynamics, and understanding the role of connectivity structure, delays and randomness. I will in particular exhibit a very surprising and somewhat paradoxical role of noise and heterogeneity, having the ability to synchronize neuronal activity.
These are only first steps lifting a small part of the veil on a wild and largely unexplored territory at the crossroads of probability theory, random matrices, dynamical systems and geometry.

September 20
Speaker: Sam Hopkins (MIT)
Title: The coincidental downdegree expectations (CDE) property for posets
Abstract: Reiner, Tenner, and Yong recently introduced the coincidental downdegree expectations (CDE) property for posets. A
poset P is CDE if the expected downdegree is the same for the uniform distribution on P and for the distribution on P
that weights each element proportional to the number of maximal chains passing through that element. Reiner, Tenner,
and Yong showed that many posets familiar to algebraic combinatorialists, such as certain intervals of the weak Bruhat
order, are CDE. In earlier independent work with Chan, Haddadan, and Moci (which was motivated by some surprising
combinational results arising from a new approach to BrillNoether theory) we found a large family of CDE intervals of
Young’s lattice. In this more recent work I extend these results to the shifted setting. These shifted results complete
the casebycase proof that all minuscule lattices are CDE, as was conjectured by ReinerTennerYong. Time permitting
I will explain some applications of the study of CDE posets: product formulas for certain kinds of setvalued tableaux
(i.e., product formulas for certain squarefree coefficients of stable Grothendieck polynomials), and homomesy results (in
the sense of Propp and Roby) for rowmotion and gyration acting on sets of order ideals.

September 27
Speaker: Kabir Ramola (Brandeis)
Title: Disordered Networks and Jamming Transitions in Amorphous Media
Abstract:
Why are amorphous materials (such as sand) able to flow like liquids, while also being able to sustain stresses like solids? Several puzzling characteristics of such granular media are the subject of active research. One such phenomenon is that of 'jamming' whereby a collection of macroscopic particles acquires global rigidity with external compression. In such cases contact forces between the individual particles play an important role in the transmission of stress through the system. System spanning contact networks determine the response of such materials to external perturbations. These networks are inherently 'disordered' as the constituent particles arrange themselves into random spatial patterns. In this talk I will explain how graph theoretical techniques applied to these networks can be used to develop predictive frameworks for granular materials.

October 11
Speaker: Thomas McConville (MIT)
Title: Zonotopes whose cellular strings are all coherent
Abstract: A monotone path on a polytope is a maximal path along the edges of the polytope in some fixed direction. For example, the monotone paths on a simplex with n+2 vertices correspond to subsets of {1,..,n} and monotone paths on an ndimensional cube correspond to permutations of {1,...,n}. In both of these cases, the set of monotone paths naturally index the vertices of some other polytope, a cube and a permutahedron respectively. In joint work with Rob Edman and Pakawut Jiradilok, we discovered some other polytopes that have a polytope of monotone paths and obtained a partial characterization of zonotopes with this property of low corank.

October 18
Speaker: Gabor Lippner (Northeastern University)
Title: Measurable graph theory
Abstract: "Graphings" are a natural generalization of finite graphs on probability measure spaces. These objects arise naturally as limits of finite graphs, as well as from the study of invariant random processes on discrete groups.
Measurable graph theory studies graphings from the perspective of classical graph theory. It lies at the crossroads of ergodic theory and discrete mathematics. I will explain how to generalize standard notions (matchings, chromatic number, expansion, etc) to graphings and survey recent results on the surprising behavior of these notions in the measurable setting.

November 1
Speaker: Ira Gessel (Brandeis)
Title: An Introduction to Symmetric Functions
Abstract: This talk will be an introduction to the theory of symmetric functions, including the important bases for symmetric functions, the scalar product, connections with representations of symmetric groups, and plethysm.

November 8
Speaker: Ira Gessel (Brandeis)
Title: An Introduction to Combinatorial Species
Abstract:The theory of combinatorial species is a method for counting labeled and unlabeled structures such as graphs of various types. In particular it provides a powerful alternative approach to problems traditionally solved by Pólya theory. I will discuss the basic definitions and examples, and describe the fundamental operations on species. I will also discuss the three fundamental generating functions associated with a species, in particular, the cycle index, which is a symmetric function associated with an action of the symmetric group.

November 15
Speaker: Ira Gessel (Brandeis)
Title: The KonvalinkaAmdeberhan conjecture and plethystic inverses
Abstract: Motivated by Billey, Konvalinka, and Matsen's formula for unlabeled tanglegrams, Tewodros Amdeberhan and Matjaž Konvalinka independently conjectured that a generalization of their formula always gives integers. I will prove their conjecture using the fact that the plethystic inverse of an integral symmetric function is integral, and I will discuss the conjectured Schur positivity of these symmetric functions, which are related to the Lyndon (primitive necklace) symmetric functions.

November 22
Speaker: Konstantin Matveev (Brandeis)
Title: TBA
Abstract: TBA

November 29
Speaker: Justin Troyka (Dartmouth)
Title: TBA
Abstract: TBA

December 6
Speaker: Tim Dwyer (Dartmouth)
Title: TBA
Abstract: TBA
Here are some indications for reaching Brandeis, and the math department.
Previous semesters:
Spring 2016,
Fall 2015,
Spring 2015,
Fall 2014,
Spring 2014,
Fall 2013.