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Events for September 2019 – January 2020

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Sep. 03, 2019 10am (MATH 350)	Geometry/Analysis Taylor Klotz (CU Boulder) X A certain class of control systems exhibit a surprising property that addition of extra differential equations suddenly allow the system to be put into a linearized normal form. This is known as extended dynamic feedback linearization (EDFL). Vassiliou showed that a sufficient condition for a control system with symmetry to be EDFL is another type of linearization known as cascade feedback linearization (CFL), in which the system has two “hidden” linearizable systems. This talk will give an overview of the CFL process as well as some new existence results for CFL systems. Cascade Feedback Linearization
Sep. 03, 2019 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Coxeter group is called a $Y$ -group if its Coxeter diagram $Y_{p q r}$ consists of a central node connected to three branches of lengths $p$ , $q$ , and $r$ . I will describe, in the special case $p=1$ , a diagram calculus for the generalized Temperley--Lieb algebra corresponding to this Coxeter group, and discuss how this relates to Kazhdan--Lusztig cells. The generalized Temperley--Lieb algebra of type E(1,q,r)
Sep. 04, 2019 4pm (MATH 350)	Grad Student Seminar Lucas Gagnon (CU Boulder) X A matroid is an abstraction of mathematical problems involving independence and overlap. Matroid structures occur in surprisingly disparate areas of mathematics including linear and abstract algebra, finite geometry, combinatorics, and topology. Moreover, general results about matroids and matroid invariants often specialize in a meaningful way to each of these cases. Unfortunately, matroids can also be frustratingly abstract and there are an absurd number of non-trivially equivalent definitions of a matroid, making matroid theory a challenging subject to begin learning about. In this talk, I will attempt to circumvent these obstacles in order to give an accessible and motivated introduction to matroids, with examples from some of the many areas of math in which they appear. Matroid Prime
Sep. 05, 2019 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X Many natural minimal homeomorphisms occur on homogeneous spaces (e.g., the Cantor set, the circle, etc). Floyd gave the first example of a minimal dynamical system on a nonhomogeneous space. I will discuss this example and further examples, which build on work of Gjerde and Johansen. C*-algebraic application will also be discussed. This is joint work with Ian Putnam and Karen Strung. Nonhomogeneous minimal dynamical systems
Sep. 10, 2019 11am (MATH 350)	Number Theory Ralph Heiner Buchholz (Defence Science and Technology Group, Australia) X I will discuss various ideas used to try to tackle one of the unsolved problems in Richard Guy's book UPINT. Two themes will be parametrised families of curves on the one hand and algebraic surfaces on the other. The status of D21 --- the search for a perfect triangle
Sep. 10, 2019 1pm (MATH 220)	Keith Kearnes (CU) Minimal Abelian Varieties of Algebras
Sep. 10, 2019 2pm (MATH 220)	Jordan DuBeau (CU) X We prove a connection between Laver tables and rank-into-rank embeddings. Introduction to Laver tables 1
Sep. 10, 2019 2pm (MATH 350)	Lie Theory Richard Green (CU) X [This is a continuation of last week's talk.] A Coxeter group is called a $Y$ -group if its Coxeter diagram $Y_{p q r}$ consists of a central node connected to three branches of lengths $p$ , $q$ , and $r$ . I will describe, in the special case $p=1$ , a diagram calculus for the generalized Temperley--Lieb algebra corresponding to this Coxeter group, and discuss how this relates to Kazhdan--Lusztig cells. The generalized Temperley--Lieb algebra of type E(1,q,r) [continued]
Sep. 11, 2019 4:03pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X You may have heard that it takes seven riffle shuffles to randomize a standard deck of cards, and that there is even a proof to back this up. But what does it mean for a deck to be in random order? We will use a card trick from around the turn of the twentieth century to answer this question, study symmetric group algebras, and possibly mention connections to Hochschild homology. This talk is based on the work of Dave Bayer and Persi Diaconis. Using a hundred year old card trick to study group algebras and win money
Sep. 12, 2019 2pm (MATH 350)	Functional Analysis Philipp Schmitt (University of Copenhagen) X Formal deformation quantization was introduced as a conceptual approach to the quantization problem and is, by now, well understood: The existence and classification of formal star products can be deduced from Kontsevich's formality theorem. To make the theory more accessible to physics, strict quantization asks to replace the formal parameter by a real number (that plays the role of Planck's constant), in order to obtain a field of well-behaved algebras, such as Fréchet-*-algebras. Quantization is intimately related to symbol calculi for operators and therefore has many applictions, e.g. to index theory. In this talk, I would like to give a brief introduction to the theory of formal and strict quantization. I will mainly focus on the example of the 2-sphere and explain how to construct an equivariant strict star product on a subclass of analytic functions. Time permitting, I will sketch how to generalize this construction to arbitrary semisimple coadjoint orbits of semisimple connected Lie groups. Strict quantization of the 2-sphere
Sep. 17, 2019 1pm (MATH 220)	Keith Kearnes (CU) Minimal Abelian Varieties of Algebras 2
Sep. 17, 2019 2pm (MATH 220)	Jordan Dubeau (CU) X We finish proving a connection between Laver tables and rank-into-rank embeddings. Introduction to Laver tables 2
Sep. 17, 2019 2pm (MATH 350)	Lie Theory Tianyuan Xu (CU) X In 1985, Lusztig defined a function $a : W \to ℕ$ for any Coxeter group $W$ by using the Kazhdan-Lusztig basis of the Hecke algebra of $W$ . The $a$ -function has important connections with the cell representation theory of $W$ and its Hecke algebra, but is usually difficult to compute directly. However, it is known that an element has $a$ -value 1 if and only it it is a non-identity element with a unique reduced word, and that $W$ contains finitely many elements of $a$ -value 1 if and only the Coxeter diagram of $W$ satisfies certain conditions. In this talk, we describe a similar classification, in terms of Coxeter diagrams, of all Coxeter groups with finitely many elements of $a$ -value 2. We will show that elements of $a$ -value 2 are fully commutative in the sense of Stembridge, and our main tools for the classification include Viennot's heaps and certain so-called star operations on Coxeter groups. (Joint work with Richard Green.) Classification of a(2)-finite Coxeter groups
Sep. 17, 2019 4pm (MATH 350)	Topology Andre Davis (CU) Generalized Cohomology Theories and Spectra
Sep. 18, 2019 4pm (MATH 350)	Grad Student Seminar Carly Matson (CU Boulder) X Ravi Vakil's excellent book The Rising Sea opens with the following quote from David Mumford: "[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics! In one respect the last point is accurate..." In this talk we will look at some of the fundamentals of algebraic geometry from the more concrete perspective of polynomials and we will let this guide us to the land of abstraction. If time permits, there may be secret plotting. From Affine to Zariski: Demystifying Algebraic Geometry
Sep. 18, 2019 5pm (Math 350)	MathClub Yuhao Hu (CU Boulder postdoc) X Imagine a convex curve in the plane. Fitting the curve tightly between two parallel lines, the distance between those lines tells us the width of the curve in a particular direction. If such width does not depend on the angle in which you direct the parallel lines, the curve is said to have constant width. Analogously, we have the notion of bodies of constant width. Rolling a flat board over bodies of the same constant width creates an illusion of rolling over round balls of the same radius. Beyond such counter-intuitiveness, there are more interesting mathematical aspects related to/inspired by shapes of constant width. Shapes of Constant Width and Beyond
Sep. 19, 2019 4pm (MATH 350)	Topology Robert Bruner (Wayne State University) X A(1)-modules are relevant to the calculation of K-theory through the Adams spectral sequence. Many interesting cohomology theories correspond in this way to sub-Hopf algebras B of the Steenrod algebra. Margolis localizations allow us to understand B-modules through their localizations. The method is especially effective for A(1)-modules as there are only two localizations to consider. The category of A(1)-modules
Sep. 24, 2019 12pm (MATH 350)	Kempner Catherine Sulem (University of Toronto) X We consider the movement of a free surface of a two-dimensional fluid over a variable bottom. We assume that the bottom has a periodic profile and we study the water wave system linearized near a stationary state. The latter reduces to a spectral problem for the Dirichlet--Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. Bloch spectral decomposition is a classical tool to address problems in periodic geometries or equivalently differential operators with periodic coefficients. We show that the spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. We find that, generically, the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. Bloch theory and spectral gaps for linearized water waves
Sep. 24, 2019 4pm (MATH 350)	Topology Colton Grainger (CU) Lazard's Theorem
Sep. 25, 2019 4pm (MATH 350)	Grad Student Seminar Jonathan Quartin (CU Boulder) X Although last week’s talk is not strictly a prerequisite, I will continue in the spirit of peeking into the world of Algebraic Geometry via polynomials (this time over the complex numbers). We will focus our attention on homogeneous polynomials, which will land us in a space with so called “points at infinity”- Projective space. Algebraic Geometry: from affine to projective
Oct. 01, 2019 1pm (MATH 220)	Peter Mayr (CU) X We present a functorial construction which, starting from a congruence of finite index in an algebra, yields a new algebra built from the congruence blocks. As an application we show that supernilpotence of congruences is decidable. Algebras from congruences 1
Oct. 01, 2019 2pm (MATH 220)	Michael Wheeler (CU) Hyperprincipal Realizations of Ultrafilters, Part 1
Oct. 01, 2019 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X The categorification of the Hopf algebra of symmetric functions by the representation theory of the symmetric group is a fundamental result in combinatorial representation theory. While some of our favorite graded Hopf algebras have some representation theoretic interpretations via finite dimensional algebras, very few have come from a tower of groups (as in the symmetric group case). This talk gives a tour of recent successes in realizing graded Hopf algebras via towers of groups and their representation theory. We begin with the established case of symmetric functions in noncommuting variables and then quickly moving on to newer cases (including the Malvenuto—Reutenauer algebra). This work is joint with F. Aliniaeifard. Categorifying combinatorial Hopf algebras
Oct. 01, 2019 4pm (MATH 350)	Topology Juan Moreno (CU) Complex Oriented Cohomology Theories
Oct. 02, 2019 4pm (MATH 350)	Grad Student Seminar Mark Pullins (CU Boulder) X In the past two talks, we have explored the Zariski topology and projective space. In this talk, we will begin in familiar territory and review the Zariski topology and varieties. Then, we will introduce linear algebraic groups from a (hopefully) more intuitive perspective and explore the wonderful (and mildly surprising) structural properties gifted from the Zariski topology. Please note that the past two talks are in no way required! Linear Algebraic Groups: From Spaces to Groups
Oct. 02, 2019 5pm (Math 350)	MathClub Harrison Stalvey (CU Boulder) X Epistemology is the philosophy of knowledge. What is knowledge? Where does it come from? How does it develop? Two particular epistemologies are constructionism and constructivism, which view knowledge as a construction, not a discovery, even if it is influenced by social factors. Under the assumption that knowledge is constructed, it is natural to wonder about the psychology behind learning. We will explore some epistemology and psychology as they pertain to a concept central to mathematics: the function. In particular, we will survey the history of the concept as an illustration of constructionism. We will also discuss a particular constructivist learning theory that has been used to describe how an understanding of mathematics can develop in the mind of the learner. Then we will apply this learning theory to the concept of function. This talk will conclude with examples of various avenues of research in mathematics education and what that research might look like. Epistemology, Psychology, and the Concept of Function
Oct. 03, 2019 2pm (MATH 350)	Functional Analysis Mitsuru Wilson (CU Boulder) X This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators. Symbolic calculus on homogeneous vector bundles (Part I)
Oct. 08, 2019 1pm (MATH 220)	Peter Mayr (CU) Algebras from congruences 2
Oct. 08, 2019 2pm (MATH 350)	Lie Theory Lucas Gagnon (CU) X Classifying all conjugacy classes and/or irreducible representations in the family of unipotent upper triangular matrices $UTn(Fq)$ is a wild problem, meaning that it is an essentially hopeless endeavor. In this talk, I will take up the superficially similar question of normal subgroups in $UTn(Fq)$ and answer it with a bijection between normal subgroups and vector spaces over $𝔽_{q}$ which are parametrized by a family of Catalan-like combinatorial objects. Following this construction, I will show how supercharacter theory can turn our newfound understanding of the normal subgroup structure of $UTn(Fq)$ into a deeper representation theoretic understanding of this family of groups. Finding all normal subgroups of the unipotent upper triangular groups
Oct. 08, 2019 2pm (MATH 220)	Michael Wheeler (CU) Hyperprincipal Realizations of Ultrafilters, Part 2
Oct. 08, 2019 4pm (MATH 350)	Topology Andrew Stocker (CU) Complex Bordism, MU and Complex Orientations
Oct. 09, 2019 4pm (MATH 350)	Grad Student Seminar Sarah Arpin (CU Boulder) X Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions (CGL06). These are large expander graphs, and the hard problem is to find an efficient algorithm for routing between two vertices of the graph. In particular, we consider two related graphs that help us understand the big picture of the full supersingular $ℓ$ -isogeny graph: the `spine' $𝒮$ , which is the subgraph of $Gℓ(Fp)$ given by the j-invariants in $𝔽_{p}$ , and the graph $Gℓ(Fp)$ , in which both curves and isogenies must be defined over $𝔽_{p}$ . This is joint work with Catalina Camacho-Navarro, Kristin Lauter, Joelle Lim, Kristina Nelson, Travis Scholl, Jana Sotáková. https://arxiv.org/abs/1909.07779 No pre-requisites are required, but a willingness to take things on faith is requested. "Why, sometimes I've believed as many as six impossible things before breakfast." ~Lewis Carroll Adventures in Supersingularland: An Exploration of Supersingular Elliptic Curve Isogeny Graphs
Oct. 10, 2019 12:10pm (MATH …	Kempner Piotr M. Hajac (IMPAN and University of Colorado Boulder) X In topology, pushouts are formal recipes for collapsing and gluing topological spaces. For instance, shrinking the boundary circle of a disc to a point yields a sphere, shrinking the equator of a sphere to a point gives two spheres joined at the point, collapsing the boundary of a solid torus to a circle, or gluing two solid tori over their boundaries, produces a three-sphere. In noncommutative topology, such procedures are expressed in terms of pullbacks of C-algebras. It turns out that one can visualize a pullback of C-algebras of graphs as a pushout of these graphs thus providing much needed intuition to the abstract setting of operator algebras. The goal of this talk is to discuss how to make this visualization rigorous by conceptualizing abundant examples from noncommutative topology that lead to a new concept of morphisms of graphs. In particular, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. (Based on joint works with Alexandru Chirvasitu, Sarah Reznikoff and Mariusz Tobolski.) From pushouts to pullbacks: a sample of noncommutative topology
Oct. 10, 2019 2pm (MATH 350)	Functional Analysis Mitsuru Wilson (CU Boulder) X This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators. Symbolic calculus on homogeneous vector bundles (Part II)
Oct. 15, 2019 4pm (MATH 350)	Topology Akira Tominaga (Tokyo University) Quillen's Theorem
Oct. 16, 2019 4pm (MATH 350)	Grad Student Seminar Erik Knutson (CU Boulder) X The Steenrod algebras are graded Hopf algebras over fields of prime order which arise in the study of topological invariants of spaces. The goal of this talk will be to learn just enough about homological algebra and the Steenrod algebra generated by the Steenrod squares over everyone's favourite two element field to play a little game using diagrammatic representations of Steenrod submodules. Playing in the Steenrod Sandbox
Oct. 16, 2019 5pm (Math 350)	MathClub Rob Tubbs (CU Boulder) X As an introduction to an important branch of number theory, diophantine approximation, we will examine several real numbers and show that they are irrational. We will also discuss whether they are surds, which are irrational numbers that are roots of rational numbers. An absurdist look at the nature of some peculiar decimals
Oct. 17, 2019 2pm (MATH 350)	Functional Analysis Shen Lu (CU Boulder) X For a fixed prime $p$ , a noncommutative solenoid as defined by Frederic Latremoliere and Judith Packer is a twisted group C*-algebra $Cleft( Z[1/p] × Z[1/p], σ )$ , where $Z[1/p] := j/pk ∈ Q : j ∈ Z, k ∈ N$ is an additive discrete group and $σ$ is a $T-$ valued group $2$ -cocycle (multiplier) on $Z[1/p] × Z[1/p]$ . In this talk, we first review the classification of all NC solenoids in terms of their defining multipliers using $K-$ theory. From there, we discuss two constructions for projective modules over the irrational NC solenoids and how they are related: one by constructing the Heisenberg equivalence bimodule of Rieffel utilizing the $p-$ adic numbers, and the other by writing the NC solenoids as direct limits of noncommutative tori. Noncommutative solenoids and their finitely generated projective modules
Oct. 22, 2019 11am (MATH 350)	Number Theory Min Ru (University of Houston) X In recent years, there have been some breakthroughs in extending Schmidt's subspace theorem to non-linear hypersurfaces, made by Corvaja-Zannier, and Evertse-Feretti, etc. In this talk, I will discuss my recent joint work with P. Vojta based on their works, as well as its applications. Schmidt's subspace theorem and its generalization
Oct. 22, 2019 12:10pm (MATH …	Kempner Jan Mycielski (CU Boulder) X A description of the structure of knowledge in the brains of animals, with an outline of the phylogeny of human intelligence. The theory supports some old ideas in the philosophy of mind and it may have applications in artificial intelligence. I will define finite algebraic structures, that I call mental models. They are intended to represent in a realistic way the descriptive component of knowledge (as opposed to its evaluating component). I will define the notions of mental object and of concepts in a mental model. Concepts can be real, like these representing objects and processes of everyday life or studied in science, or they may be imaginary, like these in tales and pure mathematics. Our definitions yields a precise form of Ockham's principle of economy of concepts. The theory suggests some ideas in ontology and pedagogy. Theories of the mind that ignore biology or brain science, or do not propose sufficiently clear notions of meaning, will be criticized. Mathematics of human intelligence and related ideas
Oct. 22, 2019 4pm (MATH 350)	Topology Jonathan Quartin (CU) The Moduli Stack of Formal Groups
Oct. 24, 2019 2pm (MATH 350)	Functional Analysis Paul Baum (Penn State University) X This talk will outline exactly how and why the Atiyah-Singer index theorem can be viewed as a corollary of Bott periodicity. The precise argument proceeds by introducing an abelian group K_0(.) obtained by considering pairs (M, E) where M is a closed Spin-c manifold and E is a a vector bundle on M. "Spin-c manifold" will be carefully defined. Most of the oriented manifolds which occur in practice are Spin-c. The above is joint work with Erik van Erp. ATIYAH-SINGER AS A COROLLARY OF BOTT PERIODICITY : PART 1 Sponsored by the Meyer Fund
Oct. 29, 2019 12:10pm (MATH …	Kempner Paul Baum X Let $Γ$ be a compact Hausdorff second countable topological group. ``second countable ``the topology of $Γ$ has a countable base" Irr( $Γ$ ) denotes the set of equivalence classes of irreducible representations of $Γ$ . The vector space for each irreducible representation of $Γ$ is a finite dimensional vector space over the complex numbers. Irr( $Γ$ ) is a countable set --- and we give it the discrete topology. Let $X$ be a locally compact Hausdorff topological space with a given (continuous) action of $Γ$ on $X$ . For each $x$ in $X$ , $Γ_{x}$ denotes the isotropy group of $x$ . If $x$ and $y$ are in the same orbit for the given action of $Γ$ on $X$ , then Irr( $Γ_{x}$ ) and Irr( $Γ_{y}$ ) are canonically in bijection. The usual quotient, denoted $X/Gamma$ , is obtained by collapsing each orbit (for the given action of $Γ$ on $X$ ) to a point. The extended quotient, denoted $X//Γ$ , is constructed by replacing the orbit of $x$ (for the given action of $Γ$ on $X$ ) by Irr( $Γ_{x}$ ). This talk will describe how extended quotient enters into the representation theory of Lie groups and reductive p-adic groups. For Lie groups this is due to Mackey-Higson-Afgoustidis. For reductive p-adic groups, this is due to Aubert-Baum-Plymen-Soll Extended quotient in representation theory or Non-commutative geometry impacts representation theory
Oct. 29, 2019 4pm (MATH 350)	Topology Kate Ponto (University of Kentucky) X The Lefschetz fixed point theorem gives an invariant that is zero when an endomorphism is homotopic to a map with no fixed points. The converse is not true unless we refine the Lefschetz number to a richer invariant called the Reidemeister trace. When you come to these invariants as a homotopy theorist (of a particular variety) you notice that the Reidemeister trace is a topological Hochschild homology invariant and is additive. This suggests that you should look for connections to algebraic K-theory since it is the universal home for additive invariants. In this talk I'll (start to) explain the connection between traces defined from fixed point invariants and traces defined for computing K-theory Traces in K-theory and fixed point theory Sponsored by the Meyer Fund
Oct. 30, 2019 3pm (MATH 350)	Algebraic Geometry Alicia Lamarche (University of South Carolina) X When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety X, to what extent can Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(X) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full ?etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin. Derived Categories, Arithmetic, and Rationality Questions Sponsored by the Meyer Fund
Oct. 30, 2019 4pm (MATH 350)	Grad Student Seminar Howie Jordan (CU Boulder) X Would you like to play a game? It's the day before Halloween, so let's get a little spooky. Open this door with the key of imagination, and step into a world beyond your usual understanding. A world where the axiom of choice fails, but not horrendously so. A world where every set of real numbers is Lebesgue measurable. A world where we can't turn a single sphere into two with the same volume or well-order every set. But it will all start with a simple game The goal of this talk is to provide an introduction to an alternative axiom of set theory, one which contradicts the axiom of choice but shares some of that axiom’s implications, as well as its own unique, interesting, and even paradoxical consequences. We will compare this axiom to choice, see how they contradict, and if time permits we may even discuss implications of our considerations for foundations of mathematics. You Have No Choice; It is Already Determined
Oct. 30, 2019 5pm (Math 350)	MathClub Sean O'Rourke (CU Boulder) X How many times do you have to shuffle a deck of cards to mix it up? The answer depends on what we mean by shuffling. I will discuss this question and explain how shuffling a deck of cards can be modeled mathematically. How many times should you shuffle a deck of cards?
Oct. 31, 2019 10am (Math 350)	Thesis Defenses Tyler Schrock X Conjugacy is the natural notion of isomorphism in symbolic dynamics. A major open problem in the field is that of finding an algorithm to determine conjugacy of shifts of finite type (SFTs). We consider several related computational problems restricted to k-block codes. We show verifying a proposed k-block conjugacy is in P, finding a k-block conjugacy is GI-hard, reducing the representation size of a SFT via a 1-block conjugacy is NP-complete, and recognizing if a sofic shift is a SFT is in P. Then we consider the complexity of isomorphism among finite groups, where we consider isomorphism between quotients of genus 2 groups. The centrally indecomposible genus 2 groups split into two cases: flat and sloped. We show determining isomorphism between quotients of genus 2 groups by non-central subgroups is in P and determining isomorphism between quotients of the genus 2 groups H^\flat_1 by central subgroups is in P. On the complexity of isomorphism in finite group theory and symbolic dynamics
Oct. 31, 2019 2pm (MATH 350)	Functional Analysis Paul Baum (Penn State University) X This talk will take the outline provided in Part 1 and fill in details so as to provide a complete proof. ATIYAH-SINGER AS A COROLLARY OF BOTT PERIODICITY : PART 2 Sponsored by the Simons Foundation
Oct. 31, 2019 3pm (MATH 350)	Probability Mei Yin (University of Denver) X Exponential random graphs are powerful in the study of modern networks. By representing a complex global configuration through a set of tractable local features, these models seek to capture a wide variety of common network tendencies. This talk will look into the asymptotic structure of weighted exponential random graphs and formulate a quantitative theory of phase transitions. The main techniques that we use are variants of statistical physics. Based on joint work with multiple collaborators. Perspectives on exponential random graphs
Nov. 05, 2019 10am (MATH 350)	Geometry/Analysis Mitsuru Wilson (CU Boulder) X I will give an introductory talk on pseudo-differential operators and their symbols. I will also present my recent results for the case of associated vector bundles of a homogeneous space wherein I generalized Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions for a pseudo-differential operator is its symbol. I will draw analogies from $ℝ^{n}$ to the compact Lie group case and to vector bundles. Global symbolic calculus on homogeneous vector bundles.
Nov. 05, 2019 1pm (MATH 220)	Peter Mayr (CU Boulder) X It is easy to see that the direct product of two finitely generated groups or finitely generated lattices is finitely generated again. We investigate under which conditions generating sets for subdirect products are finite. Finiteness conditions for subdirect products 1
Nov. 05, 2019 2pm (MATH 350)	Lie Theory Mandi Schaeffer-Fry (MSU) X A theorem of Thompson studies the structure of finite groups with exactly one irreducible character degree not divisible by a given prime. I will discuss two recent variations of Thompson’s theorem. Namely, in joint work with E. Giannelli and C. Vallejo, we prove a variation for pairs of primes, and with E. Giannelli and N. Rizo, we prove a variation in the case that exactly two degrees are not divisible by a given prime. On Some Extensions of a Theorem of Thompson
Nov. 05, 2019 3pm (MATH 350)	Algebraic Geometry Abhiram Natarajan (Purdue) X Joint seminar with the Theoretical CS seminar, and the Topology seminar. One object that is pervasive throughout this talk is random polynomials, i.e. polynomials sampled from a specific distribution over the space of polynomials of fixed degree and dimension. We are going to be interested in bounds on the average Betti numbers of the zero sets of random polynomials in various situations. The first part of the talk is about the topological complexity of real algebraic sets restricted to sets that arise in o-minimal geometry. Given any sequence of smooth and compact hypersurfaces in R^{n-1}, we prove that (up to extracting subsequences) there exists a regular definable (in a particular o-minimal structure over R) hypersurface $Γ$ in $R P^{n}$ such that each hypersurface in the sequence appears as a component of the zero set of some polynomial on $Γ$ . On the other hand, we show that given $Γ$ , for "most" polynomials, a Bezout-type bound holds for the restriction of $Γ$ on the zero set of the polynomial. In the second part of the talk, we study the arrangement of hypersurfaces defined in projective space by the zero sets of s randomly and independently chosen (Kostlan) homogeneous polynomials. Our first theorem here uses a random spectral sequence which gives information about the top Betti number of the union of the zero sets of the polynomials. The next result here deals instead with the asymptotic structure of the union of the zero sets when we have s quadrics. It turns out that in this case, the problem of understanding the number of connected components of the union of zero sets is related to the connectivity of a certain random graph model, and can be studied in a precise way. Specifically, our second theorem gives an upper bound on the average number of connected components in a random arrangement of quadrics' zero sets. This is joint work with Saugata Basu and Antonio Lerario. Quantitative bounds on the topology of "random" sets in o-minimal geometry and real algebraic hypersurface arrangements
Nov. 05, 2019 4pm (MATH 350)	Topology Caroline Matson (CU Boulder) Heights of Formal Groups
Nov. 12, 2019 12:10pm (MATH …	Kempner Krzysztof Burdzy (University of Washington) X In lieu of an abstract I offer a problem. Consider three billiard balls of the same radius and mass, undergoing totally elastic reflections on a billiard table with no walls (the whole plane). All three balls can be given non-zero initial velocities. What is the maximum (supremum) possible number of collisions among the three balls? The supremum is taken over all initial positions and initial velocities. I will discuss this problem and its generalization to any finite family of balls in one, two and higher dimensions. Joint work with Jayadev Athreya and Mauricio Duarte. On the number of collisions of billiard balls
Nov. 12, 2019 1pm (MATH 220)	Peter Mayr (CU) Finiteness conditions for subdirect products 2
Nov. 12, 2019 4pm (MATH 350)	Topology Robin Deeley (CU Boulder) The Stratification of the Moduli Stack of Formal Groups
Nov. 13, 2019 5pm (Math 350)	MathClub Peter Mayr (CU Boulder) X Consider all computer programs of a fixed ``length'', say n. There are only finitely many of them.Some may run forever in an infinite loop. But among those that eventually halt, there must be some that run for the maximum number of steps. These programs are the busy beavers of length n, and the number of steps they run is denoted by BB(n). The function BB grows incredibly fast. So far its exact values are known only for n = 1,2,3,4. In fact it follows from the undecidability of the Halting Problem for Turing machines that there is no algorithm to compute BB(n) in general. Even worse, it was recently shown that BB (1919) cannot be determined using our usual foundations of Math (Zermelo-Fraenkel set theory). Clearly BB(1919) is some concrete finite number. However Math as we know it is not strong enough to prove which one. The busy beaver at the end of Math
Nov. 14, 2019 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. Wieler solenoids and the K-theory of the associated stable and stable Ruelle algebras
CANCELED Nov. 14, 2019	Probability Dan Stroock (MIT) X Suppose $X$ and $Y$ are independent, identically distributed real valued random variables and that $θ ∈ (0,1)$ . Then $θ X + 1-θ2 Y$ has the same distribution as $X$ and $Y$ if and only if these are centered Gaussian random variables. Equivalently, if $μ$ is a Borel probability measure on $ℝ$ , then $μ(ξ) = μ(θ ξ) μ(1-θ2 ξ)$ if and only if $μ$ is a centered Gaussian measure. In this lecture I will first prove this result in the case when $θ = 2-1/2$ and then give some applications of it. Some Applications of Gaussian Measures
Nov. 19, 2019 2pm (MATH 350)	Lie Theory Noah White (UCLA) X The RSK correspondence is a bijection between permutations (more generally nonnegative integer matrices) and certain pairs of tableaux. I will outline a geometric incarnation of this bijection coming from the representation theory of the general linear group. I will describe the link to the Calogero Moser system of particles on a line and to a conjecture of Bonnafe and Rouquier related to Kazhdan--Lusztig cells of finite reflection groups. This is joint work with Adrien Brochier and Iain Gordon. Particle collisions and RSK
Nov. 19, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) The Landweber Exact Functor Theorem
Nov. 21, 2019 2pm (MATH 350)	Functional Analysis Andrew Stocker (CU Boulder) X Dynamical systems that are expansive form a large class of examples, including all Smale spaces, shift spaces, and finitely presented systems. For Smale spaces it is well known that we can build C*-algebras from the stable, unstable, and homoclinic equivalence relations. However, we wish to extend these constructions to the class of finitely presented systems, for which we will need to use the more refined notion of local conjugacy to study the dynamics. We will be following Klaus Thompsen's work on relatively expansive dynamical systems. Expansive Dynamical Systems
Dec. 03, 2019 11am (MATH 350)	Number Theory Hanson Smith (CU) X A number field $K$ is monogenic if the ring of integers admits a power integral basis, i.e., if $OK=Z[α]$ for some $α ∈ OK$ . This talk will survey recent results on monogeneity with a focus on the speaker's recent results on radical extensions and torsion fields of abelian varieties. A Survey of Monogeneity
Dec. 03, 2019 1pm (MATH 220)	Peter Mayr (CU Boulder) X In the 1980s commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator. Since this commutator has particularly nice properties for algebras in varieties with modular congruence lattices, the structure theory for abelian, nilpotent, solvable,.. algebras is best developed in this setting. Unfortunately a variety of semigroups is congruence modular if and only if it consists of groups. We give an overview of properties of the binary and higher commutators for semigroups. In particular we investigate what the derived notions of nilpotence, supernilpotence, and solvability mean and how they relate to classical concepts in semigroup theory. Commutators for semigroups
Dec. 03, 2019 2pm (MATH 220)	Howie Jordan (CU Boulder) X The Axiom of Choice is one of the most pervasive principles of foundational mathematics, with much study focusing on the consequences and equivalences of it as well as the “disasters” present in its absence. In this talk, we will introduce an alternative principle, Determinacy, and see that the absence of Choice doesn’t have to be such a disaster after all. Determinacy is a game theoretic axiom that sits in an interesting position compared with Choice, directly contradicting it but sharing some of Choice’s implications and coming with its own novel consequences. We will explore some parts of the picture of set theory under Determinacy as compared with that under Choice, consider various forms of Determinacy, and comment on the consequences of our considerations for foundations of mathematics and logic. Determinacy and Choice
Dec. 03, 2019 2pm (MATH 350)	Lie Theory Sarah Salmon (CU) X This is an ongoing project with Dr. Eirini Chavli (University of Stuttgart), Dr. Maud DeVischer (City University of London), Dr. Alison Parker (University of Leeds), and Dr. Ulrica Wilson (Morehouse College). The walled Brauer algebra is a subalgebra of the Brauer algebra. As a diagram algebra, the walled Brauer algebra can be represented using diagrams of the Brauer algebra that satisfy certain conditions. We will discuss previous work has been done to describe the center of this algebra in the cases where the algebra is semisimple. It was conjectured that the center in the nonsemisimple case can be described the same way. We will describe current work on this conjecture. On the center of the walled Brauer algebra
Dec. 03, 2019 3pm (MATH 350)	Algebraic Geometry Patricio Gallardo (Washington University St. Louis) X It is known, thanks to the work of Deligne and Mostow from 1986, that some ball quotients can be interpreted as moduli spaces of points in the line. Such interpretation has been expanded to K3 surfaces thanks to the work of Kondo and Dolgachev. Within this context, we will describe ongoing work with Matt Kerr and Luca Schaffler, which allows us to give geometric interpretations to the toroidal compactifications of such ball quotients. Geometric interpretation of toroidal compactifications of Deligne-Mostow ball quotients
Dec. 03, 2019 4pm (MATH 350)	Topology Eva Belmont (Northwestern University) X Motivic homotopy theory over R is interesting in part because of its connection to ordinary stable homotopy theory and to C_2-equivariant homotopy theory. In this talk I will review some of these connections, and discuss work in progress with Dan Isaksen to compute R-motivic stable homotopy groups of spheres using an Adams spectral sequence. One of our main applications is to a variant of the Mahowald invariant which can be computed using knowledge of the R-motivic Adams spectral sequence. The R-motivic Adams spectral sequence Sponsored by the National Science Foundation
Dec. 04, 2019 4pm (MATH 350)	Grad Student Seminar Hanson Smith (CU Boulder) X In this talk we'll survey the development and study of number fields from a historical perspective. By focussing on questions and conjectures, some solved and some unsolved, we'll gain an appreciation for number fields and their utility in algebraic number theory. This talk will assume very little background besides a basic knowledge of rings and fields. Problems with Number Fields
Dec. 04, 2019 5pm (Math 350)	MathClub David Grant (CU Boulder) X In digital communications, all messages are sequences are zeroes and ones, but because of physics (for examples, radio waves from a cell tower bouncing off the Flatirons), sometimes a 0 gets interpreted as a 1, and vice versa. To combat errors, engineers build redundancy into transmitted messages. For example, sending 101 as 1111111111 000000000 1111111111 allow a few errors in reception, while still allowing a message to be faithfully decoded. Indeed if we received that transmission as 1101110111 0011000001 1111001101 we can use “majority decoding” to correctly determine that the original message was indeed 101. The problem is that with TOO much redundancy transmission becomes slow and wastes power. This is where the MATH comes in: how do you add enough redundancy to code words, so that the message can be recovered despite a few errors, but not SO much that the process is inefficient? These possible transmitted message are called {\it codewords} and sets of the them are called {\it codes}. We’ll discuss properties of codes like the Hamming distance, which measures the number of places in which two codewords differ, and a “weight enumerator” of the code that records how many codewords have a given distance from the 0 vector. The freaky thing is that if we define the “dual code” $C⊥$ to $C$ as all the code words that have a trivial dot product (mod 2) with all the codewords of $C$ , then the weight enumerator is $C⊥$ can easily be read off from the weight enumerator of $C$ , by an amazing theorem of Jessie MacWilliams. We will present her famous theorem and provide as many details of the proof as we can. CAN YOU HEAR ME NOW?
Dec. 10, 2019 2pm (MATH 350)	Lie Theory Emily King (CSU) Lie Algebras, Coxeter Groups, and Quantum Information Theory
Dec. 10, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU) Morava Stabilizer Groups and Lubin Tate Theory
Jan. 14, 2020 10am (MATH 350)	Geometry/Analysis Svetlana Roudenko (Florida International University) X We consider Zakharov-Kuznetsov (ZK) equation, a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation, which was introduced in physical context by V. Zakharov and E. Kuznetsov back in 1972. One of the questions they asked was existence of solitons in higher dimensions (the KdV is restricted as the one-dimensional model) and their stability. In this talk I will discuss behavior of solitary waves in the 3d ZK equation, proving that solutions in the energy space that are orbitally stable are also asymptotically stable, that is, as time goes to infinity, they converge to a rescaling and shift of the solitary wave Q(x-t,y,z) in a certain rightward moving window. This is a joint work with Luiz Gustavo Farah, Justin Holmer, and Kai Yang. Soliton stability in higher-dimensional generalization of KdV equation. Sponsored by the Meyer Fund
Jan. 14, 2020 2pm (MATH 220)	Eric Samperton (University of Illinois at Urbana-Champaign) X Algebraic invariants of a manifold can often be formulated as constraint satisfaction problems, where local constraints are induced by the topology via the combinatorics of a triangulation of the manifold. There are many interesting complexity theoretic questions concerning such invariants that are basically wide open. In the first part of this talk, I’ll explain why I believe 3-dimensional manifolds are the most interesting from this perspective. I’ll then quickly discuss what is known about some important decision problems in 3-manifold topology (homeomorphism, unknot recognition, sphere recognition, etc), before turning to the things I think about the most: invariants of 3-manifolds coming from topological quantum field theories. Computational complexity and invariants of 3-dimensional manifolds
Jan. 14, 2020 2pm (MATH 350)	Lie Theory Shawn Burkett (Kent State University) X Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius--Wielandt group is a triple $( G,H,L)$ where $H/L$ is almost a Frobenius complement for $G$ ; A Camina pair is a pair $( G,N)$ where $N$ is almost a Frobenius kernel for $G$ ; A Camina triple is a triple $( G,N,M)$ where $( G,N)$ and $( G,M)$ are almost Camina pairs. In this talk we discuss triples $( G,N,M)$ where $( G,N)$ and $( G,M)$ are almost Frobenius groups. A Frobenius group analog for Camina triples Sponsored by the Meyer Fund
Jan. 14, 2020 4pm (MATH 350)	Topology Jonathan Campbell (Duke University) X In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two n-dimensional polytopes, $P$ , $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble the pieces into $Q$ . The scissors congruence problem, aka Hilbert Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent iff they have the same area. In three dimensions, there is volume AND another invariant, the Dehn Invariant. In higher dimensions, very little is known. I'll give an introduction to this very classical problem, and explain how homotopy theory can be used to get purchase on it. Prerequisites: The discussion of Hilbert's Third Problem and Dehn's invariant will be widely accessible. No knowledge of algebraic K-theory will be assumed. This is all joint work with Inna Zakharevich. Hilbert's Generalized Third Problem and Homotopy Theory
Jan. 15, 2020 3pm (Math350)	Math Phys Markus J. Pflaum organizational meeting, overview
Jan. 19, 2020 9:30am (MATH 3…	Several talks (by people) X Coffee and bagels for participants start at 9:30. We plan to start at 10am. We plan to end at 4pm. Birkhoff Seminar Sponsored by the Meyer Fund
Jan. 21, 2020 10am (Math 350)	Noncomm Geometry Vladimir Baranovsky (University of California, Irvine) X For an algebraic or holomorphic symplectic variety M with a chosen deformation quantization of functions we consider a coherent sheaf F and try to find its deformation quantization as well. It turns out that existence of such quantization imposes nontrivial conditions on the Chern Character of F. These will be derived in the talk from the Algebraic Index Theorem. Chern character of quantizable sheaves
Jan. 21, 2020 11am (MATH 220)	Functional Analysis Elizabeth Gillaspy (University of Montana) X Higher-rank graphs (k-graphs) are a combinatorial model for C-algebras; indeed, much of the structure of k-graph C-algebras is encoded in the underlying combinatorial data of the k-graph. However, different k-graphs can give rise to isomorphic or Morita equivalent C-algebras. In this talk, we present several ways to modify the structure of a k-graph which preserve the Morita equivalence class of the associated C-algebra. Our constructions are inspired by the analogous work for graph C*-algebras of Bates and Pask, as well as by the textile system approach to describing k-graphs. This is joint work with C. Eckhardt, K. Fieldhouse, D. Gent, I. Gonzales, and D. Pask. Moves on higher-rank graphs preserving Morita equivalence Sponsored by the Meyer Fund
Jan. 21, 2020 1pm (MATH 220)	Dmitriy Zhuk (Moscow State University) Quantified Constraint Satisfaction Problems Sponsored by the Meyer Fund
Jan. 21, 2020 2pm (MATH 350)	Functional Analysis Karen Strung (Czech Academy of Sciences, Prague) X Given a Hilbert C-correspondence over C(X), one can construct the associated Cuntz-Pimsner algebra. In the case that the correspondence is full, nonperiodic, and minimal, the resulting C-algebra is simple and unital. An example of such a correspondence is obtained by taking the right Hilbert C(X)-module of continuous sections of a vector bundle over X and twisting the left multiplication by a minimal homeomorphism. As is the case of crossed products by minimal homeomorphisms, one can identify "orbit-breaking" C-subalgebras. In this talk I will discuss these Cuntz--Pimsner C-algebras and their orbit-breaking subalgebras, their relationship to one another, and when they can be classified by Elliott invariants. This is joint work with Adamo, Archey, Forough, Georgescu, Jeong and Viola. Simple constructions from Hilbert C(X)-correspondences Sponsored by the Meyer Fund
Jan. 21, 2020 2pm (MATH 220)	Alexandr Kazda (Charles University Prague) The category of valued relational structures Sponsored by the Meyer Fund
Jan. 21, 2020 4pm (MATH350)	Topology Hessel Posthuma (Amsterdam) X In this talk I will explain how to define so-called “higher genera” for a proper cocompact action of a Lie group satisfying some additional assumptions. These higher genera are numbers, whose definition is inspired by index theory for Dirac type operators in Noncommutative Geometry, and generalize the signature and A-hat genus for compact manifolds. The main theorem states that the higher signatures are equivariant homotopy invariants and that vanishing of the higher A-hat numbers is a necessary condition for the existence of an invariant spin structure together with a metric of positive scalar curvature. This talk is based on joint work with P. Piazza. Higher genera for proper action of Lie groups Sponsored by the Meyer Fund
Jan. 22, 2020 3pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) Recapitulation of Hilbert spaces and their uses in QM and QFT
Jan. 22, 2020 4pm (MATH 350)	Grad Student Seminar Andrew Campbell (CU Boulder) X In 1898 Ladislaus Bortkiewicz showed that the number of Prussian soldiers killed accidentally by horse kicks in a year follows a Poisson distribution. Since then the Poisson distribution and Poisson point processes have found numerous applications in engineering, physics, finance, biology, ecology, etc. We will ignore almost all of these and instead look at the Poisson process in random geometric graphs and random linear operators. On the number of Prussian soldiers killed each year by horse kicks
Jan. 28, 2020 12:10pm (MATH …	Kempner Kate Stange and Sebastian Bozlee (CU Boulder) X During Fall 2019, ICERM held a special semester on Illustrating Mathematics, including computer visualization, 3d printing, and virtual reality, among other media, with an eye toward advancing research, as well as art, outreach and teaching. We will give a brief overview of the program and then present a few highlight projects. Mathematical topics will include algebraic number fields, Fuchsian groups, Thurston geometries and elliptic curves. Illustrating Mathematics
Jan. 28, 2020 3pm (MATH 350)	Algebraic Geometry Srimathy Srinivasan (University of Colorado) X We will discuss the notion of good reduction of reductive groups, it’s implications and what is known so far. I will also give an outline involving main ideas on the proof of a conjecture of Chernousov, Rapinchuk, Rapinchuk for good reduction of unitary groups of quaternionic skew-hermitian forms. Good reduction of unitary groups of quaternionic skew-hermitian forms
Jan. 28, 2020 4pm (MATH 350)	Topology Bob Gompf (University of Texas at Austin) X One of the most surprising discoveries in 4-manifold topology was the existence of smooth manifolds homeomorphic, but not diffeomorphic, to Euclidean 4-space. For fundamental reasons, this phenomenon can only occur in 4 dimensions. We will survey the subject, from its origin to recent developments regarding symmetries of such manifolds. This talk should be accessible to a broad audience. Exotic Smooth Structures on R^4
Jan. 29, 2020 3pm (Math350)	Math Phys Robin Deeley (CU Boulder) An Introduction to C*-Algebras
Jan. 29, 2020 4pm (MATH 350)	Grad Student Seminar Juan Moreno (CU Boulder) X The idea behind obstruction theory is that, given a fiber bundle, the existence of a section is determined by a particular cohomology class of the base space. This class is an obstruction in the sense that if it is zero in the cohomology ring, then a section exists. In this talk, we will consider the particular problem of finding linearly independent sections of vector bundles. This problem turns out to be equivalent to finding a single section of an associated bundle (Stiefel bundle). We will find that the corresponding obstruction to the existence of this section generally lies in a 'twisted' version of cohomology. Obstruction theory and local coefficients
Jan. 29, 2020 5pm (Math 350)	MathClub Elizabeth (Boo) Grulke (CU Boulder) X How students think of mathematics is related to their approach to learning and falls into one of two categories, fragmented or cohesive conceptions of math. This mixed-method study investigated using online discussions to encourage development of cohesive math conceptions in a face-to-face math course at a large metropolitan university. Students participated in either concept-connecting (n=71) or context-focused (n=79) online discussions. While students’ conceptions of mathematics in both groups became more cohesive, their surface approaches to learning also significantly increased. However, deep approaches to learning significantly increased only for the concept-connecting group. Women were found to enjoy and find the discussions more useful than men and women in concept-connecting discussions demonstrated higher exam scores and overall course grades. A Mixed-Method Study of Online Discussion in Mathematics

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