Abstracts
Costanza Benassi
Classical Spin Systems and Random Loop Models
Classical spin systems have been the object of great interest to both physicists and mathematicians. They can be explicitly reformulated as models of interacting random loops, and many questions are still open on their features. In this talk, I will introduce both models and discuss some conjectures about the presence of macroscopic loops and the joint distribution of their lengths.
Natasha Blitvic
Probability and Combinatorics: A Non-commutative Perspective
Probability and combinatorics have long gone hand-in-hand, from endowing discrete objects with a random structure to harnessing tools of probability and analysis to solve enumeration problems. In this talk, we will attempt to tell a similar story, but from a non-commutative viewpoint. Familiar combinatorial objects and techniques will provide inspiration and tools with which to address (non-commutative) probabilistic questions and we will explore what probability may, and may not, tell us about the combinatorics.
Jason Hancox
Levy Processes on *-bialgebras
We will introduce C*-bialgebras and Lévy Processes on C*-bialgebras. These are a generalization of compact topological semigroups and stochastic processes on compact topological semigroups that have 'independent increments' respectively. We will construct a family of C*-bialgebras and characterize the Lévy Processes on this family. To finish we will consider the Toeplitz algebra and investigate some processes on its C*-bialgebra structure and find concrete realizations of these processes on a commutative quotient subalgebra.
Ying-Fen Lin
Chordal graphs and Schur multipliers
I will first introduce partially defined Schur multipliers whose domains are operator systems associated with graphs. I will then describe necessary and sufficient conditions for the existence of extensions to fully defined positive Schur multipliers. Application of this result to the extension problem for partially positive definite functions on discrete groups will be given.
Helge Schäfer
The Number Of Cycles In Random Permutations Without Long Cycles
We consider random permutations without long cycles. It is shown that the number of cycles converges to a Gaussian limit as in the classical case. We provide asymptotic expansions for expected value and variance. This is joint work with V. Betz.
Einar Steingrímsson
Combinatorics vs. Probability: Counting vs. Estimating
Sometimes the difference between a combinatorialist and a (discrete) probabilist is just that the latter divides by N, while both start by counting the objects of some finite set.
In my neck of the combinatorics woods, counting is done in several different ways. There is the "plain" counting of coming up with a formula for the number of objects in a set, and then there is counting by generating functions (essentially Taylor series), where counting is done in a "wholesale" manner, somewhat similar to doing linear algebra with matrix expressions to avoid the mess of dealing with individual matrix entries. A third way to count avoids the counting altogether by exhibiting a bijection to a set of objects whose numbers we already know.
I will illustrate some aspects of these methods with examples, involving such things as the expected number of fixed points in a random permutation, the probability of a random walk in one dimension returning to the origin and the question of whether men or women have more sisters.
I will also describe the work of probabilists (Madras and Liu, 2010) that sheds some light on a hard, unsolved counting problem, that of determining how many permutations of length n avoid the pattern 1324, that is, how many such permutations have no four entries appearing in the same order of size as 1,3,2,4. This problem has also seen recent progress made by combinatorialists (Conway and Guttmann, 2015) using analytic methods in dealing with generating functions, to estimate asymptotics.
References
- N. Madras and H. Liu (2010) Random pattern-avoiding permutations. In Algorithmic Probability and Combinatorics (M. E. Lladser et al., eds), Vol. 520 of Contemporary Mathematics, AMS.
- A. R. Conway and A. J. Guttmann. On 1324-avoiding permutations. Adv. in Appl. Math., 64(0):50–69, 2015.
