# Colloquia and Seminars

Our department hosts several research seminars throughout the academic year. On this page, you will find announcement for the seminars, as well as a list of past seminars. During the Fall quarter 2020, ALL seminars will be held exclusively online using Zoom. Please email the organizer of any seminar you wish to attend online for a link to the meeting, or consult the Mathematics Department's portfolio page.

Join us on 10/09/2020 at 2pm for the following graduate colloquium, by using the Zoom ID 983 9927 0785.

Graphs, growth and geometry

Abhijit Champanerkar, PhD

College of Staten Island and The Graduate Center
The City University of New York

Abstract:  We study the growth rate of the number of spanning trees of a sequence of planar graphs which diagrammatically converge to a biperiodic planar graph. We relate this growth rate to the Mahler measure of a 2-variable polynomial and hyperbolic volume of link complements. We use this circle of ideas to study an interesting conjecture in knot theory.

• Past Colloquia

Join us on 9/25/2020 at 9am for the following graduate colloquium, by using the Zoom ID 983 9927 0785.

Mathematical models versus reality! The case of COVID-19 modelling in South Africa

Department of Mathematics and Applied Mathematics
University of Johannesburg, South Africa

Abstract:  The novel coronavirus (COVID-19 or SARS Cov-2) pandemic continues to be a global health problem whose impact has been significantly felt in South Africa when compared to the rest of the continent. In this presentation, we look at how mathematical models were used to influence policy and how some of the models resulting in panic. We also consider how the non-consideration of foundational mathematical theories can be disastrous when linking models to reality. The lack of basic mathematical principles, the use dashboards built on assumptions that the users may not have a good understanding of and the scarcity of data have huge implications in how models relate to reality. We focus mostly on deterministic models to model the transmission dynamics of COVID-19 in South Africa and discuss the experiences of the pandemic in South Africa from the modelling perspective. Of particular interest is highlighting why model predictions differed and the potential impact of the differences.

Friday, January 24th, 2020, 2:00-3:00 p.m. in CMK 309:

Double Affine Weyl Groups and Fusion Algebras for Affine Lie Algebras

Alejandro Ginory

Rutgers University

Abstract: Certain categories of modules for affine Lie algebras are not closed under the usual tensor product. For a "good" class of affine Lie algebras, called untwisted, there is a product structure called the fusion product on these categories that is analogous to the tensor product. In this talk, we will show how the fusion product structure (at the level of characters) can be described by using (double) affine versions of Weyl groups. Using this description, we uncover modular invariance phenomena and explain how in the case of so-called "twisted" affine Lie algebras the fusion product (on characters) has, somewhat surprisingly, negative structure constants.

Friday, November 15, 2019, 2:00-3:00 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.

Friday, November 8, 2019, 2:00-3:00 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, 2:00-3:00 p.m. in CMK 309:

Binary relations on partially-ordered 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 first-order logic. Via reducing first-order logic (a complicated theory involving quantifiers, among other things) to the innocent-looking 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, lattice-ordered 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, 2:00-3:00 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, 2:00-3:00 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 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 24th 2019, 2:00-3:00 p.m. 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.

Friday, May 24th 2019, 2:00-3:00 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 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 17 2019, 2:00-3:00 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 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 10 2019, 2:00-3:00 p.m. 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 26 2019, 2:00-3:00 p.m. 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 19 2019, 2:00-3:00 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, 2:00-3:00 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.

The Graduate Colloquium is organized by Dr. Kanade and will take place exclusively remotely, on Zoom. Please email the organizer for the password to the Zoom session, or consult the Mathematics Department's portfolio page.

### Algebra and Logic Seminar

Join us on Friday 10/16/2020 at 9am for the Algebra and Logic Seminar using the Zoom ID 95200393472.

Gaudin model, Feigin-Frenkel center, and Grassmannian

Kang Lu, PhD

University of Denver

Abstract:  Gaudin models were first introduced by M. Gaudin in 1976 where the Gaudin (quadratic) Hamiltonians were given. It was known there are higher (order) Hamiltonians. However, no explicit constructions were described. In the seminal work of B. Feigin, E. Frenkel and N. Reshetikhin (1994), they found that the higher Gaudin Hamiltonians can be obtained from the Feigin-Frenkel center. In this talk, we will give an introduction to Gaudin models and explain how the Feigin-Frenkel center and Gaudin models are related. We discuss the basic Bethe ansatz for Gaudin models and explain how it is related to Grassmannian using an example of sl_2.

Upcoming seminars:

• Kang Lu, Oct. 16
• Hao Li, Oct. 23
• Past Algebra and Logic Seminars

Join us on Friday 10/9/2020 at 9am for the Algebra and Logic Seminar using the Zoom ID 95200393472.

An introduction to cluster algebras and their applications

Eric Bucher, PhD

Michigan State University

Abstract: Cluster algebras were first invented by Fomin and Zelevinsky in 2003 to study total positivity of canonical bases. Since their inception, these mathematical objects have popped up in a large variety of seemingly unrelated areas including: Teichmuller theory, Calabi-Yau categories, integrable systems, and the study of high energy particle physics. In this talk we will lay the basic groundwork for working with cluster algebras as well as discuss a few of their applications to the above areas. This talk is intended to be introductory so no background or definitions will be assumed. The intent is to have everyone walk away having learned about this new and fascinating algebraic object.

Tuesday January 24th 2020, 9-9:50am in CMK 211:

Stable Hypergraph Regularity

Rehana Patel

African Institute for Mathematical Sciences-Senega

Abstract:  I will discuss an extension, to the case of hypergraphs, of results of Malliaris and Shelah on regularity lemmas for stable graphs.  This is joint work with N. Ackerman and C. Freer; similar results have been obtained independently by A. Chernikov and S. Starchenko.

Tuesday January 14th 2020, 3-4pm in CMK 201:

Ramsey-like cardinals

Victoria Gitman

CUNY

Abstract: Typically measurable and larger large cardinals are defined in terms of the existence of elementary embeddings from the universe $V$ into a transitive submodel, while smaller large cardinals are defined by combinatorial Ramsey-type properties. It turns out that most smaller large cardinals $\kappa$ can be characterized by the existence of elementary embeddings on mini-universes of size $\kappa$. The Ramsey-like cardinals arose out of the general study of properties of elementary embedding characterizations of smaller large cardinals. I will talk about elementary embedding characterizations of classical smaller large cardinals, such as weakly compact and Ramsey cardinals, and generalize these characterizations to introduce new hierarchies of large cardinals, the Ramsey-like cardinals.

Friday, November 8, 2019, 9:00-9:50 p.m. in CMK 309:

Big Ramsey degrees in universal profinite ordered  k-clique 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 k-clique free graphs, where k ≥ 3. It is based on the Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on profinite ordered graphs, we prove that every finite ordered k-clique free graph H has the big Ramsey degree T(H) in the universal profinite ordered k-clique free graph under the finite Baire-measurable colouring.

Friday, November 1st 2019, 9:00-9:50 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 Bunched-Implication 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 ideal-determined variety. Moreover, we define the notion of a double-division 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

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 set-theoretic 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 κ-filtered Boolean algebras. Each lecture will conclude with some open problems.

May 17, 2019

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 generalize it in order to understand the representation theory of L_k(g).

May 10, 2019

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

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

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

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 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, 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 Stanovský and João Araújo.

January 25, 2019

Near-fields, double transitivity and quasigroups II

Ales Drapal

Charles University

Abstract:  I will start with the definition of N(*_c), N a left near-field, 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

Near-fields, 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 near-fields. 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 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

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

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.

The Algebra and Logic seminar is organized by Dr. Linshaw and will take place exclusively remotely, on Zoom. Please email the organizer for a password to the Zoom session, or consult the Mathematics Department's portfolio page.

### Analysis and Dynamics Seminar

Join us on Friday 9/25 at 10am for the Analysis and Dynamics seminar using the Zoom ID 979 1161 6273.

One-sided versus two-sided stochastic processes

Aernout van Enter

University of Groningen

Abstract: Stochastic processes can be parametrised by time (such as occurs in  Markov chains), in which case conditioning is one-sided (the past) or by one-dimensional space (which is the case, for example, for Markov fields), where conditioning is two-sided (right and left). I will discuss some examples, in particular generalising this to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided)  produce the same objects and when they are different. We show moreover the role one-dimensional entropic repulsion plays in this setting. Joint work with R. Bissacot, E. Endo and A. Le Ny.

Our list of upcoming speakers include:

• 9/25: Rick Kenyon (Yale U.)
• 10/16: Aernout van Enter (U. of Groningen)
• 10/23: Kat Perry (Soka U.)
• 10/30: Chris Marx (Oberlin U.)
• 11/6: Sunder Sethuraman (U. of Arizona)
• Past Analysis and Dynamics Seminars

Friday, September 25th, 2020, 10:00-10:50am.

Richard Kenyon, PhD

Yale University

Abstract: This is joint work with Istvan Prause. Many well-known random tiling models such as domino tilings and lozenge tilings lead to variational problems for functions h: R^2->R which minimize a functional depending only on the gradient of h. Other examples of such variational problems include minimal surfaces and surfaces satisfying the “p-Laplacian”.  We give a representation of solutions of such a problem in terms of kappa-harmonic functions: functions which are harmonic for a laplacian with a varying conductance kappa.

Friday, February 7, 2020, 10:00-10:50 a.m. in CMK 207:

Automorphisms of symbolic systems

Scott Schmieding

University of Denver

Abstract:  We'll begin with some background on the notion of a subshift, which are fundamental objects in the realm of symbolic dynamics. We'll then discuss the automorphism group of a subshift, which consists of all self-symmetries of a given subshift. We'll focus on the case of shifts of finite type, a key class of subshifts for which these groups have been heavily studied, and give some background and problems in the area. Finally, we'll discuss some recent work with Yair Hartman and Bryna Kra in which we introduce a certain stabilization of the automorphism group, and outline some of our results in this new stabilized setting.

Friday, November 8 and November 15, 2019, 10:00-10:50 a.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 finite-dimensional Hilbert spaces. I’ll also point out the importance of this result for quantum mechanics.

Friday, November 1st 2019, 10:00-10:50 a.m. in CMK 207:

Beyond Orthonormal bases: an introduction to finite frames

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 two-part 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

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 two-part 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 self-similar Sierpinski carpet fractal billiard and the so-called T-fractal billiard. We then focus on a specific example of a periodic orbit of the T-fractal billiard that is unlike any other. Finally, current work on fractal interval exchange transformations is presented in the context of the T-fractal 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 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.

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 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

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 bi-infinite 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

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 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, 2019

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 “micro-tornadoes”, “micro-stalagtites”, and “micro-hurricanes” 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 Theory

Jim 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 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, 2018

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, 2018

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,  2018

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.

The Analysis and Dynamics seminar is organized by Dr. Yin and will take place exclusively remotely, on Zoom. Please email the organizer for a password to the zoom session, or consult the Mathematics Department's portfolio page.

### Inclusive Teaching Seminar

We will soon start a new seminar on inclusive teaching.

The Inclusive Teaching seminar is organized by Dr. Sara Botelho-Andrade and Dr. Sabine Lang. It will take place exclusively remotely, on Zoom. Please email the organizers for the link to the zoom session, or consult the Mathematics Department's portfolio page.