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 2019, the Algebra & Combinatorics Seminar is scheduled biweekly on Thursdays from 1-2pm in lecture hall Fisher 138.

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 21Title: On the parity of the partition function
Speaker: Fabrizio ZanelloAbstract: 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 4Title: Noli turbare circulos meos!, or, Magic type labelings of cycle products
Speaker: Dalibor FroncekAbstract: A Cartesian product CmCn 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 12Title: Two New Families of Cubic Surfaces in Characteristic Two
Speaker: Anton BettenAbstract: 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
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Fall 2018

Date: Sep 13Title: Research Round-Robin
Speaker: Discrete Math facultyAbstract: The faculty in the Discrete Math group at Michigan Tech will give 5-minute descriptions of their research areas.

Affiliation: MTU
Date: Sep 27Title: Paley type and negative Latin square type partial difference sets in Abelian groups
Speaker: Zeying WangAbstract: 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 25Title: Iterated differences in the q-binomial coefficients
Speaker: William J. KeithAbstract: 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 15Title: A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs
Speaker: Melissa KeranenAbstract: 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 25Title: Partial difference sets in abelian groups
Speaker: Zeying WangAbstract: 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 15Title: Unimodality in the q-analogue of Frame-Robinson-Thrall
Speaker: William J. KeithAbstract: 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 29Title: The Good Will Hunting Problem
Speaker: Donald L. KreherAbstract: 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 19Title: 2-Block intersection graphs in triple systems
Speaker: Melissa KeranenAbstract: 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 ∣ BiBj ∣ = 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 26Title: (k,j)-colored partitions and the hooklength formula
Speaker: Emily AnibleAbstract: 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 2017

Date: Jan 23Title: A class of trivalent vertex-transitive graphs
Speaker: Don KreherAbstract: Bojan Mohar of Simon Fraser University posed the problem to classify trivalent vertex-transitive graphs X whose edge set E(X) can be partitioned into a 2-factor F and a 1-factor I such that Aut(X) preserves the partition (I,F). We shall see in this introductory talk that even in the case when F consists of a single cycle, this problem is not as straightforward as it might seem at first.

Affiliation: MTU
Date: Feb 6Title: Some new Kirkman signal sets
Speaker: Melissa KeranenAbstract: A decomposition of the blocks of an STS(v) into partial parallel classes of size m is equivalent to a Kirkman signal set KSS(v, m). We give decompositions of STS(4v-3) into classes of size v-1 when v ≡ 3 (mod 6). We also give decompositions of STS(v) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to 3 (mod 6). These results produce new families of KSS(v, m).

Affiliation: MTU
Date: Feb 27Title: Linear codes with complementary duals from regular graphs invariant under finite groups
Speaker: Bernardo RodriguesAbstract: We determine all linear codes of length 50 over Fp (p a prime) which admit the projective special unitary group U3(5) as an automorphism group. By group representation theory means we prove that these can all be realized as submodules of the permutation module FΩ where Ω corresponds to the vertex set of the Hoffman-Singleton graph.

Prof. Rodrigues' more detailed LaTeX abstract can be found in pdf here.

Affiliation: University of KwaZulu-Natal
Date: Mar 13Title: Partitions into distinct parts and unimodality
Speaker: Fabrizio ZanelloAbstract: We discuss the (non)unimodality of the rank-generating function Fλ of the poset of partitions with distinct parts contained inside a given partition λ. This work, in collaboration with Richard Stanley (European J. Combin., 2015), is in part motivated by an attempt to place into a broader context the unimodality of Fλ(q)=∏ni=1(1+qi), the rank-generating function of the "staircase" partition λ=(n,n-1,…,1), for which determining a combinatorial proof remains an outstanding open problem.

We present a number of results and conjectures on the polynomials Fλ, and also discuss a few interesting recent developments. These include a (prize-winning) paper by Levent Alpoge, who solved our conjecture on the unimodality of Fλ when λ=(n,n-1,…,n-c) (the "truncated staircase"), for n ≫ c.

Affiliation: MTU
Date: Mar 31, 1pm

Fisher 101

Title: Old and New Results on Universal Cycles Obtained in Collaboration with Undergraduate and Graduate Students
Speaker: Anant GodboleAbstract: A universal cycle generalizes the notion of deBruijn cycles to combinatorial structures such as graphs, subsets, set partitions, Venn diagram allocations, etc. In a very real sense, universal cycles are combinatorial designs, though they have not been recognized as such, or studied by the MSC 05BXX community in depth. I will present 10 years' worth of work, with undergraduates, REU students, and graduate students that has contributed significantly to the body of knowledge on existence of universal cycles for the above-mentioned structures as well as for hypergraphs, naturally labeled posets, words with restrictions, lattice paths, etc.

Note the special Friday time and room: Prof. Godbole is this week's Colloquium speaker. The Colloquium data is hosted here as being hopefully of interest to Combinatorics Seminar attendees.

Affiliation: East Tennessee State University
Date: Apr 3, 11am

EERC 216

Title: Covering and Packing Threshold Progressions
Speaker: Anant GodboleAbstract: As one progresses from each member of a family of objects A being "covered" (with high or low probability) by at most one object in a random collection C, to being covered at most g times, a hierarchy of thresholds emerge. These may be smooth in their transition from one level to the next, or might feature a gap. Examples include packing and covering of t sets by k sets; Sidon sets/additive bases; weakly union free families; coverage in the permutation pattern lattice; and, a well-known motivating example.

This is joint work with Thomas Grubb, Kyutae Han, Bill Kay, Zach Higgins, and Zoe Koch.

Please note the special morning time and room for this week's Seminar.

Affiliation: East Tennessee State University
Date: Apr 10Title: Hamada's conjecture in affine spaces
Speaker: Mustafa GezekAbstract: In 1973, Hamada made the following conjecture: Let D be a geometric design having as blocks the d-subspaces of PG(n,q) or AG(n,q), and let m be the p-rank of D. If D' is a design with the same parameters as D, then the p-rank of D' is greater than or equal to m, and the equality holds if and only if D' is isomorphic to D. In 1986, Tonchev, and more recently Harada, Lam and Tonchev, Jungnickel and Tonchev, and Clark, Jungnickel and Tonchev found designs having the same parameters and p-rank as certain geometric designs, hence provide counterexamples to the "only-if" part of Hamada’s conjecture. In this work, we discuss some properties of the three known nonisomorphic 2-(64,16,5) designs of 2-rank 16, one being the design of the planes in the 3-dimensional affine geometry over the field of order 4, and try to find an algebraic way to use the similarities between these designs in a search for counterexamples to Hamada's conjecture in affine spaces of higher dimension.
Affiliation: MTU
Date: Apr 17Title: Resolvable Steiner 2-designs and maximal arcs in projective planes
Speaker: Vladimir TonchevAbstract: The topic of this talk is a combinatorial characterization of resolvable Steiner 2-designs embeddable as maximal arcs in finite projective planes. Applications to the known planes planes of order 16 are discussed.

Affiliation: MTU

Fall 2016

Date: Sep 8Title: Partitions into a small number of part sizes
Speaker: William J. KeithAbstract: Overpartitions - partitions in which the last instance of a given part size may be overlined, or not - are a current topic of interest in partition theory. If vi(n) is the number of partitions of n in which exactly i part sizes appear, then the number of overpartitions of n is

p̅(n)=2v1(n)+4v2(n)+8v3(n)+ . . .,

so it is of interest to study the vi(n). We will do so with a wide array of the types of tools found in partition theory: combinatorial bijections, generating function identities, and modular forms. The talk will be accessible to graduate students.

Affiliation: MTU
Date: Sep 22Title: Partitions into a small number of part sizes, Part II
Speaker: William J. KeithAbstract: Part II of last session's talk.
The slides for this talk can be downloaded here.

Affiliation: MTU
Date: Sep 29, 5pmTitle: Plane Tilings
Speaker: Richard StanleyAbstract: Given a set of plane shapes (tiles), together with a region R of the plane, a tiling of R is a filling of R with the tiles without overlap in their interiors. A jigsaw puzzle is a familiar (though not very mathematical) example. We will survey some interesting mathematics associated with plane tilings. In particular, we will discuss how mathematics can be used to investigate the following questions: Is there a tiling? If so, how many are there? If it is infeasible to find the exact number of tilings, then approximately how many tilings are there? If a tiling exists, is it easy to find? Is it easy to prove that a tiling does not exist? Is it easy to convince someone that a tiling does not exist? What does a "typical" tiling with the given tiles and region look like? What are the relations among different tilings? What special properties, such as symmetry, could a tiling possess? These questions involve such subjects as combinatorics, group theory, probability theory, number theory, and computer science.

This is the public lecture for the second annual Kliakhandler Lecture Series. Please note a special time and room: 5pm - 6pm in Dow 641.

Affiliation: MIT
Date: Sep 30Title: A survey of alternating permutations
Speaker: Richard StanleyAbstract: A permutation a1, a2, . . . is alternating if a1 > a2 < a3 > a4 < a5 > . . . . If En is the number of alternating permutations of 1, 2, . . . , n, then

n ≥ 0 xn / n! = sec x + tan x.

We will discuss several aspects of the theory of alternating permutations. Some occurrences of the numbers En, such as counting orbits of group actions and volumes of polytopes, will be surveyed. The behavior of the length of the longest alternating subsequence of a random permutation will be analyzed, in analogy to the length of the longest increasing subsequence. We will also explain how various classes of alternating permutations, such as those that are also fixed-point free involutions, can be counted using a certain representations of the symmetric group Sn whose dimension is En.

This is the Department's Colloquium talk this week, and is the second, specialist-oriented lecture of this year's Kliakhandler Lecture Series. Please note a special time and room: it runs Friday 1-2pm in Fisher 138.

Affiliation: MIT
Date: Nov 4Title: Applications of linear algebraic methods in combinatorics and finite geometry
Speaker: Qing XiangAbstract: Most combinatorial objects can be described by incidence, adjacency, or some other (0,1)-matrices. So one basic approach in combinatorics is to investigate combinatorial objects by using linear algebraic parameters (ranks over various fields, spectrum, Smith normal forms, etc.) of their corresponding matrices. In this talk, we will look at some successful examples of this approach; some examples are old, and some are new. In particular, we will talk about the recent bounds on the size of partial spreads of H(2d-1, q2) and on the size of partial ovoids of the Ree-Tits octagon.

This is the Departmental Colloquium talk for this week: please note the special date. It runs Friday November 4th 1-2pm in Fisher 127.

Affiliation: University of Delaware
Date: Nov 10Title: Great Expectations. . . and projective planes
Speaker: Juan MiglioreAbstract: The so-called ``Lefschetz Properties'' have attracted a great deal of attention in recent years. Informally, they have the following form. Given a fixed set of dimensions of some objects, you make a ``general choice'' of something (to be explained in the talk) that produces a new dimension. There is an expected value for this dimension, and the game is to understand under what circumstances this expected value fails to be achieved. There is a striking variety of situations where this kind of behavior has been studied. In this talk we give two very different examples, both having to do with projective planes. After recalling the necessary facts about projective planes, we describe a very geometric setting (joint work with D. Cook II, B. Harbourne and U. Nagel) and a much more algebraic setting (joint work with D. Cook II, U. Nagel and F. Zanello).

Affiliation: Notre Dame
Date: Nov 30Title: The average number of rational points on genus two curves is bounded
Speaker: Levent AlpogeAbstract: It is a theorem of Faltings (resolving a conjecture of Mordell) that a smooth projective curve of genus g > 1 defined over a number field must have finitely many rational points. In this talk we ask, informally, how 'finitely many.' There is speculation as to whether the number of rational points on, say, a genus 2 curve over, say, ℚ should be uniformly bounded (the current record, which has stood since 2008, is 642). We will show that at least the average number of points on these curves is bounded.

This is the Departmental Colloquium talk for this week: please note the special date. It runs Wednesday November 30th 1-2pm in Fisher 127.

Affiliation: Princeton
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Spring 2016

Date: Jan 21Title: Decomposing the blocks of a Steiner triple system into partial parallel classes
Speaker: Melissa KeranenAbstract: 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 18Title: Independent sets in geometries
Speaker: Stefaan De WinterAbstract: 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 23
1:05pm
Fisher 125
Title: Infinite Dimensional Lie Algebras
Speaker: Jie SunAbstract: 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 WangAbstract: 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
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Fall 2015

Date: Sep 4Title: The ternary Pace code
Speaker: Jürgen BierbrauerAbstract:

Note the special Friday date: Prof. Bierbrauer's retirement reception will be held Thursday during the usual Seminar slot. The talk will be held in Fisher 329 instead of the usual room.

Affiliation: MTU
Date: Sep 24Title: R-sequenceable groups and orthogonal directed cycles, Part 1
Speaker: Don KreherAbstract: In his 1974 solution to the map colouring problem for all compact 2-dimensional manifolds except the sphere, Gerhard Ringle was led to the following group-theoretic problem:

  When can the non-identity elements of a group of order n be cyclically arranged in a sequence g0, g1,g2,. . .,gn-1 such that the quotients gi-1gi+1, i=0,1,2,. . .,n (with subscripts modulo n) are all distinct?

The complete Cayley graph X on a group G is the complete directed graph where the edge (x,y) is labeled by x-1y. The edges with a given label z in G form a 1-factor Fz and {Fz: z ∈ G} is a 1-factorization of X. A subgraph H of X is an orthogonal subgraph if it contains exactly one edge of each of the one-factors. This year Brian Alspach got me interested in the following problem:

    For which groups G does the complete Cayley graph X admit an orthogonal directed cycle?

It is not difficult to see that the problem of Alspach and the problem of Ringle are the same. I believe we (Alspach, Kreher, Pastine) will soon be able to show that every abelian group is R-sequenceable with possibly a small list of exceptions.

Affiliation: MTU
Date: Oct 8
5:00 pm
ME-EM 112
Title: Kliakhandler Lecture The Indian Genius, Ramanujan: His Life and the Excitement of his Mathematics
Speaker: George AndrewsAbstract: This talk focuses on the famous Indian genius, Ramanujan, and will provide an account of his amazing, albeit short life. We shall try to lead gently from some simple problems from elementary number theory to a discussion of some of Ramanujan's achievements. The mathematics used in the talk will be restricted to arithmetic. I hope to provide some idea of his profound impact on contemporary mathematics. I will conclude with an account of some of the various literary and theatrical enterprises that have attempted to tell the story of Ramanujan.

Note the special time and room: this is the public Kliakhandler lecture given by this year's invited speaker, the eminent partition theorist George Andrews.

Affiliation: Penn State
Date: Oct 9
1:05pm
Fisher 329
Title: Kliakhandler Colloquium: Partitions, Dyson and Ramanujan
Speaker: George AndrewsAbstract: Freeman Dyson's first published paper, Some Guesses in the Theory of Partitions (Eureka, 1944), looked at Ramanujan's congruences for the partition function combinatorially. Dyson was an undergraduate at the time. The paper raised many more questions than it answered. However, the questions asked in this paper have led to a cornucopia of deep and surprising theorems; research on Dyson's original ideas continues to this day. The object of this talk will be to provide an account of what led up to Dyson's questions, and what the questions were. We shall conclude with a gentle account of some of the combinatorics that has arisen from this undergraduate masterpiece.

Note the special Friday date and room: this is the Colloquium talk given by Prof. Andrews.

Affiliation: Penn State
Date: Oct 29Title: Orientable Distance Magic Graphs
Speaker: Bryan FreybergAbstract: Let G=(V,E) be a graph of order n. A distance magic labeling of G is a bijection ℓ:V → {1,2,. . .,n} for which there exists a positive integer k such that ∑x ∈ N(v)(x) = k for all , where N(v) is the open neighborhood of v. In this talk we will generalize the notion of distance magic labeling for oriented graphs.

Affiliation: MTU
Date: Nov 4
1:05pm
Fisher 133
Title: Reduced words: counting and exploiting
Speaker: Bridget TennerAbstract: The reduced words of a Coxeter group element are fundamental to understanding the group's architecture and to defining such objects as the Bruhat order on the group and several graph structures. These words, and the Coxeter relations connecting them, lead to a range of enumerative problems. These address both well-known objects and others that are less so. Some of these problems were answered many years ago, others only more recently, and still others remain open.

Note the special Wednesday date and room: this is the week's Colloquium talk.

Affiliation: DePaul
Date: Nov 19Title: Distance magic and group distance magic graphs
Speaker: Dalibor FrončekAbstract: Let G be a graph with n vertices and f a bijection f:V(G) → {1,2,. . .,n}. We define the weight of vertex x as the sum of the labels of its neighbors, that is,

w(x) = ∑x y ∈ E(G) f(y).

At IWOGL 2010, Arumugam presented a list of open problems on distance magic labelings. We present solutions to some of them as well as some other recent results.

However, it turns out that this type of magic labeling is very restrictive and consequently even many classes of vertex transitive graphs are not distance magic.

As an example, we prove that for d ≡ 0, 1, 3 (mod 4) the hypercube Qd with 2d vertices is not distance magic. On the other hand, we disprove a conjecture by Acharya, Rao, Singh and Parameswaran, who believed that hypercubes are not distance magic except Q2 and present a distance magic labeling for Q6. This was recently generalized by Gregor and Kovar who found a distance magic labeling of Qd for any d ≡ 2 (mod 4).

Such negative results then raise a question whether it would not be more natural to perform the addition in n rather than in . Graphs that satisfy the above definition with the provision that the addition is performed in n will be called n-distance magic.

To support this idea, we show some examples of graphs that are not distance magic yet are n-distance magic. We show that when we perform addition in 2d rather than in , then Qd is 2d-distance magic if and only if d is even.

We present some results on Γ-distance magic labelings of products of cycles and pose several open problems.

Affiliation: UMN-Duluth
Date: Dec 3Title: R-sequenceable groups and orthogonal directed cycles, Part 2
Speaker: Don KreherAbstract: Part 2 of Prof. Kreher's Sep 24 talk of this semester.

Affiliation: MTU
Date: Dec 10Title: On the Density of the Odd Values of the Partition Function
Speaker: Samuel JudgeAbstract: There has been a long standing conjecture that the partition function is equidistributed modulo 2. Alas, to this point, it is not even known if the density of the odd values is positive. In this talk, we present a new way to attack this problem. Namely, we show that positive density (of the odd values) of certain powers of the partition function (5, 7, 11, 13, 17, 19, 23, 25) implies positive density of the partition function itself, using multiple tactics, ranging from simple algebra to a new technique involving modular forms recently introduced by Radu.

Affiliation: MTU

Spring 2015

Date: Jan 29Title: An introduction to partition theory, part 1
Speaker: William J. KeithAbstract: 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 12Title: An introduction to partition theory, part 2
Speaker: William J. KeithAbstract: 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.

Slides, Day 2

Affiliation: MTU
Date: Feb 19Title: An introduction to partition theory, part 3
Speaker: William J. KeithAbstract: Continuation. Partition congruences: dissection techniques; the rank and crank; the open question of the parity of the partition function and its tertiarity.

Slides, Day 3

Affiliation: MTU
Date: Mar 5Title: An introduction to partition theory, part 4
Speaker: William J. KeithAbstract: 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.

Slides, Day 4

Affiliation: MTU
Date: Mar 19Title: Incidence structures, codes, and Galois geometry
Speaker: Vladimir D. TonchevAbstract: 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 2Title: Incidence structures, codes, and Galois geometry
Speaker: Vladimir D. TonchevAbstract: Continuation and conclusion of last session's talk.

Affiliation: MTU
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Fall 2014

Date: Sep 17Title: A census of Hughes-Kleinfeld semifields
Speaker: Jürgen BierbrauerAbstract: Semifields (or non-associative division rings) are algebraic structures which resemble fields. In fact, just forget the associativity and commutativity of field multiplication and you arrive at the axioms of a semifield. The geometric meaning is the following: semifields coordinatize the projective planes which are translation planes and also dual translation planes.

The nuclei of a semifield measures how far it is from being associative; the autotopism group is the pertinent group of symmetries. The Hughes-Kleinfeld semifields (1960) are non-commutative but very close to being associative. In the talk we will see an elementary algebraic mechanism which allows us to decide  

• the number of Hughes-Kleinfeld semifields of a given order, and
• the orders of the autotopism groups.

This includes the smallest proper semifield. It is of order 16 and has precisely 108 autotopisms. The corresponding projective plane has 214 × 27 collineations.

Affiliation: MTU
Date: Oct 1Title: Universal extensions of slm|n over Z2-graded algebras
Speaker: Jie SunAbstract: In this talk central extensions of the Lie superalgebra slm|n(A) are constructed, where A is a Z2-graded superalgebra over a commutative ring K. The Steinberg Lie superalgebra stm|n(A) plays a crucial role. We show that stm|n(A) is a central extension of slm|n(A) for m+n ≥ 3. We will study the kernel of this central extension and discuss the universality of this construction.

Affiliation: MTU
Date: Oct 15Title: Colored partitions with restrictions on the number of colors appearing
Speaker: William J. KeithAbstract: We will define overpartitions, k-colored partitions, and an object that combines and generalizes the two. The talk will start with an introduction to why these objects are interesting and some of the questions surrounding them, and then give a tour of the speaker's recent research into the new type.

Affiliation: MTU
Date: TBA(Delayed)

Title: On Hadamard matrices and their applications
Speaker: Vladimir TonchevAbstract: This talk surveys some problems and results concerning Hadamard matrices and their generalizations, such as regular Hadamard matrices, Bush type matrices, Hadamard difference sets, generalized Hadamard matrices over groups, and related combinatorial structures and error-correcting codes.

Affiliation: MTU
Date: Oct 29Title: Progress on 3-GDDs with five groups
Speaker: Don KreherAbstract: We study the edge decompositions of Kg0,g1,g2,g3,g4 into triangles. Such decompositions are also known as 3-GDDs with five groups. So far we have settled the existence of 3-GDDs with five groups when there are only one or two group sizes. (Joint work with Charles J. Colbourn and Melissa S. Keranen.)

Affiliation: MTU
Date: Nov 7Title: Flag Algebras and Applications to Permutations
Speaker: Bernard LidickýAbstract: Flag algebras is a method, recently developed by Razborov, designed for attacking problems in extremal combinatorics. There are recent applications of the method also in discrete geometry or permutation patterns. The aim of talk is to give a gentle introduction to the method and show some of its applications, mainly to permutations.

The talk is based on a joint work with J. Balogh, P. Hu, O. Pikhurko, B. Udvari, and J. Volec.

Note special date: Prof. Lidický will be speaking at the Colloquium.)

Affiliation: Iowa State
Date: Nov 14
10 am
Fisher 222
Title: Decompositions of complete bipartite graphs into prisms
Speaker:Dalibor FrončekAbstract: A generalized prism, or more specifically an (0,j)-prism of order 2n (where n is even) is a cubic graph consisting of two cycles u0,v1,. . .,vn-1 and v0,v1,. . .,un-1 joined by two sets of spokes , namely u1v1, u3v3,. . ., un-1vn-1 and u0vj, u2vj+2,. . ., un-2vj-2.

The question of factorization of complete bipartite graphs into (0,j)-prisms was completely settled by the author and S. Cichacz. Some partial results on decompositions of complete bipartite graphs have also been obtained by S. Cichacz, DF, and P. Kovar, and on decompositions of complete graphs S. Cichacz, S. Dib, and DF. The problem of decomposition of complete graphs into prisms of order 12 and 16 was completely solved by S. Cichacz, DF and M. Meszka.

We will present a complete solution for the decomposition of complete bipartite graphs into (0,0)-prisms (that is, the usual prisms).

We will also show why the method used for this problem works particularly well in Duluth, MN and Houghton, MI.

Please note special date, time, and room.

Affiliation: UMN-Duluth
Date: Dec 3Title: The Hamilton-Waterloo Problem with triangle factors and C3x factors
Speaker: Melissa KeranenAbstract: The Hamilton-Waterloo Problem in the case of C(m) factors and C(n) factors asks if Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. In this talk, I will discuss some general constructions for such decompositions and apply them to the case where m=3 and n=3x.

Affiliation: MTU
Date: Dec 10Title: Partial difference sets in small Abelian groups
Speaker: Stefaan de WinterAbstract: In [1] Ma provided a list of parameter sets of regular (v,k,λ,μ) partial difference sets with k ≤ 100 in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. In [2] Ma excluded existence for 13 of these parameter sets. In this talk I will explain how a recent result of De Winter, Kamischke and Wang on strongly regular graphs can be used to develop a few new techniques to study regular partial difference sets in Abelian groups. As an application we exclude the existence of a regular partial difference set for all but two of the undecided upon parameter sets from Ma's list. This is joint work with Z. Wang.

References

[1] S.L. Ma, A survey of partial difference sets, Designs, Codes, Cryptogr. 4, 221-261, 1994.

[2] S.L. Ma, Some necessary conditions on the parameters of partial difference sets, J. Statist. Plann. Inference 62, 47-56, 1997.

Affiliation: MTU

Spring 2014

Date: April 24Title: m-Regular partitions and eta-function symmetries
Speaker: William J. KeithAbstract: A famous result in partition theory is Ramanujan's congruences, that the number of partitions of 5n+4 is divisible by 5, those of 7n+5 divisible by 7, and those of 11n+6 divisible by 11. These are now understood as members of an infinite family of such congruences, unified by the symmetries of modular forms. More recent work has been devoted to finding congruences for the m-regular partitions, those in which parts may not be divisible by m. These are now numerous, but we do not yet have a similar unifying structure. This talk will outline each of these ideas, demonstrate that it is now fairly easy to prove many conjectured congruences with current techniques, and lay out a few ideas, tentative as yet, for constructing such general theorems.
Affiliation: MTU
Date: April 17Title: Computing minimum strong rainbow colorings of block graphs
Speaker: Melissa KeranenAbstract: Let G be an undirected graph that is simple and finite. A path in G is rainbow if no two edges of it are colored the same. A graph is rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, G is strong rainbow connected. The minimum number of colors needed to make G strong rainbow colored is known as the strong rainbow connection number and is denoted by src(G). A strong rainbow coloring of G using src(G) colors is called a minimum strong rainbow coloring. A block graph is an undirected graph where every maximum biconnected component, known as a block, is a clique. Given a block graph G, we give an algorithm that constructs a minimum strong rainbow coloring of G in linear time. We also give a simpler linear time algorithm for computing src(G).
Affiliation: MTU
Date: March 20Title: Rank sizes and lattice properties of differential posets
Speaker: Pat ByrnesAbstract: Differential posets are a generalization of Young's lattice introduced by Stanley. These posets have a simple definition, yet there are many basic open questions remaining to be answered. We will discuss some results and open questions on rank sizes and lattice properties of differential posets.
Affiliation: Century College
Date: March 6Title: On Hamilton decomposition of Cayley graphs over elementary abelian groups of characteristic 3
Speaker: Adrian PastineAbstract: In this work we give an algorithm to decompose Cayley graphs over the elementary abelian group of characteristic 3 with 27 elements. Ideas on how to proceed with higher dimensions are given, but not yet proved.
Affiliation: MTU
Date: February 27Title: Vertex transitive graphs of prime square order are Hamilton decomposable
Speaker: Don KreherAbstract: The talk will show that when p is an odd prime, all vertex transitive graphs on p^2 points can be edge decomposed into Hamilton cycles.
Affiliation: MTU
Date: February 13Title: Central extensions of Lie algebras
Speaker: Jie SunAbstract: Extensions are used to build bigger groups from smaller groups. 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: root systems and the corresponding Coxeter-Dynkin diagrams. Central extensions of Lie algebras are special extensions which play important roles in both the structure theory and the representation theory of Lie algebras. In this talk, I will survey some recent developments of this topic and mention a few open problems.
Affiliation: MTU
Date: Jan 23Title: Pure O-sequences, f-vectors of pure simplicial complexes, and other level h-vectors
Speaker: Fabrizio ZanelloAbstract:Gorenstein and level Hilbert functions, and their monomial and squarefree-monomial coun- terparts, play an important role in combinatorics and commutative algebra, also in light of their connections with a number of other topics.

This talk will attempt to review some of the old and new developments that have been shaping this field during the past 35 years: from R. Stanley’s seminal contributions in the late Seventies, to the algebraic progress of the Nineties, to the comeback center stage of the combinatorial aspect of the story during the last few years. In particular, we will discuss some of the most recent results concerning pure O-sequences (i.e., monomial level Hilbert functions) and f-vectors of pure simplicial complexes (i.e., squarefree pure O-sequences), as well as some of their fascinating interactions with other areas.

The talk will include a discussion of future research directions, and a selection of conjectures and open problems accessible to young researchers interested in combinatorics or commutative algebra.

Affiliation: MTU

Fall 2013

Date: December 12Title:
Speaker: Dalibor FrončekAbstract:

Affiliation: UMN-Duluth
Date: November 21Title: Classifying spreads and packings in finite projective three-spaces
Speaker: Anton BettenAbstract:Spreads in projective spaces are systems of lines that are pairwise disjoint and cover all points. The interest in classifying spreads originates from the desire to classify translation planes, and it has been observed multiple times that the two problems are equivalent (Andre 1954, Bruck/Bose 1964). Packings of projective three-space are built up from spreads. A packing is a set of pairwise (line-)disjoint spreads that together use up all the lines. These objects arose as early as 1850 when Reverend Kirkman posed the problem of the 15 schoolgirls, which spawned a new mathematical discipline that is now known as Design Theory.

We plan to report on progress on the problem of classification of spreads and packings of PG(3,q):

A) The spreads containing a regulus when q=8 or q=9.

B) The packings of PG(3,3).

The Computer Science tools that we use are Exact Cover and Rainbow Cliques in graphs.

Affiliation: Colorado State University
Date: November 7Title: An extremal characterization of projective planes
Speaker: Stefaan De WinterAbstract:

Affiliation: MTU
Date: October 31Title: Projective polynomials and semifields
Speaker: Juergen BierbrauerAbstract:

Affiliation: MTU
Date: October 24Title: Decomposition of complete graphs into kayak paddles
Speaker: Leah TollefsonAbstract:

Affiliation: MTU
Date: October 17Title: The Robinson-Schensted-Knuth algorithm: a gentle introduction
Speaker: William J. KeithAbstract: The RSK algorithm is a beautiful and useful theorem, fundamental in enumerative combinatorics. It describes a bijection between permutations of n elements, and ordered pairs of standard Young tableaux of the same shape. We will describe and prove the bijection, and illustrate its use proving a theorem for which higher cases remain open.

Affiliation: MTU

Spring 2013

Date: April 17Title: Algebraic applications of the LLL algorithm
Speaker: Benjamin FedorkaAbstract:

Affiliation: MTU
Date: April 10Title: An algebraic problem related to semifields
Speaker: Juergen BierbrauerAbstract:

Affiliation: MTU
Date: April 3Title: Linear representations of subgeometries
Speaker: Stefaan De WinterAbstract:

Affiliation: MTU
Date: March 27Title: Hamilton decomposition of vertex transitive graphs of order p squared, p prime
Speaker: Don KreherAbstract:

Affiliation: MTU
Date: February 27 Title: Permutation Codes
Speaker: Juergen BierbrauerAbstract:

Affiliation: MTU