*Date:* Feb. 2, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Jinyoung Park, Courant Institute, New York University.

*Title*: Thresholds

*Abstract*: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

*Title*: Gosper's algorithm and Bell numbers

*Abstract*: Computers are now very good at evaluating finite sums in closed form. They can do them almost instantly and can often prove that a sum cannot be
evaluated in closed form, like the partial sums of k!. Just like in calculus, we can fix summands that don't yield closed forms by multiplying them by a suitable "summation factor."
I will show that this idea, applied to sums of falling factorials, produces sequences of polynomials closely related to the Bell numbers.

*Title*: Noncommutative generalized Catalan numbers and their generating functions

*Abstract*: In 1999 M. Aigner defined Catalan-like numbers as North-West
entries of powers of semi-infinite three-diagonal matrices. In this
talk I will generalize this construction to various matrices over
noncommutative rings and discuss properties of the corresponding
generating functions. I will also consider orthogonal polynomials
associated with these "numbers". Most of the results are new even in
the commutative case. Joint work with A. Berenstein and M. Gekhtman.

*Title*: Solving the Race in Backgammon

*Abstract*: Backgammon is perhaps the oldest game that is still played today. It is a game that combines luck with skill, where two players take turns rolling dice and decide how to move their checkers in the best possible way. It is the ultimate math game, where playe
rs who possess a little bit of mathematical knowledge can have a big advantage over their opponents. Players also have the opportunity to double the stakes of a game using something called the doubling cube, whichwhen used optimallyleads to players winning more in the lon
g run. Optimal use of the doubling cube relies on a player's ability to estimate their winning chances at any stage of the game.
When played to completion, every game of backgammon eventually becomes a race, where each player attempts to remove all of their checkers before their opponent does. The goal of our research is to be able to determine the optimal doubling cube action for any racing position,
and approximate the game winning chances for both sides. By calculating the Effective Pip Count for both players and identifying the positions' Variance Types, we arrive at a reasonably simple method for achieving this which is demonstrably superior to other popular methods

*Title*: Partitioning a Square into Similar Rectangles

*Abstract*:
OEIS Sequence A359146
gives the number of different aspect ratios that are possible when partitioning a square into n similar rectangles. We will discuss several algorithms that have been used to compute the first few terms of this sequence,
and see a theoretical result that connects the problem to continued fractions. As a fun challenge to do before the talk,
you can try to compute a(3), the number of different ratios that are possible when dividing up a square into 3 similar rectangles (not necessarily of the same size).

*Date:* April 13, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Edinah Gnang, Johns Hopkins University

*Title*: A Functional Approach to Graph Labeling

*Abstract*: We describe a unified framework for graph labeling problems based on the theory of functional directed graphs. We show that this approach sheds light on algebraic, enumerative and combinatorial design aspects of graph labeling and coloring problems.

*Title*: Ramanujan series for 1/π . Automatic proofs.

*Abstract*: We develop a method for proving automatically Ramanujan series for
1/π using modular equations. For the case of alternating series these equations are of a much lower degree than those required in the methods of other authors. For example, it was thought that a complete explicit proof of the Chudnovskys' fastest series for
1/π would require a supercomputer. We will show that this is not true

*Date:* Thu., Sept. 14, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Neil Sloane, The OEIS Foundation and Rutgers University

*Title*: New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences

*Abstract*:
Topics to be discussed: Scott Shannon's Circle Counting Problems, Dissecting Polygons into Rectangles (with Gavin Theobald), New Gilbreath Conjectures, Eric Angelini's Sum and Erase Sequence, and a Report on Status of OEIS.

*Date:* Thu., Sept. 28, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Ido Kaminer, The Technion

*Title*: The Ramanujan Machine 2.0: algorithm-assisted discovery of an intrinsic order among mathematical constants

*Abstract*: Recent years have shown a rise in the number of discoveries in fields of mathematics that are being assisted by computer algorithms, primarily for
exploring large parameter spaces that humans would take too long to investigate. Now that computers and algorithms become more powerful, their ability to augment human intuition is expected to lead to new kinds of discoveries.

*Date:* Thu., Oct. 12, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Persi Diaconis, Stanford University

*Title*: GAMBLER'S RUIN WITH K GAMBLERS

*Abstract*: Consider (say) three gamblers with initial capital A, B , C. Each time a pair of gamblers are picked (uniformly at random), a fair coin is flipped and $1 is transfered.
Eventually, one of the gamblers goes broke and the other two continue with the usual coin tossing until one is left with all A+B+C. Of interest: how long does this all take, what is the
distribution of the 'elimination order' (where order 3,1,2 means that gambler C is eliminated first then A leaving B with all the money)? If the game goes on a long time and no one
is eliminated where is it likely to be? AND how does all this depend on A,B,C. These questions are of interest in things like poker tournaments(world series of poker).
In joint work with Stew Ethier, Laurent Saloff-Coste and Kelsey Huston-Edwards we have simulations, heuristics, bad asymptotics and some theorems. This has led to some nice 'useful' theorems of others. I'll try to explain to a non-specialist audience in 'English'.

*Date:* Thu., Oct. 26, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*Robert Dougherty-Bliss, Rutgers University

*Title*: Hardinian Arrays

*Abstract*: Kauers and Koutschan recently performed an automated search of sequences in the
OEIS that might satisfy previously unknown recurrences. Among many promising
hits was a 2014 sequence about king-moves on an array submitted by R.H. Hardin.
I will show how to confirm and extend the conjectured recurrence using
determinant evaluations and computer algebra.

*Date:* Thu., Nov. 9, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Marni Mishna, Simon Fraser University

*Title*: Characterizing Transcendence in Combinatorics

*Abstract*: The problem of understanding the structure of transcendental objects has fascinated mathematicians for well over a century. Combinatorics provides an intuitive framework to study power series. A combinatorial family is associated to a power series in R[[t]] via its enumerative generating function wherein the number of objects of size n is the coefficient of t^n. Twentieth century combinatorics and theoretical computer science provided characterizations of classes with rational and algebraic generating functions. Finding natural extensions of these correspondences has been a motivating goal of enumerative combinatorics for several decades. This talk will focus on two well studied classes of transcendental functions: the differentiably finite and differentially algebraic.

*Date:* Thu., Nov. 16, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
John Campbell, Dalhousie University

*Title*: Recent applications concerning WZ theory

*Abstract*:
Recent applications of Zeilberger's algorithm and the WZ method within areas such as numerical analysis and computer algebra are to be explored and discussed.

slides

lecture (passwd: john211532)

*Date:* Thu., Nov. 30, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Miranda Holmes-Cerfon, University of British Columbia

*Title*: Sphere packings, singularities, and statistical mechanics

*Abstract*: What are all the ways to arrange N hard spheres into a rigid packing? And what can the solution tell us about how materials crystallize? I will introduce an algorithm to enumerate rigid sphere packings (clusters) and describe some of the data it produces, which include many clusters with geometrically unusual properties. Among these are an abundance of "singular" clusters, those that are linearly flexible but nonlinearly rigid, so called because they correspond to singular solutions to a set of algebraic equations. These are also the clusters one sees with unusually high probability, in experiments which consider colloidal particles interacting with a short-ranged potential. I will explain why these clusters are so prevalent, drawing links between statistical mechanics and the volumes of semialgebraic sets, and show how these calculations applied to our sphere packing data bring insight into the pathways to crystallization of sticky spheres.

*Date:* Thu., Dec. 14, 2023, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Yuri Matiyasevich, Steklov Institute of Mathematics at St. Petersburg, Russia

*Title*: Nontrivial zeros of the Riemann zeta function know a lot

*Abstract*: Bernhard Riemann gave an exact formula for the
number of primes below given bound via a particular sum over the zeros of
zeta function. In the course of large scale computer calculations
the speaker discovered new relationships between these zeros and prime numbers.

slides [print version]
See also animation of N values

See also this website: Finite Dirichlet series with prescribed zeroes.

lecture