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.
Graduate Colloquium
Join us on 11/12/2021 at 2pm for the following graduate colloquium.
Introduction to vertex algebras from a number theoretic viewpoint
Antun Milas, PhD
SUNY Albany
Abstract: In this introductory talk, we will discuss vertex algebras, vertex algebra modules, and their graded dimensions (also known as "characters"). I will explain how abstract algebra, combinatorics of qseries, and modular functions interact in surprising ways. Many examples will be presented.
This talk should be appropriate for graduate students interested in algebra and number theory.

Past Colloquia
Join us on 10/28/2021 at 2pm for the following graduate colloquium.
How I learned to stop worrying and love the group: some unexpected connections between symbolic dynamics and geometric group theory
Ronnie Pavlov, PhD
University of Denver
Abstract:
A topic of significant recent interest in symbolic dynamics is how the word complexity function of a subshift X affects its group Aut(X) of automorphisms. In particular, several recent papers show that slow enough growth of word complexity forces Aut(X) to be "small" in some sense. In recent work with Scott Schmieding, we proved an interesting result of this form; slow enough growth of word complexity forces Aut(X) to be an extension of some locally finite group by Z.
In the course of this work, we learned a bit about the fascinating field of geometric group theory, of which I was previously unaware. I'll give a very brief introduction to this area, and explain a bit about how it leads to our results. All terms will be defined and I'll assume no background aside from some very elementary group theory.
Join us on 4/23/2021 at 2pm for the following graduate colloquium.
From knots to modularity
Robert Osburn, PhD
University College Dublin
Abstract: Knots are objects which appear in nature, science and the arts. We see them while untying our shoelaces, looking under a microscope or admiring the Book of Kells. Knot invariants are quantities defined for each knot which are the same for equivalent knots.
Modular forms are analytic objects with intrinsic symmetric properties. They played a key role in the proof of Fermat’s Last Theorem and occur in many diverse areas such as mathematical physics, algebraic geometry, combinatorics, logic and black holes.
Over the past two decades, there have been intriguing connections between these two seemingly disparate areas. In this talk, we discuss historical developments and recent striking interactions between quantum knot invariants and a new spectrum of modular forms, namely mock modular and quantum modular forms.
Join us on 3/5/2021 at 2pm for the following graduate colloquium.
Logarithms in conformal field theories and indecomposable representations of superalgebras
Victor Gurarie, PhD
University of Boulder
Abstract: Conformal field theories have long been paradigms in constructing solutions to quantum field theories via representations of their symmetry groups. The program of classifying solutions to conformally invariant quantum field theories in two dimensional space was especially successful, with many exact solutions constructed and classified beginning in 1980s. Nonetheless, a group of conformal field theories with logarithms in their correlation functions so far defied complete classification. At the same time, these conformal field theories are known to describe many interesting physical phenomena. I will describe connection between these logarithms and the indecomposable representations of superalgebras and discuss what we could learn if the classification of the conformal field theories with logarithms were to be carried out.
Join us on 2/26/2021 at 2pm for the following graduate colloquium.
Applications of vector field topology in surface and volume meshing
Ed Chien, PhD
Boston University
Abstract:
In many applied settings, e.g., finiteelement simulation and modeling, discretizations of the domain at hand are vitally important for determining performance further down the pipeline. Oftentimes, a practitioner is after quadrilateral surface meshes and hexahedral volumetric meshes, where mesh elements are topological squares and cubes, respectively. These meshes must also have elements that are not too distorted (geometrically) and need to faithfully represent the domain boundaries. I will detail these problems and discuss two works where the classical PoincareHopf theorem is generalized and applied in algorithms for quad and hex mesh generation. Time permitting, related works from the field of geometry processing will be discussed, and perspectives on an ongoing transition to this applied field from a pure math Ph.D. will be given.
Join us on 2/05/2021 at 2pm for the following graduate colloquium.
Geodesics, bigeodesics, and coalescence in first passage percolation
Ken Alexander, PhD
University of Southern California
Abstract:
In first passage percolation, independent identically distributed bond passage times are attached to the bonds of the lattice ℤ^{d}; these may alternatively be viewed as random lengths. This creates a random distance on the lattice: the geodesic from x to y is the lattice path which minimizes the sum of passage times, and this minimum is the distance T(x,y). We are interested in infinite geodesics for this distance: a θray is a path to ∞ with asymptotic direction θ for which every finite segment is a geodesic, and a bigeodesic is an analogous doubly infinite geodesic path. It is known that under mild hypotheses, for each starting point x and direction θ, there is a θray from x. In d=2 it is a.s. unique, and furthermore, for all x,y the θrays from x and y eventually coalesce, and there are no bigeodesics with θ as either asymptotic direction. We show that in general dimension, under somewhat stronger hypotheses, a weak form of coalescence called bundling occurs: we take all θrays starting next to a hyperplane H_{0}, translate the hyperplane forward by distance R to give H_{R}, and consider the density of sites in H_{R} where one of the θrays first crosses H_{R}. We show this density approaches 0 as R→∞, with nearoptimal bounds on the rate. Essentially as a consequence, we show that there are no bigeodesics in any direction.
Join us on 1/22/2021 at 2pm for the following graduate colloquium.
Synchronization properties in complex networks
Zahra Aminzare, PhD
University of Iowa
Abstract: Synchronized activities are crucial in many biological systems such as brain function, circadian rhythms generation, and animal locomotion. Finding conditions that foster synchronization is critical to understanding these biological systems. Motivated by insect locomotion, I first introduce a class of coupled dynamical systems that can model networks of neurons and explain their synchronization properties. Then, employing techniques from applied dynamical systems, stochastic differential equations, and algebraic graph theory, I present several conditions that foster complex synchronization patterns in such networks.
Join us on 11/06/2020 at 2pm for the following graduate colloquium, by using the Zoom ID 983 9927 0785.
On the classification of modular categories
Julia Plavnik, PhD
Indiana University
Abstract: Modular categories are intricate organizing algebraic structures appearing in a variety of mathematical subjects including topological quantum field theory, conformal field theory, representation theory of quantum groups, von Neumann algebras, and vertex operator algebras. They are fusion categories with additional braiding and pivotal structures satisfying a nondegeneracy condition. The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter.
In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. I will give an overview about the current situation of the classification program for modular categories and mention some open directions to explore.
Join us on 10/30/2020 at 2pm for the following graduate colloquium, by using the Zoom ID 983 9927 0785.
Arrow's Theorem and Voting
Thomas French, PhD
University of Denver
Abstract: It's clear that the plurality voting system we use doesn't always produce fairseeming outcomes. We will discuss what criteria we might want a voting system to meet in order to be considered fairthen immediately dash those hopes by proving Arrow's Theorem, which says that no voting system can meet all those criteria. If there's time, we can also talk about the upcoming presidential electionand which states to watch if you want to predict the outcome of the election as soon as possible!
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 YorkAbstract: 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 2variable polynomial and hyperbolic volume of link complements. We use this circle of ideas to study an interesting conjecture in knot theory.
Join us on 9/25/2020 at 9am for the following graduate colloquium, by using the Zoom ID 983 9927 0785 (password: f20_colloq)
Mathematical models versus reality! The case of COVID19 modelling in South Africa
Farai Nyabadza, PhD
Department of Mathematics and Applied Mathematics
University of Johannesburg, South AfricaAbstract: The novel coronavirus (COVID19 or SARS Cov2) 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 nonconsideration 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 COVID19 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, 2:003:00 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 socalled "twisted" affine Lie algebras the fusion product (on characters) has, somewhat surprisingly, negative structure constants.
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.
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.
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 11/18/2021 at 9am for the Algebra and Logic Seminar.
Heterotic sigma models via BV quantization and formal geometry
Matt Szczesny, PhD
Boston University
Abstract: We construct a rigorous BV quantization of the halftwisted heterotic sigma model with target a complex manifold X equipped with a holomorphic vector bundle E, study its local anomalies, and relate the algebra of observables to chiral differential operators acting on E. The latter is a sheaf of vertex algebras on X originally introduced by Malikov, Schechtman, Vaintrob, and Gorbounov. This is joint work with Owen Gwilliam, James Ladouce, and Brian Williams.

Past Algebra and Logic Seminars
Join us on Friday 10/21/2021 at 9am for the Algebra and Logic Seminar.
How I learned to stop being scared and love the Bailey machinery
Shashank Kanade, PhD
University of Denver
Abstract: I'll give an introduction to the Bailey machinery for proving RogersRamanujantype identities. Except for linear algebra, no other prerequisites will be assumed. Depending on the interest and time, I will then provide an overview of (a subset) of the following directions 
 series representations of principal characters of the integrable, highestweight modules. (Joint work with Russell)
 Doublepole Nahmtype representations of AndrewsGordon, AndrewsBressoud and false theta identities. (Joint work with Milas and Russell, building on earlier works of JenningsShaffer and Milas).
 AndrewsSchillingWarnaar's marvelous generalization of the Bailey machinery to the root system that leads to characters of certain representations of the VOA.
Join us on Friday 10/15/2021 at 9am for the Algebra and Logic Seminar.
Chiral homology of elliptic curves
Jethro van Ekeren, PhD
Universidade Federal Fluminense
Abstract: In this talk I will discuss results of an ongoing project (joint with Reimundo Heluani) on the chiral homology of elliptic curves with coefficients in a conformal vertex algebra. We relate the nodal curve limit with the Hochschild homology of the Zhu algebra. Along the way we find interesting links between these structures, Poisson homology, associated varieties of vertex algebras, and classical identities of RogersRamanujan type (this last part joint work with George Andrews).
Join us on Friday 10/8/2021 at 9am for the Algebra and Logic Seminar.
Representations of the Determinant Lie Algebra
Yuly Billig, PhD
Carleton University
Abstract: Determinant Lie algebra is a central extension of the Lie algebra of divergence zero vector fields on a 2dimensional torus. It is a Z^2graded Lie algebra with all components outside the Cartan subalgebra having dimension 1. Our goal is to study simple weight modules with finitedimensional weight spaces for this Lie algebra which are supported on a halfplane. In order to achieve this goal we introduce the notion of a jet Lie algebra and get a correspondence between representations of the Determinant Lie algebra and its jet Lie algebra.
It turns out that the jet Lie algebra in this case is a vertex Lie algebra, so we can consider its universal enveloping vertex algebra and its simple quotient. The vertex algebra that we construct has a Heisenberg subalgebra and we obtain a Walgebra by taking the commutant of the Heisenberg subalgebra inside the vertex algebra.We show that this Walgebra belongs to the family of universal Walgebras constructed by Andrew Linshaw. This Walgebra looks very interesting since it is a limit of Walgebras for affine sl(2) as the
central charge goes to zero. This limit is rather delicate since the Heisenberg subalgebra does not exist at zero central charge. We conjecture that the simple quotient of this Walgebra has the same character as the Walgebra for sl(2). We are still searching for an explicit model for this simple quotient. We used a computer to test our conjecture up to degree 14.
This is a joint work with Kenji Iohara and Olivier Mathieu.
Join us on Friday 10/1/2021 at 9am for the Algebra and Logic Seminar.
Chiral De Rham Complex on the Upper Half Plane and Modular Forms
Xuanzhong Dai, PhD
Fudan
Abstract: Chiral de Rham complex is a sheaf of vertex algebras on any complex manifold or nonsingular variety, which is introduced by Malikov, Schechtman and Vaintrob in 1998. For any congruence subgroup $\Gamma$, we consider the vertex algebra of the $\Gamma$invariant global sections of chiral de Rham complex on the upper half plane that are holomorphic at all the cusps, denoted by $\Omega^{ch}(\mathbb H,\Gamma)$. We show that $\Omega^{ch}(\mathbb H,\Gamma)$ is a topological vertex algebra and give an explicit lifting formula from modular forms to it. As an application, we will modify the RankinCohen bracket such that the modified bracket is compatible with the vertex algebra structure of $\Omega^{ch}(\mathbb H,\Gamma)$. We also see that the Hecke operators act on $\Omega^{ch}(\mathbb H,\Gamma)$ and the actions reduce to the actions on modular forms.
Join us on Friday 9/24/2021 at 11am for the Algebra and Logic Seminar.
Quantum topology and categorification
Josh Sussan, PhD
CUNY Medgar Evers
Abstract: The Jones polynomial may be constructed using the representation theory of quantum sl(2). We will give a review of this construction and then an overview of the categorification program, along with applications to low dimensional topology.
Join us on Friday 5/7/2021 at 11am for the Algebra and Logic Seminar.
Shifted Yangians and polynomial Rmatrices
Huafeng Zhang, PhD
Université de Lille
Abstract: Associated to a complex finitedimensional simple Lie algebra are the socalled shifted Yangians, which include Drinfeld's Yangian as a particular case. We study a category O of representations of shifted Yangians. We establish cyclicity and cocyclicity properties for tensor products of arbitrary irreducible modules with a distinguished family of modules. This implies existence, uniqueness and polynomiality of the Rmatrices for such tensor products. As an application, we prove that the tensor product of two irreducible modules is of finite representation length. (Joint work with David Hernandez)
Join us on Friday 11/20/2020 at 11am for the Algebra and Logic Seminar.
The evasive affine Gaudin model
Evgeny Mukhin, PhD
IUPUI
Abstract: Despite many efforts the affine Gaudin models are still poorly understood. We will describe the difficulties and formulate some conjectures. Our approach originates in the study of the quantum toroidal algebras and the representations of these algebras written in terms of vertex operators.
Join us on Friday 10/30/2020 at 11am for the Algebra and Logic Seminar using the Zoom ID 95200393472 (password: algebra).
Tensor Categories arising from the Virasoro algebra
Flor Orosz Hunziker , PhD
University of Colorado, Boulder
Abstract: In this talk we will discuss the tensor structure associated with certain representations of the Virasoro algebra. In particular, we will show that there is a braided tensor category structure on the category of C1cofinite modules for the Virasoro vertex operator algebras of arbitrary central charge. This talk is based on joint work with Jinwei Yang, Thomas Creutzig, Cuibo Jiang and David Ridout.
Join us on Friday 10/23/2020 at 9am for the Algebra and Logic Seminar using the Zoom ID 95200393472.
Vertex Algebras and Infinite Jet Algebras
Hao Li, PhD
SUNY, Albany
Abstract: The infinite jet algebras are closely related to the vertex algebras. By analyzing the structure of the vertex algebras, we can obtain some nontrivial information of the arc spaces, e.g., Hilbert polynomials. In reverse direction, the properties of certain arc spaces can encode the structure of socalled "classically free" vertex algebras. In this talk, I will provide some examples from FeiginStoyanovsky principal subspaces of the lattice vertex algebras to show this interesting interaction. In particular, these examples tied several qseries identities together. They also found some applications in the area of differential algebras.
Join us on Friday 10/16/2020 at 9am for the Algebra and Logic Seminar using the Zoom ID 95200393472.
Gaudin model, FeiginFrenkel 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 FeiginFrenkel center. In this talk, we will give an introduction to Gaudin models and explain how the FeiginFrenkel 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.
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, CalabiYau 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, 99:50am in CMK 211:
Stable Hypergraph Regularity
Rehana Patel
African Institute for Mathematical SciencesSenega
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, 34pm in CMK 201:
Ramseylike 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 Ramseytype properties. It turns out that most smaller large cardinals $\kappa$ can be characterized by the existence of elementary embeddings on miniuniverses of size $\kappa$. The Ramseylike 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 Ramseylike cardinals.
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.
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.
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 11/18/21 at 10am for the Analysis and Dynamics seminar.
Asymptotic scaling and universality for skew products with factors in SL(2,R)
Hans Koch, PhD
University of Texas at Austin
Abstract: The Hofstadter model (of an electron moving on a lattice) and the associated almost Mathieu model has motivated a lot of work in the past 40+ years, both in Physics and Mathematics. Among the areas involved are Schroedinger operators, dynamical systems, renormalization, fractals, arithmetic, quantum groups, integrable models, etc. After describing some existing results and numerical observations, I will focus on the critical behavior of the model near certain phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation.

Past Analysis and Dynamics Seminars
Join us on Friday 10/1/21 at 10am for the Analysis and Dynamics seminar.
Two Theories of Integration
William G. Faris, PhD
University of Arizona
Abstract: In multivariable calculus one integrates differential forms. There are actually two kinds of differential forms and two theories of integration. The first kind are the differential forms that integrate over an oriented curve or surface. The second kind are the twisted differential forms that integrate over a transversely oriented curve or surface.
Sometimes (but not always) there are nice pictures of differential forms and of twisted differential forms. These are very different from pictures of vector fields and twisted vector fields, but nevertheless there are interesting connections between these pictures. The closest relation is that between vector fields and twisted n1 forms. In physics this is the relation between a force field and a flux field. In many ways the flux field picture is more illuminating.
Join us on Friday 09/24/21 at 10am for the Analysis and Dynamics seminar.
An overview of eigenvalue statistics for random Schrödinger operators and random band matrices
Peter Hislop, PhD
University of Kentucky
Abstract: This review talk will present some recent results concerning local eigenvalue statistics (LES) for some random Schr\"odinger operators (RSO) and random band matrices (RBM). The motivation for studying LES for RSO is the conjecture that RSO in three or more dimensions are expected to exhibit a localizationdelocalization transition. It is anticipated that the LES for RSO in the localized phase is given by a Poisson point process, whereas in the delocalized phase the LES is the same as the Gaussian orthogonal ensemble in random matrix theory. Similarly, RBM are expected to exhibit a transition in LES depending on the ratio of the bandwidth to the matrix size.
Join us on Friday 05/28/21 at 10am for the Analysis and Dynamics seminar.
Stability of the Bulk Gap
Robert Sims, PhD
University of Arizona
Abstract: We prove that uniformly small shortrange perturbations do not close the bulk gap above the ground state of frustrationfree quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finitesystem Hamiltonians uniform in the system size. To obtain this result, we adapt the BravyiHastingsMichalakis strategy to the GNS representation of the infinitesystem ground state. This is joint work with Bruno Nachtergaele and Amanda Young.
Join us on Friday 05/14/21 at 10am for the Analysis and Dynamics seminar.
An overview of eigenvalue statistics for random Schrodinger operators and random band matrices
Peter Hislop, PhD
University of Kentucky
Abstract: This review talk will present some recent results concerning local eigenvalue statistics (LES) for some random Schr\"odinger operators (RSO) and random band matrices (RBM). The motivation for studying LES for RSO is the conjecture that RSO in three or more dimensions are expected to exhibit a localizationdelocalization transition. It is anticipated that the LES for RSO in the localized phase is given by a Poisson point process, whereas in the delocalized phase the LES is the same as the Gaussian orthogonal ensemble in random matrix theory. Similarly, RBM are expected to exhibit a transition in LES depending on the ratio of the bandwidth to the matrix size.
Join us on Friday 04/30/21 at 10am for the Analysis and Dynamics seminar.
Scaling limit of soliton statistics of a multicolor boxball system
Hanbaek Lyu, PhD
University of California, Los Angeles
Abstract: The boxball systems (BBS) are integrable cellular automata whose longtime behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Kortewegde Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified GreeneKleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.
Join us on Friday 04/02 at 10am for the Analysis and Dynamics seminar.
The 3 Gaps Theorem, the BoshernitzanDyson Theorem, frequencies, and applications
Roland Roeder, PhD
Indiana University–Purdue University Indianapolis
Abstract: In this talk, I will prove the 3 Gaps Theorem and mention some applications. I will then discuss a higherdimensional analog that was proved by Boshernitzan and Dyson in early 1990s and also recent new results about the frequencies at which the gaps occur. (The latter is joint work with Bleher, Homma, Ji, and Shen and most recently with Haynes.) Everything will be discussed at an elementary level.
Join us on Friday 03/19 at 10am for the Analysis and Dynamics seminar.
Trees, functional inversion and the virial expansion
Sabine Jansen, PhD
LudwigMaximiliansUniversität München
Abstract: Trees are ubiquitous. Probabilists may think of branching processes and ask about extinction or survival. The recursive structure of trees leads to functional equations for generating functions, of interest in analytic combinatorics. Trees also help organize power series expansions in various areas of analysis and mathematical physics, from numerics (Butcher trees) to renormalization (GallavottiNiccolo trees).
The talk presents yet another application, namely inverse function theorems for functionals in measure spaces for which Banach inversion is not possible. Combined with cluster expansions from equilibrium statistical mechanics, the theorem allows for a rigorous derivation, in a restricted parameter regime, of density functionals used in analytic models of materials. The talk is based on joint work with Tobias Kuna and Dimitrios Tsagkarogiannis (arXiv:1906.02322 [mathph])and considerably improves earlier results based on LagrangeGood inversion (J., Tate, Tsagkarogiannis, Ueltschi, CMP 2014).
Join us on Friday 02/26 at 10am for the Analysis and Dynamics seminar.
Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries
Yizao Wang, PhD
University of Cincinnati
Abstract: We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by [0, 1]. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier result by Derrida et al. (2004) for totally asymmetric simple exclusion process in the same phase. In the low/high density phases, the limit fluctuations are Brownian motion. Most interestingly, at the boundary of the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian meander (or a timereversal of the latter, depending on the side of the boundary). Our proofs rely on a representation of the joint generating function of the asymmetric simple exclusion process with respect to the stationary distribution in terms of joint moments of a Markov processes, which is constructed from orthogonality measures of the AskeyWilson polynomials.
Join us on Friday 02/05 at 10am for the Analysis and Dynamics seminar.
Radial Transfer matrices for higher dimensional graphs and absolutely continuous spectrum
Christian Sadel, PhD
Pontificia Universidad Católica de Chile
Abstract: The transfer matrix method for obtaining spectral properties is well established for one dimensional Schrödinger operators and Jacobi operators. In the setup of any locally finite graph (for example Z^d) I will define affine spaces of (radial) transfer matrices. Then a certain special averaging formula for Jacobi operators by Carmona will be generalized to this setup. From this we obtain criteria for absolutely continuous spectrum and will apply them to certain models with random matrix potentials.
Join us on Friday 10/30/2020 at 10am for the Analysis and Dynamics seminar.
Parameter dependence of the density of states for Schrödinger operators
Chris Marx
Oberlin College
Abstract: In this talk we will explore the dependence of the density of states for Schrödinger operators on the potential. The density of states characterizes the averaged spectral properties of a quantum system. Formally, it can be obtained as an infinite volume limit of the spectral density associated with finitevolume restrictions of a quantum system. Such limit is known to exist for certain quantum mechanical models, most importantly for Schrödinger operators with periodic and random potentials.
Following ideas by J. Bourgain and A. Klein, we will consider the density of states outer measure (DOSoM) which is well defined for all Schrödinger operators. We will explicitly quantify the parameter dependence of the DOSoM by proving a modulus of continuity with respect to the potential (in L^{∞}norm and weakstar topology). This result is obtained for all discrete Schrödinger operators on infinite graphs and captures the geometry of the graph at infinity. Applications of this result to random and ergodic operators will be presented.
This talk is based on joint work with Peter Hislop (University of Kentucky).
Join us on Friday 10/16 at 10am for the Analysis and Dynamics seminar using the Zoom ID 979 1161 6273.
Onesided versus twosided 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 onesided (the past) or by onedimensional space (which is the case, for example, for Markov fields), where conditioning is twosided (right and left). I will discuss some examples, in particular generalising this to gmeasures 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 (onesided or twosided) produce the same objects and when they are different. We show moreover the role onedimensional entropic repulsion plays in this setting. Joint work with R. Bissacot, E. Endo and A. Le Ny.
Friday, September 25th, 2020, 10:0010:50am.
Gradient variational problems
Richard Kenyon, PhD
Yale University
Abstract: This is joint work with Istvan Prause. Many wellknown 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 “pLaplacian”. We give a representation of solutions of such a problem in terms of kappaharmonic functions: functions which are harmonic for a laplacian with a varying conductance kappa.
Friday, February 7, 2020, 10:0010:50 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 selfsymmetries 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, 10:0010:50 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 finitedimensional Hilbert spaces. I’ll also point out the importance of this result for quantum mechanics.
Friday, November 1st 2019, 10:0010:50 a.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.
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
Teaching Seminar
Date: Tuesday, Sept 21
Time: 4  5 pm
Location: CMK 201
Discussion topic: Join us for our first seminar meeting of the Fall quarter this Tuesday (9/21) for a discussion on the challenges of changing course modalities this Fall. Many of us are returning to inperson learning this quarter and facing new circumstances. Are your students wearing their masks? How do you handle students who need to quarantine? Is it harder to keep students engaged? We want to hear your firstweek experiences and share ideas on how to manage these challenges. Let's chat.

Past Inclusive Teaching Seminars
We will record past seminar titles and abstracts in this section.
The Inclusive Teaching seminar is organized by Dr. Sara BotelhoAndrade 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.
Other Seminars
Our department holds various additional seminars which may not be offered regularly. Please contact the organizer(s) or follow the provided link of any seminar you wish to attend for further information.
 NonClassical Logic Seminar, N. Galatos (organizer), https://sites.google.com/view/nonclassicallogicwebinar/home
 Rocky Mountain Representation Theory Seminar, A. Linshaw (organizer), https://sites.google.com/view/rockymountainreptheory/home