Colloquia and Seminars
Our department hosts several research seminars throughout the academic year. Graduate seminars feature research from graduate students, and visiting scholars speak at the algebra and logic and past analysis and dynamics seminars.
Graduate Colloquium
Friday, November 15, 2019, 2:003:00 p.m. in CMK 309:
Introduction to rotation theory
Yiqing Geng
University of Denver
Abstract: Rotation theory is an interesting topic in mathematics as it combines different fields of mathematics such as topology dynamical system, real analysis etc. This presentation will focus on a classic theorem in rotation theory called Weyl's theorem. One of the most fundamental dynamical systems by studying maps of the circle to itself. We will start from looking at properties and facts about unit circle as a metric space then we will go to the details of Weyl's theorem.
NOTE: This talk is presented by a masters student towards a partial fulfillment of the requirements for the degree.

Past Colloquia
Friday, November 8, 2019, 2:003:00 p.m. in CMK 309:
Resumes + CV’s That Get Results!
Patricia Hickman
University of Denver
Abstract: Explaining your work and experience on a resume or CV can be challenging. This interactive workshop will focus on how to write a resume/CV that will be visually appealing and easy to scan as well as highlight your skills. Specifics include learning the differences between a resume and CV, formatting and techniques for writing about your experience. Take advantage of this opportunity to spruce up your resume or CV! Presented by Patty Hickman/Director Graduate Career & Professional Development.
Friday, October 25th 2019, 2:003:00 p.m. in CMK 309:
Binary relations on partiallyordered sets
Nick Galatos
University of Denver
Abstract:
Binary relations can be found everywhere in mathematics (and in every discipline for that matter). We are all able to manipulate binary relation and intuitively familiar with many of the laws that hold when we combine relations (by union, intersection, composition, inverse, etc). The mathematical study of the algebra of relations is mainly pioneered by A. Tarski, who also connected it to firstorder logic. Via reducing firstorder logic (a complicated theory involving quantifiers, among other things) to the innocentlooking equational theory of algebras of relations, he proved the undecidability of the latter. The complications do not end there: where Cayley succeeded with axiomatizing symmetric groups and Stone with axiomatizing Boolean algebras of powersets, Tarski failed, and this was not due to lack of ingenuity.
We present a generalization of the notion of the algebra of relations on a set, by introducing an ordering relation and considering only those relations that are compatible with the order. This results into bringing an intuitionistic/constructive character to the study, since the resulting "weakening relation algebras" are not based on Boolean algebras. We prove that the new algebras, while being much more encompassing (for example, latticeordered groups can be embedded in appropriate ones), they still enjoy a lot of the nice properties of relation algebras (they are semisimple) and that they admit a simple description of their congruences (analogous to normal subgroups in group theory and to filters in Boolean algebras). (Joint work with P. Jipsen.)
Friday, October 18th 2019, 2:003:00 p.m. in CMK 309:
Quantum Entanglement
Stan Gudder
Uiniversity of Denver
Abstract: Entanglement is an important resource in quantum computation. Entanglement is a little mysterious and Einstein called it “spooky action at a distance”. We first present a simple criterion for determining when a pure state is entangled or not. We next define an entanglement number that measures the amount of entanglement for a pure state. Finally, we define an entanglement number for mixed states.
Friday, May 24th 2019, 2:003:00 p.m. 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 noncommutative 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 algebraictopology 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 handson nature, graphs are extremely efficient in unraveling the structure and Ktheory of graph algebras. We will exemplify this phenomenon by showing a CWcomplex structure of the VaksmanSoibelman quantum complex projective spaces, and how it explains their Ktheory.
Friday, May 24th 2019, 2:003:00 p.m. in CMK 309:
The Method of 4Shadows
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.
Friday, May 24th 2019, 2:003:00 p.m. 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 noncommutative 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 algebraictopology 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 handson nature, graphs are extremely efficient in unraveling the structure and Ktheory of graph algebras. We will exemplify this phenomenon by showing a CWcomplex structure of the VaksmanSoibelman quantum complex projective spaces, and how it explains their Ktheory.
Friday, May 17 2019, 2:003:00 p.m. 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 socalled 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 socalled simple equations. Our main results involve proving decidability and undecidability for broad classes of such structures.
Friday, May 10 2019, 2:003:00 p.m. in CMK 309:
Tukey Order, Small Cardinals, and Oﬀdiagonal 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 26 2019, 2:003:00 p.m. in CMK 309:
Decomposing Graphs into Edges and Triangles
Adam Blumenthal
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 19 2019, 2:003:00 p.m. 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 5 2019, 2:003:00 p.m. 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
Friday, November 8, 2019, 9:009:50 p.m. in CMK 309:
Big Ramsey degrees in universal profinite ordered kclique free graphs
Kaiyun Wang
Shaanxi Normal University
Abstract: In this talk, we build a collection of new topological Ramsey spaces of trees, extending Zheng's work to the setting of finite kclique free graphs, where k ≥ 3. It is based on the HalpernL\"{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of HuberGeschkeKojman on profinite ordered graphs, we prove that every finite ordered kclique free graph H has the big Ramsey degree T(H) in the universal profinite ordered kclique free graph under the finite Bairemeasurable colouring.

Past Algebra and Logic Seminars
Friday, November 1st 2019, 9:009:50 p.m. in CMK 309:
Heyting residuated lattices
Nick Galatos
Uiniversity of Denver
Abstract: Separation logic is used in computer science in pointer management and memory allocation. Its basic metalogic is BunchedImplication logic, a substructural logic whose algebraic semantics are Heyting residuated lattices. We describe the congruences on Heyting RL's and show that they form an idealdetermined variety. Moreover, we define the notion of a doubledivision conucleus on a Heyting RL and show that it preserves discriminator terms of specific form.
October 18 and 25, 2019
Higher amalgation of algebraic structures
David Milovich
Welkin Sciences at Colorado Springs
Abstract: Given a class V of algebraic structures, say that structures A, B ∈ V with underlying sets A, B overlap in V if A and B have a common substructure C ∈ V with underlying set A ∩ B. Say that a set S of overlapping structures amalgamates in V if there is structure D ∈ V such that every A ∈ S is a substructure of D. Call any such D an amalgamation of S in V. (All of the above can abstracted into category theory if desired.) Much is known already about group amalgamation: • Every two, but not every three, overlapping groups amalgamate in the class of groups. • Every three, but not every four, overlapping abelian groups amalgamate in the class of abelian groups. • Every set of overlapping locally cyclic groups amalgamates in the class of abelian groups. These results, most of them due to Hannah Neumann, were published no later than 1954. Subsequent research has extensively studied binary amalgamation but neglected higher amalgamation (that is, amalgamation as defined above of three or more algebraic structures). In the first of a pair of lectures, I will give a characterization of linear amalgamations, which are amalgamations of n overlapping structures obtained by repeatedly maximally amalgamating pairs of overlapping structures. (The relevant maximality concept is the pushout of category theory.) I will show that every finite set of overlapping vector spaces (over a common field) is linearly amalgamable, as is every finite set of overlapping divisible groups. In the second lecture, I will present applications to uncountable Boolean algebras (which, by Stone duality, are also applications to settheoretic topology). Any directed family of countable sets with union of size ≥ ℵn necessarily includes n countable sets in “general position” with respect to inclusion. This is a potential obstacle because every two, but not every three, overlapping Boolean algebras amalgamate. Fortunately, the closure properties of elementary substructures fit linear amalgamation like a glove. Combining this fact with the technique of Davies trees, I obtain a new way to build uncountable Boolean algebras from countable ones (in ZFC). Applications include new characterizations of projective Boolean algebras (whose Stone duals are the absolute extensors of dimension zero) and an answer to a question of Stefan Geschke about tightly κfiltered Boolean algebras. Each lecture will conclude with some open problems.
May 17, 2019
Simple weight modules with finitedimensional weight spaces
David Ridout
University of Melbourne
Abstract: Let g be a finitedimensional 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 gmodules with finitedimensional weight spaces. These may be understood using Mathieu’s theory of coherent families. We shall review this theory and generalize it in order to understand the representation theory of L_k(g).
May 10, 2019
RainbowCycleForbidding Edge Colorings
Andrew Owens
Auburn University
Abstract: A JLcoloring 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 JLcolorings 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
Irreducible convergence and irreducibly orderconvergence in T_0 spaces
Kaiyun Wang
Abstract: In this talk, we aim to lift liminfconvergence and orderconvergence in posets to a topology context. Based on the irreducible sets, we define and study irreducible convergence and irreducibly orderconvergence in T0 spaces. Especially, we give sufficient and necessary conditions for irreducible convergence and irreducibly orderconvergence in T0 spaces to be topological.
April 5, 2019
Walgebras and integrability
Tomas Prochazka
University of Munich, Arnold Sommerfeld Center for Theoretical Physics
Abstract: I will review what Walgebras are from the conformal field point of view. After that I'll explain the definition of affine Yangian by ArbesfeldSchiffmannTsymbaliuk 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
Title: Inner Partial Automorphisms of Inverse Semigroups I
Michael Kinyon
University of Denver
Abstract: Groups are the algebraic structures underlying symmetries, that is, structurepreserving 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 WagnerPreston 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 Stanovský and João Araújo.
February 1, 2019
Inner Partial Automorphisms of Inverse Semigroups I
Michael Kinyon
University of Denver
Abstract: Groups are the algebraic structures underlying symmetries, that is, structurepreserving 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 WagnerPreston 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 Stanovský and João Araújo.
January 25, 2019
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 2transitive 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 2transitive. If time allows, the application of N(*_c) to extreme nonassociativity will be discussed too.
January 18, 2019
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 2transitive 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
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 NashWilliams, GalvinPrikry, 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
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 NashWilliams, GalvinPrikry, 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
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 NashWilliams, GalvinPrikry, 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
Friday, November 8 and November 15, 2019, 10:0010:50 p.m. in CMK 207:
Naimark Dialation Theorem
Stan Gudder
University of Denver
Abstract: Sara pointed out the importance of the Naimark Dialation Theorem in her work. I will give a simple proof of this theorem for finitedimensional Hilbert spaces. I’ll also point out the importance of this result for quantum mechanics.

Past Analysis and Dynamics Seminars
Friday, November 1st 2019, 10:0010:50 p.m. in CMK 207:
Beyond Orthonormal bases: an introduction to finite frames
Sara Andrade
University of Denver
Abstract: In many signal processing applications orthonormal bases pose a number of limitations. Frames provide a redundant, stable way of representing a signal. Unlike orthonormal bases, frame representations are robust to erasures and allow a flexibility in design. Frame theory might be regarded as partly belonging to applied harmonic analysis, functional analysis, operator theory as well as numerical linear algebra and matrix theory. This twopart presentation will be a crash course in frame theory. In the first talk, we will cover some foundational results in frame theory and investigate the relationship between orthonormal bases and a special class of frames. The second talk will give an overview of a few applied problems in frame theory; such as frame design and phase retrieval.
October 25, 2019
Beyond Orthonormal bases: an introduction to finite frames
Sara Andrade
University of Denver
Abstract: In many signal processing applications orthonormal bases pose a number of limitations. Frames provide a redundant, stable way of representing a signal. Unlike orthonormal bases, frame representations are robust to erasures and allow a flexibility in design. Frame theory might be regarded as partly belonging to applied harmonic analysis, functional analysis, operator theory as well as numerical linear algebra and matrix theory. This twopart presentation will be a crash course in frame theory. In the first talk, we will cover some foundational results in frame theory and investigate the relationship between orthonormal bases and a special class of frames. The second talk will give an overview of a few applied problems in frame theory; such as frame design and phase retrieval.
October 18, 2019
The special atom space, Haar System and Wavelet in higher dimensions
Geraldo de Souza
Auburn University
Abstract: In this presentation, we will explore the special atom spaces introduced by De Souza in 1980 in his Ph.D thesis. The impetus of this exploration is to extend to higher dimension the definition originally proposed by De Souza. A by product of this endeavor will be the definition of the Haar wavelets and wavelets system in higher dimensions. Even though the Haar System in higher dimension has been discussed by numerous authors in the literature, the definitions proposed do not always seem natural extension of the one dimension case and often are unnecessarily cumbersome and difficult to follow. The special atoms spaces are closely connected with several knowns spaces in the literature, like Lipschitz , Bergman, Zygmund,
Lorentz spaces etc. these connections were possible because of their analytic characterization, duality, interpolation etc. Also the special atoms is related with the Haar function.September 20, 2019
Fractal Billiard
Robert Niemeyer
Metro State, Denver
Abstract: In this talk, we will understand what the main issue is with reflection in a fractal boundary and how one chooses to get around this issue. We then describe families of periodic orbits in three fractal billiard tables, the Koch snowflake fractal billiard, a selfsimilar Sierpinski carpet fractal billiard and the socalled Tfractal billiard. We then focus on a specific example of a periodic orbit of the Tfractal billiard that is unlike any other. Finally, current work on fractal interval exchange transformations is presented in the context of the Tfractal translation surface (which isn’t really a surface) and implications for classifying the dynamics thereon.
'May 31, 2019
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 wellknown that the solution to this problem (and its higherdimensional 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 Sobolevstyle 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 quantummechanic 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 isoperimetrictoSobolev connection in the context of such “magnetic” graphs.
May 24, 2019
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
Solutions to Variational Inequalities on GraphsPaul 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
Subshifts of linear complexityRonnie Pavlov
University of Denver
Abstract: A subshift X is a topological dynamical system defined by a closed shiftinvariant set of biinfinite sequences taking values in a finite alphabet. The complexity function c_n(X) counts the number of nletter 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
Ramsey Theory on trees and applications to infinite graphs
Natasha Dobrinen
University of Denver
April 12, 2019
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 nonEuclidian 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, 2019
Counterdiffusion in Biological and Atmospheric SystemsPatrick Shipman
Colorado State
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 protontransfer, 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 3dimensional 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, 2019
An Application of Descriptive Set Theory to Banach Space TheoryJim Hagler
University of Denver
October 26, 2018
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 difficulttostudy 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, 2018
Estimation and Inference of Heteroskedasticity Models with Latent Semiparametric Factors for Multivariate Time SeriesWen Zhou
Colorado State
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 subjectspecific covariate effects. Serving as the loadings, the covariate effects are further modeled through additive models. We propose a twostep procedure for estimation. First, the latent processes of factors and their nonparametric loadings are estimated via projectionbased methods. The estimation of regression coefficients is further conducted through generalized least squares. Theoretical validity of the twostep 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 twostep 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 datasets.
October 12, 2018
Symmetries of CuntzPimsner algebrasValentin Deaconu
University of Nevada
Abstract: I will recall the definition of a $C^*$correspondence and of the CuntzPimsner 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, 2018
Exponential Random Graph ModelsRyan 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 ErdösRényi model and build to a natural extension, the Exponential Random Graph Model (ERGM). This is a generalization of the ErdösRényi 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.