Algebra & Combinatorics Seminar

This is the list of speakers and talks for the Algebra & Combinatorics Seminar for Michigan Technological University. Previous semesters' speakers can be found lower on the page.

In Spring 2020, the Algebra & Combinatorics Seminar is scheduled biweekly on Thursdays from 1:05pm-1:55pm in Fisher 126.

Spring 2020

 Date: Jan 30 Title: Personal Perspectives on m-ary Partitions Speaker: James Sellers Abstract: A great deal of my research journey has involved the study of m-ary partitions. These are integer partitions wherein each part must be a power of a fixed integer m > 1. Beginning in the late 1960s, numerous mathematicians (including Churchhouse, Andrews, Gupta, and Rødseth) studied divisibility properties of m-ary partitions. In this talk, I will discuss work I completed with Rødseth which generalizes the results of Andrews and Gupta from the 1970s. Time permitting, I will then discuss several problems related to m-ary partitions, including my work with Neil Sloane on non-squashing stacks of boxes, an application of m-ary partitions to objects known as "unique path partitions" (which are motivated from representation theory of the symmetric group), as well as very recent work with George Andrews and Aviezri Fraenkel on the characterization of the number of m-ary partitions of n modulo m. Throughout the talk, I will attempt to highlight various aspects of the research related to symbolic computation. The talk will be self-contained and geared for a general mathematical audience. Affiliation: UMN-Duluth Date: Feb 27 Title: The Hamilton-Waterloo problem with C6 and C3x factors Speaker: Zazil Santizo Huerta Abstract: A solution of the Hamilton-Waterloo problem, that is, a resolvable (Cm,Cn)-decomposition of Kv into r Cm-factors and s Cn-factors, is denoted by (m,n)-HWP(v;r,s). This problem has been solved for v ≤ 17 when v is odd and for v ≤ 10 when v is even. The most difficult case is when either r or s is equal to 1. In this talk, I will give the construction of (6,9)-HWP(18;1,7), and settle the problem for v = 18t when t is odd. Furthermore, in order to extend the latter idea to the case n=3x, we proved that there exists a (6,3x)-HWP(6xt;1,3xt-2) for all odd x ≥ 3, which completes the case (6,3x)-HWP(3xt;1,(3xt-4)/2) for all x ≥ 3 and t ≥ 1. Affiliation: MTU Date: Title: Speaker: Abstract: Affiliation: Date: Title: Speaker: Abstract: Affiliation: Date: Title: Speaker: Abstract: Affiliation: Date: Title: Speaker: Abstract: Affiliation:

Fall 2019

 Date: Sep 12 Title: Two New Families of Cubic Surfaces in Characteristic Two Speaker: Anton Betten Abstract: Cubic surfaces with 27 lines are beautiful objects from classical geometry. Several infinite families are known, due to Fermat, Clebsch and Hilbert-Cohn-Vossen. We consider cubic surfaces with 27 lines over finite fields. Besides the classical families, many other examples appear. In recent joint work with Karaoglu, the speaker has classified these surfaces up to isomorphism in all fields of order at most 97, using a computer. Now comes the fun part: By analyzing the data, we are trying to find new infinite families of cubic surfaces with 27 lines. In the talk, we address the problem in characteristic two. Two new families will be constructed, bringing the total number of known families to three. This extends work of Hirschfeld from 1964.Prof. Betten's talk is rescheduled from last semester after weather difficulties. He will also be giving a Colloquium talk on Friday. Affiliation: Colorado State Date: Sep 19, 5pmDow 641 Title: Kliakhandler Lecture: Why Does Ramanujan, "The Man Who Knew Infinity," Matter? Speaker: Ken Ono Abstract: Srinivasa Ramanujan, one of the most inspirational figures in the history of mathematics, was a poor gifted mathematician from lush south India who left behind three notebooks that engineers, mathematicians, and physicists continue to mine today. Born in 1887, Ramanujan was a two-time college dropout. He could have easily been lost to the world, a thought that scientists cannot begin to absorb. He died in 1920. Prof. Ono will explain why Ramanujan matters today, and will share several clips from the film, “The Man Who Knew Infinity,” starring Dev Patel and Jeremy Irons. Professor Ono served as an associate producer and mathematical consultant for the film. This special note is for the Kliakhandler Public Lecture, which we feel will be of interest to Algebra & Combinatorics Seminar attendees. Please note the special room and time. Prof. Ono will also be giving the Kliakhandler Colloquium on Friday. Affiliation: UVa and Emory Date: Sep. 26 Title: Generating functions for Mullineux fixed points Speaker: David Hemmer Abstract: Mullineux defined an involutary bijection on the set of e-regular partitions of n. When e is prime, these partitions label irreducible symmetric group modules in characteristic e. Mullineux conjectured (since proven) that this “Mullineux map” described the effect on these labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible Sn modules preserved under this tensor product (i.e. fixed points of the Mullineux map) is related to counting irreducible modules for the alternating group An. In 1991, Andrews and Olsson worked out the generating function of these fixed points when e is prime, as evidence in support of the conjecture. We generalize their work to arbitrary e, and discover distinct answers depending on the parity of e. We will also discuss a conjectural block-theoretic version. Affiliation: MTU Date: Oct. 24 Title: Connections between Schur functions and major index distributions Speaker: William J. Keith Abstract: Let fλ,i denote the distribution of the major index over all standard Young tableaux of shape λ and descent number i. For several families of partition shapes (though not all), this distribution quite unexpectedly becomes a principal specialization of a Schur function, which allows us to immediately conclude positivity and unimodality. This talk will introduce the objects described and will prove several theorems regarding their connections, concluding with potential future work in the area. Affiliation: MTU Date: Nov 1Fisher 327B Title: Colloquium: Cameron-Liebler Sets in Projective and Polar Spaces Speaker: Morgan Rodgers Abstract: The objects we will consider originated in the study of collineation groups of PG(n,q) having equally many orbits on points and lines; line orbits of such a group have very nice combinatorial properties, which led to the definition of what are now called Cameron-Liebler line classes. For a while it was thought that there could not exist any nontrivial examples of these objects, until Bruen and Drudge gave an infinite family of examples in PG(3,q); since then, several new examples have been found including two more infinite families. After discussing some recent results and open problems concerning line sets in PG(n,q), we will look at some generalizations of these objects to collections of higher dimensional subspaces in both projective and polar spaces. In particular we will look at some characterization results for Cameron-Liebler sets of maximals in the finite classical polar spaces.Please note the Friday date (still at 1:05pm) and different room: Prof. Rodgers is visiting under Zeying Wang and giving this week's Mathematical Sciences Colloquium. Affiliation: Cal State Fresno Date: Title: Speaker: Abstract: Affiliation: Date: Title: Speaker: Abstract: Affiliation:

Spring 2019

 Date: Feb 21 Title: Lusztig Slices in the Affine Grassmannian Speaker: Daniel Rowe Abstract: We will define the Affine Grassmannian for the group GLn: a coset space of invertible matrices over formal Laurent series modulo invertible matrices over formal power series. The Affine Grassmannian can be interpreted as a space of lattices, which can in turn be identified with a space of nilpotent matrices. We will introduce these ideas and explain a theorem that gives an interesting isomorphism between sub-varieties of the Affine Grassmannian and sub-varieties of nilpotent matrices. Affiliation: NMU Date: Mar 21 Title: On the parity of the partition function Speaker: Fabrizio Zanello Abstract: We outline a possible new approach to one of the basic and seemingly intractable conjectures in number theory, namely that the partition function p(n) is equidistributed modulo 2. The best results to date, obtained incrementally over several decades by Serre, Ono, Soundararajan and many others, don't even imply that p(n) is odd for √x values of n ≤ x. We present an infinite class of conjectural identities modulo 2, and show how to, in principle, prove any such identity. We describe a number of important consequences of these identities: For instance, if any t-multipartition function is odd with positive density and t ≢ 0 (mod 3), then p(n) is also odd with positive density. All of these facts seem virtually impossible to show unconditionally today. Our arguments employ several complex-analytic and algebraic methods, ranging from a study modulo 2 of some classical Ramanujan identities and other eta product results, to a unified approach to the parity of the Fourier coefficients of a broad class of modular forms recently introduced by Radu. Much of this research is joint with my former PhD student S. Judge and/or with W.J. Keith (see my papers in J. Number Theory, 2015 and 2018; Annals of Comb., 2018). Affiliation: MTU Date: Apr 4 Title: Noli turbare circulos meos!, or, Magic type labelings of cycle products Speaker: Dalibor Froncek Abstract: A Cartesian product Cm ☐ Cn of two cycles Cm and Cn can be seen as a toroidal m × n grid with mn vertices of degree four and 2mn edges. We can bijectively label edges, vertices, or both by consecutive positive integers 1, 2, ..., s or by elements of an Abelian group Γ of order s (where s is the number of labeled elements) and define the weight of an element (that is, an edge or a vertex) as the sum of the labels of the adjacent and/or incident elements. When the weights of all elements in question are equal, we call the labeling magic (of some kind). When the weights are all different, the labeling is called antimagic. I will present some old and new results on various kinds of magic labelings of cycle products and pose several open questions. The results are based on collaboration with Sylwia Cichacz and Jack, James, and Michael McKeown. Keywords: Graph labeling, magic type labeling, magic graphs, supermagic graphs Affiliation: UMN-Duluth Date: Apr 12 Title: Two New Families of Cubic Surfaces in Characteristic Two Speaker: Anton Betten Abstract: Cubic surfaces with 27 lines are beautiful objects from classical geometry. Several infinite families are known, due to Fermat, Clebsch and Hilbert-Cohn-Vossen. We consider cubic surfaces with 27 lines over finite fields. Besides the classical families, many other examples appear. In recent joint work with Karaoglu, the speaker has classified these surfaces up to isomorphism in all fields of order at most 97, using a computer. Now comes the fun part: By analyzing the data, we are trying to find new infinite families of cubic surfaces with 27 lines. In the talk, we address the problem in characteristic two. Two new families will be constructed, bringing the total number of known families to three. This extends work of Hirschfeld from 1964.Please note the Friday date; Prof. Betten is giving a Colloquium talk related to the subjects of the Seminar. Affiliation:Colorado State

Fall 2018

 Date: Sep 13 Title: Research Round-Robin Speaker: Discrete Math faculty Abstract: The faculty in the Discrete Math group at Michigan Tech will give 5-minute descriptions of their research areas. Affiliation: MTU Date: Sep 27 Title: Paley type and negative Latin square type partial difference sets in Abelian groups Speaker: Zeying Wang Abstract: Recently we proved that if there is a Paley type partial difference set (in short, PDS) in an Abelian group G of order m, where m = p12k1p22k2...pn2kn, n ≥ 2, p1, p2,...,pn are distinct odd prime numbers, then for any 1 ≤ i ≤ n, pi is a prime congruent to 3 modulo 4 whenever ki is odd. Also we found some new necessary conditions for the existence of negative Latin square type PDS in Abelian groups of order p2xq2y, where gcd(p,q)=1 and p,q are odd positive integers. In this talk I will first introduce and define all necessary concepts and provide some historical background. Then I will present the main ideas used in our proofs and state our main results. I will conclude the talk with some ongoing research, and ideas for future research. Affiliation: MTU Date: Oct 25 Title: Iterated differences in the q-binomial coefficients Speaker: William J. Keith Abstract: We study the iterated differences in n of p(M,N;n), the number of partitions of n with at most M parts, each of size at most N. For the unrestricted partition function Odlyzko showed that the k-th differences alternate in sign up to some n(k) and are thereafter positive. In this talk it will be shown that for small N, differences in n of p(M,N;n) alternate in sign for indefinitely large M and all n. Open conjectures will be discussed. Affiliation: MTU Date: Nov 15 Title: A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs Speaker: Melissa Keranen Abstract: The Hamilton-Waterloo Problem (HWP) asks for a decomposition of Kv (or Kv - F when v is even) into 2-factors where each 2-factor is isomorphic to either a given 2-factor P or a given 2-factor Q. In the uniform case, all of the cycles in P have the same size and all of the cycles in Q have the same size. In this talk, I will discuss the Hamilton-Waterloo problem for equipartite graphs. Results will be used to find solutions to the HWP on the complete graph in both the uniform and non-uniform cases. Affiliation: MTU

Spring 2018

 Date: January 25 Title: Partial difference sets in abelian groups Speaker: Zeying Wang Abstract: Recently we proved a theorem for strongly regular graphs that provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then used this result to develop some new techniques to study regular partial difference sets in Abelian groups. Our main results so far are the proof of non-existence of PDS in Abelian groups with small parameters, a complete classification of PDS in Abelian groups of order 4p2, and a proof that no non-trivial PDS exist in Abelian groups of order 8p3. In this talk I plan to give an overview of these results with a focus on our most recent work on the PDS in Abelian groups of order 8p3, where p is a prime number ≥ 3. Affiliation: MTU Date: February 15 Title: Unimodality in the q-analogue of Frame-Robinson-Thrall Speaker: William J. Keith Abstract: The unimodality of the q-binomial coefficient and its various proofs are two of the most beautiful achievements of partition theory and its related combinatorics. Proving the same property for differences and sums of q-binomial coefficients is an even greater challenge, one for which only a few results exist. In this talk, we will bring together many threads in combinatorics: permutations and pattern avoidance, the Robinson-Schensted correspondence, the Frame-Robinson-Thrall formula for standard Young tableaux and its q-analogue. We will prove unimodality for several families of q-binomial formulas, and discuss the next questions that remain. The pace will be gentle and suitable for graduate students. Depending on pacing and audience feedback, this may be one or two sessions. Affiliation: MTU Date: March 29 Title: The Good Will Hunting Problem Speaker: Donald L. Kreher Abstract: In the movie "Good Will Hunting", the main character Will Hunting (Matt Damon) - a janitor - solves a blackboard problem, which had been assigned as a challenge to an applied theories class. In this lecture we will use elementary linear algebra and a little combinatorics to show that this problem can be easily solved. This will be followed by some thoughts on a graph theory solution to recurrence relations. The talk will be elementary. Only a half course in introductory linear algebra such as MA2330 is required. Undergraduates are encouraged to attend. Affiliation: MTU Date: April 19 Title: 2-Block intersection graphs in triple systems Speaker: Melissa Keranen Abstract: A TS(v,λ) is a pair (V,B) where V contains v points and B contains 3-element subsets of V so that each pair in V appears in exactly λ blocks. A 2-block intersection graph (2-BIG) of a TS(v,λ) is a graph where each vertex is represented by a block from TS(v,λ) and each pair of blocks Bi, Bj ∈ B are joined by an edge if ∣ Bi ∩ Bj ∣ = 2. Using known constructions for TS(v,λ), we show that there exists a TS(v,λ) for v ≡ 0 or 4 (mod 12) whose 2-BIG is Hamiltonian.Joint with John Asplund, Dalton State College Affiliation: MTU Date: April 26 Title: (k,j)-colored partitions and the hooklength formula Speaker: Emily Anible Abstract: We investigate an extension of k-colored partitions, the (k,j)-colored partitions, at an indeterminate number of colors, and their relationship to the Han/Nekrasov-Okounkov hooklength formula under truncation to hooks of size at most j. We find the formulas match at the constant and linear terms for all n. Further, we attempt to match the two formulas at the quadratic term for j=2 by adding a simple offset to C1-b,j. We find pleasing relations to the harmonic numbers, and conjecture generating functions to describe squaring the number of frequencies of at least i in partitions of n.This presentation is the result of an undergraduate research project. Undergraduates are warmly encouraged to attend.Slides for this presentation are here. Affiliation: MTU

Spring 2016

 Date: Jan 21 Title: Decomposing the blocks of a Steiner triple system into partial parallel classes Speaker: Melissa Keranen Abstract: Does there exist a Steiner triple system with t triples such that for any m with m|t, the triples can be decomposed into matchings of size m? In this talk I will discuss results on STS(4v-3) where v ≡ 1 or 3 (mod 6). In particular, I will present constructions for decompositions into matchings of size v-1 or 2(v-1)/3 when v ≡ 3(mod 6), or v ≡ 1 (mod 6), respectively. Affiliation: MTU Date: Feb 18 Title: Independent sets in geometries Speaker: Stefaan De Winter Abstract: Independent sets or cocliques play an important role in graph theory, and finding the maximal size of an independent set for specific classes of graphs is a major research topic. However, for bipartite graphs the standard measure of size for an independent set does not necessarily make a lot of sense, as each of the parts will be an (uninteresting) independent set. A different way to measure how ``large'' an independent set is will be introduced. Then I will discuss ``large'' independent sets in bipartite graphs from a geometric point of view (after all, every bipartite graph is equivalent to a point-line geometry). Affiliation: MTU Date: Feb 231:05pm Fisher 125 Title: Infinite Dimensional Lie Algebras Speaker: Jie Sun Abstract: A group becomes a Lie group if a compatible manifold structure is added to the group structure. The manifold structure makes it possible to talk about the tangent space at a point, in particular the tangent space at the identity. This tangent space inherits a rich algebraic structure from the group structure on the manifold, called a Lie algebra. Many Lie algebras can be classified by combinatorial objects. Examples of infinite dimensional Lie algebras include affine Kac-Moody Lie algebras which have applications in many areas of mathematics and physics. Central extension is an important topic for infinite dimensional Lie algebras. In this talk, we will look at generalizations of affine Kac-Moody Lie algebras and locally finite Lie algebras, and discuss recent developments on central extensions of these algebras. No prior familiarity with Lie algebras will be assumed. Note special date and room: this is an interview talk for possible promotion to the tenure track for Prof. Sun. Affiliation: MTU Date: Feb 25 1:05pm Fisher 125 Title: Automorphisms of strongly regular graphs with applications to partial difference sets Speaker: Zeying Wang Abstract: Recently we proved a theorem for strongly regular graphs that provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then used this result to develop a new technique to study regular partial difference sets in Abelian groups. In 1994 S.L. Ma provided a list of parameter sets of regular partial difference sets of size at most 100 for which existence was known or had not been excluded. As an application of our results we excluded the existence of a regular partial difference set for all but two of the remaining 18 parameter sets from Ma's list. As a second application we provide a complete classification of partial difference sets in Abelian groups of order 4p2, p an odd prime. It turns out that the known examples are the only examples. These are, up to complements, the trivial examples, the PCP examples, and a sporadic example in an Abelian group of order 36. Only a few general classification results for partial difference sets are known. In this talk I will first introduce all necessary concepts and provide some historical background. Then I will present the main ideas used in our proofs and state our main results. I will conclude the talk with some ongoing research, and ideas for future research. Note special date and room: this is an interview talk for possible promotion to the tenure track for Prof. Wang. Affiliation: MTU Date: Mar 3 Title: Speaker: Abstract: Affiliation: Date: Mar 17 Title: Speaker: Abstract: Affiliation: Date: Mar 31 Title: Speaker: Abstract: Affiliation: Date: Apr 14 Title: Speaker: Abstract: Affiliation:

Spring 2015

 Date: Jan 29 Title: An introduction to partition theory, part 1 Speaker: William J. Keith Abstract: A topics-course level introduction to partition theory, suitable for graduate students and colleagues interested in picking up the basics of the subject. Day 1: the basic definitions and generating functions. Ferrers diagrams, conjugation, bijective and generating function proof. Partitions into odd and distinct parts; the pentagonal number theorem.Slides, Day 1 Affiliation: MTU Date: Feb 12 Title: An introduction to partition theory, part 2 Speaker: William J. Keith Abstract: Continuation. The q-factorial and the q-binomials. Lattice paths and partitions in boxes. The Young lattice, unimodality, and open questions regarding symmetric chain decomposition. The first Borwein Conjecture. Affiliation: MTU Date: Feb 19 Title: An introduction to partition theory, part 3 Speaker: William J. Keith Abstract: Continuation. Partition congruences: dissection techniques; the rank and crank; the open question of the parity of the partition function and its tertiarity. Affiliation: MTU Date: Mar 5 Title: An introduction to partition theory, part 4 Speaker: William J. Keith Abstract: Conclusion of the series. Modular forms techniques; proving identities and congruences with the theorems of Gordon, Hughes, and Newman, and Sturm. The m-regular partitions. Partitions into a small number of part sizes. Open questions on both of these. Affiliation: MTU Date: Mar 19 Title: Incidence structures, codes, and Galois geometry Speaker: Vladimir D. Tonchev Abstract: The lecture discusses a new invariant for finite incidence structures based on linear codes and Galois geometry, which has both an algebraic and a geometric description, and is motivated by the longstanding Hamada's conjecture about the minimum p-rank of the classical geometric designs. The new invariant was used recently in a joint work of the speaker with Dieter Jungnickel to prove a Hamada type characterization of the classical geometric designs having as blocks the d-subspaces of an n-dimensional projective or affine geometry over a finite field of order q. MSC2010: 05B05, 11T71, 51E20,94B27 Keywords: incidence structure, combinatorial design, finite geometry, p-rank, linear code, trace code, Galois closed code, Hamada conjecture.Slides are available here. Affiliation: MTU Date: Apr 2 Title: Incidence structures, codes, and Galois geometry Speaker: Vladimir D. Tonchev Abstract: Continuation and conclusion of last session's talk. Affiliation: MTU Date: Apr 16 Title: Speaker: Abstract: Affiliation: Date: Apr 23 Title: Speaker: Abstract: Affiliation: