# 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.

Dropping inverses in lattice-ordered groups

Nick Galatos

University of Denver

Abstract:

Lattice-ordered groups (aka l-groups) are groups and lattices on the same set such that multiplication is compatible with the lattice order. l-groups have a rich algebraic theory and also play a prominent role in the algebraic study of substructural logics. Holland's embedding theorem ensures that every l-group can be embedded in a symmetric l-group: the l-group of order-preserving permutations on a totally-ordered set.

The compatibility of the multiplication with the order implies that multiplication distributes over both joins and meets of the lattice; also, the lattices of l-groups are distributive (join and meet distribute). Therefore, the inverse-free reducts of l-groups (obtained by dropping the inversion operation) are distributive lattice-ordered monoids (DLMs): monoids and lattices on the same set where all the above-mentioned distributivity properties hold.

We prove [1] that an inverse-free equation is valid in all l-groups iff it is valid in all DLMs; this is surprising as it contrasts with the abelian case where a discrepancy was known to exist. Furthermore, we show that, even more surprisingly, every l-group equation (with inverses) is equivalent to an inverse-free equation, i.e. we can "clear denominators" even in the non-abelian case, in contrast to the group case. The combination of these two facts allows for the reduction of the full equational theory of l-groups to that of DLMs.

We further show that DLMs have the finite model property (hence their equational theory is decidable), and that they are generated as a variety by the DLM of all order-automorphisms of the rationals.

Finally, we establish a correspondence between the validity of DLM equations and the existence of right-compatible orders on free monoids. As a byproduct we obtain that every right order on the free monoid extends to a right order on the free group.

[1] From distributive l-monoids to l-groups, and back again, A. Colacito, N. Galatos, G. Metcalfe, S. Santchi, Journal of Algebra 601 (2022), 129–148.

• Past Colloquia

Path signatures in topology, dynamics and data

Vidit Nanda

The University of Oxford

Abstract: The signature of a path in d-dimensional Euclidean space resides in the tensor algebra of that space; it is obtained by systematic iterated integration of the components of that path against one another. This straightforward definition conceals a host of deep theoretical properties and impressive practical consequences. Here I will describe the topological origins of path signatures, their subsequent application to stochastic analysis, and how they facilitate efficient machine learning from persistence diagrams in topological data analysis. This last bit is joint work with Ilya Chevyrev and Harald Oberhauser.

Fair algorithms and causal models

Joshua Loftus

London School of Economics

Abstract:  I will discuss a line of recent work using probabilistic causal models to understand algorithmic fairness. Rather than attempting to make minimal assumptions and provide robust inferences, this approach uses strong assumptions for the sake of interpretability, transparency, and falsifiability. Although the application focus is on fairness, causal models can be applied in similar ways toward achieving other values or objectives in responsible machine learning or data-driven decisions more broadly.

Schur Functors

John Baez

University of California, Riverside

Abstract: The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A "rig" is a "ring without negatives", and the free rig on one generator is N[x], the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of "symmetric 2-rig", and it turns out the category of Schur functors is the free symmetric 2-rig on one generator.  Thus, in a certain sense, Schur functors are the next step after polynomials.

Calculus Reordered: A History of the Big Ideas

David Bressoud
Dewitt Wallace Professor Emeritus

Macalester College

Abstract: This talk will use the calculus curriculum to illustrate how history and research in mathematics education can inform our teaching of mathematics. The standard order of the four big ideas of calculus—limits then derivatives then integrals then series—is historically and pedagogically problematic. Drawing on history and recent research in undergraduate mathematics education, I will make the case for calculus introduced first as problems of accumulation (integration), then ratios of change (differentiation), then sequences of partial sums (series), and finally the algebra of inequalities (limits).

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 q-series, 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.

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.

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.

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.

Applications of vector field topology in surface and volume meshing

Ed Chien, PhD

Boston University

Abstract:

In many applied settings, e.g., finite-element 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 Poincare-Hopf 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.

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 ℤd​​; 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 θ​​-ray 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 θ​​-ray from x​​. In d=2​​ it is a.s. unique, and furthermore, for all x,y​​ the θ​​-rays 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 θ​​-rays starting next to a hyperplane H0​​, translate the hyperplane forward by distance R​​ to give HR​​, and consider the density of sites in HR​​ where one of the θ​​-rays first crosses HR​​.  We show this density approaches 0 as R→∞​​, with near-optimal bounds on the rate. Essentially as a consequence, we show that there are no bigeodesics in any direction.

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 non-degeneracy 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 fair-seeming outcomes. We will discuss what criteria we might want a voting system to meet in order to be considered fair--then 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 election--and 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 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.

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 5/20/2022 at 9am for the Algebra and Logic Seminar.

Integer partitions and characters of Lie algebra representations

Jehanne Dousse

Universite de Lyon

Abstract: A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. In the 1980's, Lepowsky and Wilson established a connection between the Rogers-Ramanujan partition identities and the representation theory of the affine Lie algebra A_1^{(1)}. Other representation theorists have then extended their method and discovered new identities yet unknown to combinatorialists.

After presenting the history of the interactions between the two fields, we will introduce a new generalisation of partitions which is better suited to make the connection with representation theory, and show how it can be used to prove refined partition identities and non-specialised character formulas.

This is joint work with Isaac Konan.

• Past Algebra and Logic Seminars

Join us on Friday 4/29/2022 at 9am for the Algebra and Logic Seminar.

On $\mathbb{N}$-graded vertex algebras associated with vertex algebroids that are cyclic Leibniz algebras

Gaywalee Yamskulna, Ph. D.

Illinois State University

Abstract: For an $\mathbb{N}$-graded vertex algebra $V=\oplus_{n=0}^{\infty}V_n$ such that $\dim V_0\geq 2$, it is known that $V_0$ is a unital commutative associative algebra, and $V_1$ is a Leibniz algebra. Compatibility relations on the algebraic structure of the vertex algebra $V$ yield a beautiful structure on $V_0\oplus V_1$ which we call a vertex $V_0$-algebroid $V_1$. It was shown by Gorbounov, Malikov, and Schechtman that for a given vertex $A$-algebroid $B$, one can construct an $\mathbb{N}$-graded vertex algebra $V=\oplus_{n=0}^{\infty}V_n$ such that $V_0=A$ and the vertex $A$-algebroid $V_1$ is isomorphic to $B$. All the constructed $\mathbb{N}$-graded vertex algebras are generated by $V_0\oplus V_1$ with a spanning property of PBW type. The vertex algebra associated with a $\beta\gamma$ system is an example of such $\mathbb{N}$-graded vertex algebra.

The $\mathbb{N}$ graded vertex algebras associated with vertex algebroids are natural and important to study. However, a classification of such $\mathbb{N}$-graded vertex algebras and their modules are far from being completed. In this talk, I will discuss our investigation into representation theory of vertex algebras associated with vertex algebroids that are cyclic Leibniz algebras.

This talk is based on joint work with C. Barnes, E. Martin, G. Seelinger, and J. Service.

Join us on Friday 2/26/2022 at 9am for the Algebra and Logic Seminar.

Reflecting (on) the modulo 9 Kanade--Russell (conjectural) identities

Ali Uncu, Ph.D.

University of Bath

Abstract: I will be reflecting on the five modulo 9 Kanade-Russell conjectures and our recent efforts towards proving them. I will also present some new modulo 45 conjectures we encountered while trying to prove the modulo 9 conjectures.

This is a joint work with Wadim  Zudilin.

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 half-twisted 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.

Join us on Friday 11/5/2021 at 9am for the Algebra and Logic Seminar.

The structure of parafermion vertex operator algebras

Qing Wang, Ph.D.

Xiamen University

Abstract: In this talk, we will present some recent results about the structure of the parafermion vertex operator algebra related to any basic classical simple Lie superalgebra, particularly, we determine the generators for this algebra. This is a joint work with Cuipo Jiang.

Join us on Friday 10/28/2021 at 9am for the Algebra and Logic Seminar.

Varieties associated to affine vertex operator algebras $L_{k}(sl_3)$ at non-admissible levels

Cuipo Jiang

Shanghai Jia Tong University

Abstract: We will talk about some recent results on varieties associated to affine vertex operator algebras at non-admissible levels, especially for type $A_2$.

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

University of Denver

Abstract: I'll give an introduction to the Bailey machinery for proving Rogers-Ramanujan-type 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 --

1. -series representations of principal characters of the integrable, highest-weight  modules. (Joint work with Russell)
2. Double-pole Nahm-type representations of Andrews-Gordon, Andrews-Bressoud and false theta identities. (Joint work with Milas and Russell, building on earlier works of Jennings-Shaffer and Milas).
3. Andrews-Schilling-Warnaar'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

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 Rogers-Ramanujan 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 2-dimensional torus. It is a Z^2-graded Lie algebra with all components outside the Cartan subalgebra having dimension 1. Our goal is to study simple weight modules with finite-dimensional weight spaces for this Lie algebra which are supported on a half-plane. 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 W-algebra by taking the commutant of the Heisenberg subalgebra inside the vertex algebra.

We show that this W-algebra belongs to the family of universal W-algebras constructed by Andrew Linshaw. This W-algebra looks very interesting since it is a limit of W-algebras 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 W-algebra has the same character as the W-algebra 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 Rankin-Cohen 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 R-matrices

Huafeng Zhang, PhD

Université de Lille

Abstract: Associated to a complex finite-dimensional simple Lie algebra are the so-called 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 R-matrices 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

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 C1-cofinite 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 so-called "classically free" vertex algebras. In this talk, I will provide some examples from Feigin-Stoyanovsky principal subspaces of the lattice vertex algebras to show this interesting interaction. In particular, these examples tied several q-series 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, 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.

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 2/25/22 at 10am for the Analysis and Dynamics seminar.

Sumit Mukherjee

Columbia University

Abstract:  In this talk we will give an exact characterization for the asymptotic distribution of quadratic forms in IID random variables with finite second moment, where the underlying matrix is the adjacency matrix of a graph. In particular we will show that the limit distribution of such a quadratic form can always be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite sum of centered chi-squares, and a Gaussian with a random variance. As a consequence, we derive necessary and sufficient conditions for asymptotic normality, and universality of the limiting distribution. This talk is based on joint work with B. B. Bhattacharya, S. Das, and S. Mukherjee.

• Past Analysis and Dynamics Seminars

Join us on Friday 2/25/22 at 10am for the Analysis and Dynamics seminar.

Sumit Mukherjee

Columbia University

Abstract:  In this talk we will give an exact characterization for the asymptotic distribution of quadratic forms in IID random variables with finite second moment, where the underlying matrix is the adjacency matrix of a graph. In particular we will show that the limit distribution of such a quadratic form can always be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite sum of centered chi-squares, and a Gaussian with a random variance. As a consequence, we derive necessary and sufficient conditions for asymptotic normality, and universality of the limiting distribution. This talk is based on joint work with B. B. Bhattacharya, S. Das, and S. Mukherjee.

Join us on Friday 2/18/22 at 10am for the Analysis and Dynamics seminar.

Planar site percolation

Zhongyang Li

University of Connecticut

Abstract:  Matching pair of graphs were introduced by Sykes and Essam in 1964, and further investigated by Kesten in 1982. I will show that for any matching pair of graphs (G_1,G_2) constructed from an infinite, locally finite, quasi-transitive, 2-connected, planar, one-ended, simple graphs, the critical percolation probabilities satisfy p_c(G_1)+p_u(G_2)=1. I will then discuss how to apply these results to prove two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm.Probab. 1 (1996) 71–82) in the case of quasi-transitive graphs. Finally I will talk about the strict inequality p_c(G)+p_u(G)>1 for any transitive graph that is not a triangulation. Based on joint work with Geoffrey Grimmett (Cambridge).

Join us on Friday 2/11/22 at 10am for the Analysis and Dynamics seminar.

Scaling Limits for Shortest Remaining Processing Time Queues

Amber L. Puha

California State University San Marcos

Abstract:  Joint Work with Sayan Banerjee, Amarjit Budhiraja, H. Christian Gromoll and Lucas Kruk

We consider a single server queue serving a single customer or job type that processes jobs according to the shortest remaining processing time (SRPT) discipline.  For this, the job with the shortest remaining processing time is served first, with preemption.  In particular, if a job arrives that requires less processing time than that remaining for the job in-service, the job in-service is placed on hold and the new arrival commences service.  This is done in a non-idling fashion so that the server doesn’t rest unless there are no jobs in system.  SRPT is of interest because of its optimality properties; it minimizes the number of jobs in system.  But, an exact analysis is intractable.

This talk concerns approximations in the form of scaling limits.  Under functional law of large numbers scaling, we find that there is an order of magnitude difference between the number of jobs in system and the amount of work in system in that the former vanishes and the later does not.  We use a distribution dependent scaling to more fully explain the behavior of the number of jobs in system.  We find that the limit depends on the tail behavior of the processing time distribution.

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.

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 n-1 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 localization-delocalization 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 short-range perturbations do not close the bulk gap above the ground state of frustration-free 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 finite-system Hamiltonians uniform in the system size. To obtain this result, we adapt the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system 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 localization-delocalization 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 box-ball system

Hanbaek Lyu, PhD

University of California, Los Angeles

Abstract: The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de 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 Greene-Kleitman 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 Boshernitzan-Dyson 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 higher-dimensional 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

Ludwig-Maximilians-Universität 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 (Gallavotti-Niccolo 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 [math-ph])and considerably improves earlier results based on Lagrange-Good 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 time-reversal 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 Askey--Wilson 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

Abstract: The transfer matrix method for obtaining spectral properties is well established for one dimensional Schrödinger operators and Jacobi operators. In the set-up 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 finite-volume 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 weak-star 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.

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.

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

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 in-person 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 first-week experiences and share ideas on how to manage these challenges. Let's chat.

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.