# Colloquia and Seminars

Our department runs several research seminars.

Our next graduate colloquium will be held on 5/24, from 2pm to 3pm, in CMK 309.

George E. Andrews

Pennsylvania State University

Abstract: This talk is devoted to discussing the implications of a very elementary technique for proving mod 4 congruences in the theory of partitions.  It starts with a tribute to the late Hans Raj Gupta and leads in unexpected ways to partitions investigated by Clark Kimberling, to Bulgarian Solitaire, and to Garden of Eden partitions.

• Past Colloquia

Friday, May 24th 2019, 2 PM - 3 PM in CMK 309:

Operator Algebras that one can see

Piotr Hajac

CU Boulder / IMPAN

Abstract:  Operator algebras are the language of quantum mechanics just as much as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. One of these endeavors is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C*-algebras) and locally compact Hausdorff spaces. Thus noncommutative C*-algebras are thought of as quantum topological spaces, and are researched from this perspective. However, such C*-algebras can enjoy features impossible for commutative C*-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new ("quantum") intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces, and how it explains their K-theory.

Friday, May 17th 2019, 2 PM - 3 PM in CMK 309:

Decidability for residuated lattices and substructural logics

Gavin St. John (PhD Dissertation Defense)

University of Denver

Abstract: Decidability is a fundamental problem in mathematical logic. We address decidability properties for substructural logics, particularly for their extensions by so-called simple structural rules. Substructural logics are a mathematical logic framework that encompasses most of the interesting nonclassical logics, and thus have an interesting comparative potential. A powerful tool to study substructural logics is given by their algebraic semantics, residuated lattices. Indeed, syntactic properties of algebraizable logics can be rendered as semantical properties for a particular variety of algebras, and vice versa. In particular, logics extended by simple structural rules algebraically correspond to varieties axiomatized by so-called simple equations. Our main results involve proving decidability and undecidability for broad classes of such structures.

Friday, May 10th 2019, 2 PM - 3 PM in CMK 309:

Tukey Order, Small Cardinals, and Oﬀ-diagonal Metrization

Ziqin Feng

Auburn University

Abstract: In 1945, Sneider proved that any compact space $X$ with a $\delta$-diagonal is metrizable. Motivated by this result, we deﬁne a space with an $M$-diagonal in what follows. Let $\mathcal{K}(M)$ be the collection of all compact subsets of $M$. A space $X$ is dominated by $M$, or $M$-dominated, if $X$ has a $\mathcal{K}(M)$-directed compact cover. We say $X$ has an $M$-diagonal if $X^2\backslash\Delta$ is dominated by $M$, where $\Delta = \{(x,x) : x \in X \}$. We investigate spaces with a $\mathbb{Q}$-diagonal, where $\mathbb{Q}$ is the space of rational numbers, and prove that any compact space with a $\mathbb{Q}$-diagonal is metrizable. This answers an open question raised by Cascales, Orihuela, and Tkachuk positively. In the proof, we use Tukey order and a few independent statements of small cardinals.

Friday, April 26th 2019, 2 PM - 3 PM in CMK 309:

Decomposing Graphs into Edges and Triangles

Iowa State University

Abstract: Let $\pi_3(G)$ be the minimum of twice the number of $K_2$'s plus three times the number of $K_3$'s over all edge decompositions of a graph $G$ into copies of $K_2$ and $K_3$. Let $\pi_3(n)$ be the maximum of $\pi_3(G)$ over graphs with $n$ vertices. This specific extremal function was studied by Győri and Tuza, and recently improved by Král', Lidický, Martins and Pehova. We extend the proof by giving the exact value of $\pi_3(n)$ for large $n$ and classify the extremal examples. We also provide a generalization to $K_2$ and $K_3$ decompositions with different weight ratios.
This is joint work with Bernard Lidický, Yani Pehova, Oleg Pikhurkho, Florian Pfender, and Jan Volec.

Friday, April 19th 2019, 2 PM - 3 PM in CMK 309:

A locally trivial talk​

Mariusz Tobolski​

IMPAN

Abstract: This talk is inspired by the synergy of mathematics and physics. On one hand, the investigation of symmetries through group actions led to the notion of a principal bundle in algebraic topology, which found applications in gauge theory in physics. On the other hand, understanding quantization as a noncommutative deformation is one of the starting points of noncommutative topology. We generalize the concept of a compact principal bundle to the realm of noncommutative topology with emphasis on the local triviality condition.

Friday, April 5th 2019, 2 PM - 3 PM in CMK 309:

Finite constraint: A combinatorial concept with Ramsey theoretic applications

Rebecca Coulson

West Point

Abstract: In their 2005 seminal paper, "Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups," Kechris, Pestov, and Todorcevic, tied together the fields of model theory, Ramsey theory, descriptive set theory, and topological dynamics, via the concept of homogeneity. A key tool used is a combinatorial concept called finite constraint. We will show that a class of graphs called metrically homogeneous graphs, of interest to model theorists and combinatorialists, is finitely constrained, and we show how this is used to derive a whole host of Ramsey theoretic and topological dynamical applications.

### Algebra and Logic Seminar

Our next algebra and logic seminar will be held on 5/17, from 9am to 9:50am, in CMK 301.

Simple weight modules with finite-dimensional weight spaces

David Ridout

University of Melbourne

Abstract: Let g be a finite-dimensional simple Lie algebra. Motivated by the representation theory of the simple affine vertex algebra L_k(g), we are led to study certain categories of simple weight g-modules with finite-dimensional weight spaces. These may be understood using Mathieu’s theory of coherent families. We shall review this theory and generalise it in order to understand the representation theory of L_k(g).

• Past Algebra and Logic Seminars

May 17, 2019, 9 - 9:50 am, CMK 301

Simple weight modules with finite-dimensional weight spaces

David Ridout

University of Melbourne

Abstract: Let g be a finite-dimensional simple Lie algebra. Motivated by the representation theory of the simple affine vertex algebra L_k(g), we are led to study certain categories of simple weight g-modules with finite-dimensional weight spaces. These may be understood using Mathieu’s theory of coherent families. We shall review this theory and generalise it in order to understand the representation theory of L_k(g).

May 10, 2019, 9 - 9:50 am, CMK 301

Rainbow-Cycle-Forbidding Edge Colorings

Andrew Owens

Auburn University

Abstract: A JL-coloring is an edge coloring of a connected graph G that forbids rainbow cycles and uses the maximum number of colors possible, |V(G)|-1. In this talk we discuss the correspondence between JL-colorings of a graph on n vertices and (isomorphism classes of) full binary trees with n leafs. Furthermore, we will explore the question of properly edge coloring connected graphs in order to avoid rainbow cycles.

April 26, 2019, 9 - 9:50 am, CMK 301

Irreducible convergence and irreducibly order-convergence in T_0 spaces

Kaiyun Wang

Abstract: In this talk, we aim to lift lim-inf-convergence and order-convergence in posets to a topology context. Based on the irreducible sets, we define and study irreducible convergence and irreducibly order-convergence in T0 spaces. Especially, we give sufficient and necessary conditions for irreducible convergence and irreducibly order-convergence in T0 spaces to be topological.

April 5, 2019, 9 - 9:50 am, CMK 301

W-algebras and integrability

Tomas Prochazka

University of Munich, Arnold Sommerfeld Center for Theoretical Physics

Abstract:  I will review what W-algebras are from the conformal field point of view. After that I'll explain the definition of affine Yangian by Arbesfeld-Schiffmann-Tsymbaliuk as an associative algebra with generators and relations. Finally I'll explain how Miura transformation can be used as a bridge between these two pictures.

February 8, 2019, 9-9:50 am, CMK 207

Title: Inner Partial Automorphisms of Inverse Semigroups I

Michael Kinyon

University of Denver

Abstract: Groups are the algebraic structures underlying symmetries, that is, structure-preserving permutations of a set. Inverse semigroups, a generalization of groups, were introduced in the1950s more or less independently by Ehresmann in France, Preston in the UK and Wagner in the Soviet Union. They are the algebraic structure underlying partial symmetries, that is, partial bijections between subsets.  For instance, just as the exemplar of a group is the symmetric group on a set, the exemplar of an inverse semigroup is the symmetric inverse monoid of all partial bijections between subsets. It is not an exaggeration to say that inverse semigroups are the most well studied class of semigroups.  For the first part of this talk or talks(?), I will start by giving a gentle introduction to inverse semigroups, outlining some of their basic structure, and going so far as to sketch the proof of the Wagner-Preston Theorem, which is the generalization to inverse semigroups of Cayley’s Theorem. Then I will turn to what I have been working on.  Recall that an inner automorphism of group G is a permutation \phi_g: G→G, g \in G,  defined  by \phi_g (x)  = g x g^{-1} for  all x \in G.The set Inn(G) = \{\phi_g |\ g \in G\} is the inner automorphism group of Gand the mapping G→Inn(G), g→\phi_g, is a homomorphism with kernel Z(G), the center of G.  It is surprising (to me, at least) that these ideas have never been generalized to inverse semigroups.  The correct generalization turns out to start with the notion of an inner partial automorphism of an inverse semigroup.  Given an inverse semigroup S, there is a natural homomorphism from S to the inner partial automorphism monoid Inn(S) and the kernel of that homomorphism is what we can (and should!) call the center of S. I’ll discuss all this and what I think are the implications for inverse semigroup theory.  Finally,  if there is time, I’ll talk about the relationship between all this and inverse semiquandles, the generalization of quandles to the partial bijection setting. This is all joint work with various people, primarily David Stanovsk ́y and Jo ̃ao Ara ́ujo.

February 1, 2019, 9-9:50 am, CMK 207

Inner Partial Automorphisms of Inverse Semigroups I

Michael Kinyon

University of Denver

Abstract: Groups are the algebraic structures underlying symmetries, that is, structure-preserving permutations of a set. Inverse semigroups, a generalization of groups, were introduced in the1950s more or less independently by Ehresmann in France, Preston in the UK and Wagner in the Soviet Union. They are the algebraic structure underlying partial symmetries, that is, partial bijections between subsets.  For instance, just as the exemplar of a group is the symmetric group on a set, the exemplar of an inverse semigroup is the symmetric inverse monoid of all partial bijections between subsets. It is not an exaggeration to say that inverse semigroups are the most well studied class of semigroups.  For the first part of this talk or talks(?), I will start by giving a gentle introduction to inverse semigroups, outlining some of their basic structure, and going so far as to sketch the proof of the Wagner-Preston Theorem, which is the generalization to inverse semigroups of Cayley’s Theorem. Then I will turn to what I have been working on.  Recall that an inner automorphism of group G is a permutation \phi_g: G→G, g \in G,  defined  by \phi_g (x)  = g x g^{-1} for  all x \in G.The set Inn(G) = \{\phi_g |\ g \in G\} is the inner automorphism group of Gand the mapping G→Inn(G), g→\phi_g, is a homomorphism with kernel Z(G), the center of G.  It is surprising (to me, at least) that these ideas have never been generalized to inverse semigroups.  The correct generalization turns out to start with the notion of an inner partial automorphism of an inverse semigroup.  Given an inverse semigroup S, there is a natural homomorphism from S to the inner partial automorphism monoid Inn(S) and the kernel of that homomorphism is what we can (and should!) call the center of S. I’ll discuss all this and what I think are the implications for inverse semigroup theory.  Finally,  if there is time, I’ll talk about the relationship between all this and inverse semiquandles, the generalization of quandles to the partial bijection setting. This is all joint work with various people, primarily David Stanovsk ́y and Jo ̃ao Ara ́ujo.

January 25, 2019, 9 - 9:50 am, CMK 207

Nearfields, double transitivity and quasigroups II

Ales Drapal

Charles University

Abstract:  I will start with the definition of N(*_c), N a left nearfield, and prove that this is a quasigroup. (That will make the talk nearly independent of part I.) From that there follows a characterization of quasigroups possessing a sharply 2-transitive group of automorphisms. This will be then generalized to a characterization of all (finite) quasigroups with a doubly transitive automorphism groups. Then there will considered situations when Aut(N*_c) is not sharply 2-transitive. If time allows, the application of N(*_c) to extreme nonassociativity will be discussed too.

January 18, 2019, 9 - 9:50 am, CMK 207

Nearfields, double transitivity and quasigroups I

Ales Drapal

Charles University

Abstract:  In 1964 Sherman K. Stein published a paper that relates quasigroups possessing a sharply 2-transitive group of automorphisms to nearfields. It’s kind of a seminal paper, the content of which is easy to understand. I will mention some recent applications and show how to characterize all quasigroups with a doubly transitive automorphism groups.

November 16, 2018, 9-9:50 am, CMK 100

Introduction to infinitary Ramsey theory III

Natasha Dobrinen

University of Denver

Abstract: We give an introductory tutorial into Ramsey theory where the objects being colored are infinite.  Topology becomes indispensable in this study as a way to restrict colorings to nicely definable sets so that the Axiom of Choice cannot product “bad” colorings.  We will cover theorems of Nash-Williams, Galvin-Prikry, and Silver, culminating with Ellentuck’s topological characterization of those subsets of the Baire space which have the Ramsey property.  Time permitting, we will cover some classical and some recently developed topological Ramsey spaces and some of their applications to ultrafilters and relational structures.

November 9, 2018, 9-9:50 am, CMK 100

Introduction to infinitary Ramsey theory II

Natasha Dobrinen

University of Denver

Abstract: We give an introductory tutorial into Ramsey theory where the objects being colored are infinite.  Topology becomes indispensable in this study as a way to restrict colorings to nicely definable sets so that the Axiom of Choice cannot product “bad” colorings.  We will cover theorems of Nash-Williams, Galvin-Prikry, and Silver, culminating with Ellentuck’s topological characterization of those subsets of the Baire space which have the Ramsey property.  Time permitting, we will cover some classical and some recently developed topological Ramsey spaces and some of their applications to ultrafilters and relational structures.

November 2, 2018, 9-9:50 am, CMK 100

Introduction to infinitary Ramsey theory I

Natasha Dobrinen

University of Denver

Abstract: We give an introductory tutorial into Ramsey theory where the objects being colored are infinite.  Topology becomes indispensable in this study as a way to restrict colorings to nicely definable sets so that the Axiom of Choice cannot product “bad” colorings.  We will cover theorems of Nash-Williams, Galvin-Prikry, and Silver, culminating with Ellentuck’s topological characterization of those subsets of the Baire space which have the Ramsey property.  Time permitting, we will cover some classical and some recently developed topological Ramsey spaces and some of their applications to ultrafilters and relational structures.

### Analysis and Dynamics Seminar

Our next analysis and dynamics seminar will be held on 5/31, from 10am to 10:50am, in CMK 301.

Isoperimetric and Sobolev inequalities for magnetic graphs

Javier Alejandro Chavez Dominguez

University of Oklahoma

Abstract: The classical isoperimetric problem on the plane, dating back to antiquity, asks for the region of maximal area having a fixed perimeter. It is well-known that the solution to this problem (and its higher-dimensional versions) is intimately related to inequalities that give the norm of the embedding of a Sobolev space into an L_p space (that is, Sobolev inequalities).

In many applications, the domains of interest are typically a discrete set of points. A very useful model is to take the domain to be a graph, that is, a finite set of vertices where some pairs of them are related (and this is denoted by having them joined with an edge). In this context, relationships between isoperimetric and Sobolev-style inequalities have also found plenty of applications (for example, the famous Cheeger inequality for graphs).

Some situations, such as the presence of a magnetic potential in some quantum-mechanic models of bonds between atoms, are modeled not just with a graph but also with an additional assignment of a complex number of modulus one for each edge of the graph: this indicates not only that two vertices are related, but also how they are related. In this talk we will present recent results making the isoperimetric-to-Sobolev connection in the context of such “magnetic” graphs.

• Past Analysis and Dynamics Seminars

Our next analysis and dynamics seminar will be held on 5/24, from 10am to 10:50am, in CMK 111.

Pullbacks of graph C*-algebras from admissible intersections of graphs

Piotr M. Hajac

CU Boulder / IMPAN

Abstract: Following the idea of a quotient graph, we define an admissible intersection of graphs. We prove that, if the graphs E_1 and E_2 are row finite and their intersection is admissible, then the graph C*-algebra of the union graph is the pullback C*-algebra of the canonical surjections from the graph C*-algebras of E_1 and E_2 onto the graph C*-algebra of the intersection graph. Based on joint work with Sarah Reznikoff and Mariusz Tobolski.

May 17, 2019 in CMK 301

Solutions to Variational Inequalities on Graphs

Paul Horn

University of Denver

Abstract: In this talk we’ll consider the support to solutions to variational inequalities on graphs, which arise from certain minimization problems.  As noted by Brezis, and Brezis and Friedman, adding what amounts to an L_1 penalty term forces the support of solutions to minimization problems on R^n to become compact.  This observation has become important recently in the study of ‘compressed modes,’ which are essentially localized eigenvectors of operators, by Osher and others.  Here, we’ll discuss some of these results and their graph theoretical analogues, with some generalizations.

May 3 and May 10, CMK 301:

Subshifts of linear complexity

Ronnie Pavlov

University of Denver

Abstract: A subshift X is a topological dynamical system defined by a closed shift-invariant set of biinfinite sequences taking values in a finite alphabet. The complexity function c_n(X) counts the number of n-letter strings appearing within elements of X. A subshift X is said to have linear complexity if c_n(X) is bounded from above by Kn for some constant K.

I will discuss properties of this class of subshifts, focusing on recent results with Nic Ormes and Andrew Dykstra which control some types of topological/measurable subsystems contained within a subshift of linear complexity. No prior knowledge is required.

April 19 and April 26, CMK 301:

Ramsey Theory on trees and applications to infinite graphs

Natasha Dobrinen

University of Denver

April 12, CMK 301

Mathematics, science, and philosophy

Marco Nathan

DU Philosophy

Abstract: Traditionally, mathematics is taken to share much in common with the natural sciences and little with philosophy. This has an intuitive explanation: the methodological core of much science is mathematical at heart. This talk explores an alternative perspective. By discussing historical developments, I show that, from a foundational standpoint, mathematics is closer to philosophy than to the natural sciences. Since the emergence of non-Euclidian geometry, which threatens to undermine their necessity, both disciplines have become increasingly subdued to the agenda of the hard sciences, with dangerous consequences. I conclude that the fate and future of philosophy and mathematics is more inextricably tied together than is often realized.

March 1, CMK 207

Counterdiffusion in Biological and Atmospheric Systems

Patrick Shipman

Abstract: In topochemically organized, nanoparticulate experimental systems, vapor diffuses and convects to form spatially defined reaction zones. In these zones, a complex sequence of catalyzed proton-transfer, nucleation, growth, aggregation, hydration, charging processes, and turbulence produce rings, tubes, spirals, pulsing crystals, oscillating fronts and patterns such as Liesegang rings. We call these beautiful 3-dimensional structures “microtornadoes”, “microstalagtites”, and “microhurricanes” and make progress towards understanding the mechanisms of their formation with the aid of mathematical models.  This analysis carries over to the study of similar structures in protein crystallization experiments and the formation of periodic structures in plants.

January 18, January 25, and February 1, CMK 207

An Application of Descriptive Set Theory to Banach Space Theory

Jim Hagler

University of Denver

October 26, CMK 207

Making qualitative data quantitative: An overview of content analysis

Andrew Schnackenberg

DU Management

Abstract: Content analysis is a research technique used to make replicable and valid inferences by interpreting and coding textual material. By systematically evaluating texts (e.g., documents, oral communication, and graphics), qualitative data can be converted into quantitative data. These data can be used for further statistical analyses to explore many important but difficult-to-study issues of interest to management researchers in areas as diverse as business policy and strategy, managerial and organizational cognition, organizational behavior, and human resources. In this presentation, we will examine content analysis, with a focus on understanding what it is and why it is useful. We will also explore some common approaches to content analysis with illustrative examples.

October 19, CMK 207

Estimation and Inference of Heteroskedasticity Models with Latent Semiparametric Factors for Multivariate Time Series

Wen Zhou

Abstract: This paper considers estimation and inference of a flexible heteroskedasticity model for multivariate time series, which employs semiparametric latent factors to simultaneously account for the heteroskedasticity and contemporaneous correlations. Specifically, the heteroskedasticity is modeled by the product of unobserved stationary processes of factors and subject-specific covariate effects. Serving as the loadings, the covariate effects are further modeled through additive models. We propose a two-step procedure for estimation. First, the latent processes of factors and their nonparametric loadings are estimated via projection-based methods. The estimation of regression coefficients is further conducted through generalized least squares. Theoretical validity of the two-step procedure is documented. By carefully examining the convergence rates for estimating the latent processes of factors and their loadings, we further study the asymptotic properties of the estimated regression coefficients. In particular, we establish the asymptotic normality of the proposed two-step estimates of regression coefficients. The proposed regression coefficient estimator is also shown to be asymptotically efficient. This leads us to a more efficient confidence set of the regression coefficients. Using a comprehensive simulation study, we demonstrate the finite sample performance of the proposed procedure, and numerical results corroborate our theoretical findings. Finally, we illustrate the use of our proposal through applications to a variety of real data-sets.

October 12, CMK 207

Symmetries of Cuntz-Pimsner algebras

Valentin Deaconu

Abstract: I will recall the definition of a $C^*$-correspondence and of the Cuntz-Pimsner algebra. I will discuss group actions on $C^*$-correspondences and crossed products. I will illustrate with examples related to graphs and to vector bundles.

September 21 and October 5, CMK 207

Exponential Random Graph Models

Ryan DeMuse

University of Denver

Abstract: Random graph models are probability measures on graph spaces that can answer questions about what features a typical graph drawn from the space exhibits. We will begin by considering the classic Erdös-Rényi model and build to a natural extension, the Exponential Random Graph Model (ERGM). This is a generalization of the Erdös-Rényi model that can capture key features present in modern networks. We will discuss the machinery and methods involved in the study of ERGMs and, time permitting, existence of normalization constants and the efficiency of sampling from ERGM distributions.