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Events for November 2022 – March 2023

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Nov. 01, 2022 11am (Math 350)	Number Theory James Rickards (CU Boulder) X An Apollonian circle packing is an arrangement of tangent circles in the plane. Packings where all curvatures are integral has been a rich topic of study in the last 20 years. Instead of fixing a packing and considering the possible curvatures in the packing, we will invert the problem: fix a curvature, and consider all packings that have a circle of this curvature. One natural way to study this set is to ask: "What is the distribution of the minimal curvatures of these packings?" This problem leads to a staircase-like distribution, which we name the "Apollonian Staircase." By exploring a connection to binary quadratic forms and hyperbolic geometry, we will describe and prove some properties of the staircase. The Apollonian Staircase
Nov. 01, 2022 1:25pm (Zoom)	Bernardo Rossi (University Linz, Austria) X Universal algebraic geometry, first introduced by B. Plotkin, extends some basic notions of classical algebraic geometry to universal algebra. For a clone $C$ on a set $A$ , a subset of $A^{n}$ is called algebraic if it is the solution set of a system of equations from the $n$ -ary part of $C$ . A basic fact in classical algebraic geometry is that the union of two algebraic sets is algebraic. This is no longer true in the present setting, and clones with this special property are called equationally additive. We prove that on a finite set with at least three elements there is a continuum number of equationally additive constantive clones. We characterize the Mal'cev algebras whose clone of polynomial functions is equationally additive in terms of properties of the binary term condition commutator, and equationally additive E-minimal algebras in terms of their TCT-type. Joint work with E. Aichinger and M. Behrisch. On when the union of two algebraic sets is algebraic
Nov. 01, 2022 2:30pm (MATH 3…	Lie Theory Manabu Hagiwara (Chiba University (Japan)) X This talk introduces the topics of coding theory and its nontrivial relationship with root systems and perfect codes, with examples. The talk will focus on introductory aspects. Mainly two topics will be discussed: one is the connection between the combinatorial structures in representation theory and deletion perfect codes. The other is the connection between the sphere-packing problem and bit-flipping perfect codes. Root Systems and Perfect Codes
Nov. 02, 2022 4:40pm (MATH 3…	Grad Student Seminar Matt Watson (CU Boulder) X In classic algebraic geometry, we want to study the zero sets of polynomials. In modern algebraic geometry, smart people develop algebraic tools and methods for studying the geometry of these zero sets. This talk is not about modern algebraic geometry. Instead, we will explore the world of complex donuts (compact Riemann surfaces) and their holomorphic and meromorphic functions. By the end of this talk, we will know a few basic results in this theory and see them in action on the Riemann sphere. Hopefully this helps to give some geometric intuition for anyone taking algebraic geometry next semester. If time permits, we may also state the Riemann-Roch Theorem in the complex curve setting and give a friendly introduction to sheaves. ???? The Geometric Intuition Behind Algebraic Geometry
Nov. 03, 2022 11:15am (Math …	Noncomm Geometry Emil Prodan (Yeshiva University) X I will show that each metamaterial has a groupoid behind it and that all physically sound Hamiltonians can be generated from the left-regular representation of the groupoid’s C-algebra. The K-theories of this groupoid algebra classify the various dynamical features that can be observed in the metamaterial. Furthermore, in the presence of defects, the space of units of the groupoid displays closed invariant subspaces leading to a decomposition of this unit space that prompts and exact sequence of C-algebras. The so called bulk-defect correspondence principle can be formulated in terms of this exact sequence and a specific Kasparov product. Furthermore, since exact sequences of C*-algebras are classified by certain Kasparov groups, we can use Kasparov’s theory to investigate and classify geometric defects of metamaterials. Examples will be provided. Metamaterials by groupoid methods and Kasparov’s K-theory
Nov. 03, 2022 3:35pm (Virtual)	Probability Jin Feng (University of Kansas) X The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated (2 scale) hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as mean field, we derive a macroscopic fluctuation structure associated with the hydrodynamic limit. The purpose of such derivation is to illustrate a new, Hamiltonian-based, functional analytic and variational approach for the understanding of hydrodynamics limit program in general. A large deviation result is established through an abstract Hamilton-Jacobi method applied to this specific setting. The principle idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is achieved through 1). A change of coordinate to the density-flux description of the problem; 2). Extending the weak KAM theory (in Hamiltonian dynamical system) to infinite particle context for explicitly identifying effective Hamiltonian; 3). Developing results for a new class of PDEs: Hamilton-Jacobi equation in the space of measures. This is a joint work with Toshio Mikami and Johannes Zimmer. Hydrodynamic limit fluctuations from nonlinear heat equation given by stochastic Carleman particles
CANCELED Nov. 08, 2022	Number Theory Ian Whitehead (Swarthmore) TBA
Nov. 08, 2022 1:25pm (Zoom)	Joshua Wiscons (California State University, Sacramento) X We discuss a geometry of involutions arising in "quasi-Frobenius" groups of finite Morley rank: groups possessing a quasi-self-normalizing subgroup H whose conjugates intersect trivially. The resulting geometry is that of a generically defined projective 3-space, and our focus is on whether it is genuine or not: a distinction that separates SO3(R) and PGL2(C). We conjecture natural conditions on quasi-Frobenius groups of finite Morley rank to identify PGL2(K), discuss progress on the conjecture (by us and, more recently, by Zamour), and highlight applications. This is joint work with Adrien Deloro. Geometry of involutions in ranked groups
Nov. 09, 2022 3:30pm (MATH350)	Math Phys Howy Jordan (CU Boulder) X Classical information theory answers fundamental questions about the storage and transmission of data. Suppose the goal is to transmit events that occur according to some specified probability distribution. How much information is inherent in the result of each event? How can the events be encoded (into bits, say) for efficient transmission of this information? Intuitively, a more likely outcome is less surprising and less informative, and so requires less in the way of storage. In this talk, we'll introduce the fundamental quantity of entropy (and its variations) which provide a foundation for precise answers to the above and related questions. We illustrate its use in the analysis of data encoding and transmission. We also set the stage for a comparison of classical information sources with quantum sources. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 An Introduction to Classical Information Theory
Nov. 09, 2022 4:40pm (MATH 3…	Grad Student Seminar Henry Maher (CU Boulder) X Category theory studies things and relationships between those things. However, in many settings, there are also relationships between those relationships; for example, there are homotopies between continuous maps between spaces, or natural transformations between functors between categories. This data is naturally described through the language of higher categories. In this talk, I aim to give a motivation of higher category theory by explaining its relationship to homotopy theory, as well as provide some examples of results from higher category theory. An introduction to higher category theory
Nov. 10, 2022 9am (MATH 350)	Algebraic Geometry Hannah Larson (Harvard / Clay) X There are certain natural cohomology classes on the moduli space M_{g,n}-bar of n-pointed, genus g stable curves known as "tautological classes." I will define these classes and explain the significance of the special situation when all classes are tautological. In joint work with Samir Canning, we find many new small values (g, n) for which all classes on M_{g,n}-bar are tautological. Via an inductive argument, these results lead to applications for arbitrary (g, n). For example, in joint work with Samir Canning and Samuel Payne, we show that the point counts of M_{g,n}-bar over finite fields are surprisingly well approximated by polynomials. Moduli spaces of stable curves
Nov. 10, 2022 2:30pm (MATH 3…	Functional Analysis Alex Kumjian (University of Nevada) X When is the canonical map from a k-graph into its fundamental groupoid an injection? We also discuss a class of examples, associated to actions of triangle groups on buildings, that led Robertson and Steger to introduce higher rank Cuntz-Pimsner algebras and thereby inspired the notion of higher rank graph (or k-graph) C*-algebras. The fundamental groupoid of a k-graph was introduced by Pask, Raeburn and Quigg in '04 and an example was given that shows that a k-graph need not embed. This is an interim report on continuing work with my colleagues Nathan Brownlowe, David Pask and Aidan Sims. Embeddability of k-graphs and a class of examples associated to triangle group actions on buildings. Sponsored by the Meyer Fund
Nov. 10, 2022 3:35pm (MATH 3…	Probability Ping Zhong (University of Wyoming) X Random variables in Voiculescu’s free probability theory can model the limits of suitable random matrix models. The Brown measure of a free random variable is a replacement for the eigenvalue counting measure of square matrices. The R-diagonal operators are a large family of free random variables that are natural limit operators of various well-known non-normal random matrix models. I will report some recent progresses on the Brown measure of the sum of two free random variables, one of which is R-diagonal. We show that subordination functions in free additive convolution can detect information about the Brown measure. These Brown measures are related to the limit eigenvalue distributions of deformed i.i.d. and deformed single ring random matrix models. Part of this talk is based on joint work with Hari Bercovici. Brown measure and eigenvalue distributions of deformed random matrices
Nov. 11, 2022 10am (MATH 350)	Functional Analysis Christopher Seaton (Rhodes College) X In symplectic geometry, the quotient of a symplectic manifold $M$ by a Lie group $G$ acting as symplectomorphisms does not usually inherit a symplectic structure. Rather, one forms the \emph{symplectic quotient}, the quotient by $G$ of a subset of $M$ defined as a fiber of the \emph{moment map} $μ : M \to 𝔤^{}$ where $𝔤$ is the Lie algebra of $G$ and $𝔤^{}$ its dual; this subset of $M$ is called the \emph{shell}. At a regular value of the moment map, the shell is a submanifold of $M$ and its quotient is a symplectic manifold. Otherwise, the shell and symplectic quotient are singular varieties. To consider the kinds of singularities that arise, one can restrict to the case when $M = V⊕ V$ for a finite-dimensional linear $G$ -representation $V$ . In this case, there is a natural homogeneous quadratic moment map $μ V⊕ Vto g$ such that $0$ is a singular value, hence the shell $N = μ-1(0)$ and symplectic quotient $N/!/ G$ are singular. For many cases of $G$ and $V$ , it has been demonstrated by Bellamy--Schedler, Becker, Terpereau, and others that $N/!/ G$ has \emph{symplectic singularities}; equivalently, $N/!/ G$ is Gorenstein with rational singularities and its smooth locus admits a holomorphic symplectic form. In some of these cases, the variety $N$ already has rational singularities. We will present recent results demonstrating that for a given $G$ , the shell $N$ has rational singularities and the symplectic quotient $N/!/G$ has symplectic singularities in the case of ``most" $G$ -modules $V$ . This includes the important case where $G$ is semisimple and $V = pg$ with $p ≥ 2$ , optimizing results of Aizenbud--Avni and generalizing a theorem of Budur. We will also discuss applications indicating that the representation varieties of surface groups have rational singularities and the corresponding character varieties have symplectic singularities. Joint work with Hans-Christian Herbig and Gerald Schwarz Rational singularities of the zero fiber of the moment map Sponsored by the Meyer Fund
Nov. 12, 2022 9am (Math350)	Math Phys Groupoidfest 2022 speakers X https://math.colorado.edu/groupoidfest2022/ Groupoidfest Sponsored by the Meyer Fund
Nov. 13, 2022 9am (Math350)	Math Phys Groupoidfest 2022 speakers X https://math.colorado.edu/groupoidfest2022/ Groupoidfest 2022 Sponsored by the Meyer Fund
Nov. 15, 2022 1:25pm (Zoom)	Jakub Oprsal (ISTA Klosterneuburg, Austria) X We will revisit Taylor's 1977 seminal paper `Varieties obeying homotopy laws' through the lens of recent developments in universal algebra and constraint satisfaction problems. The paper studies Maltsev conditions that influence homotopical properties of topological algebras in a variety satisfying such a condition. We recall that a topological algebra is a topological space together with operations that are continuous. Among other insights, we will provide a characterisation of idempotent varieties whose congruence lattices satisfy the meet-semidistributive law in terms of properties of topological algebras in the variety. Topological characterisation of varieties with meet-semidistributive congruences
Nov. 16, 2022 3:30pm (Math350)	Math Phys Daniel Spiegel (CU Boulder) X Following up on Howy Jordan’s talk, I’ll define the quantum mechanical von Neumann entropy and cover some of its basic properties. This talk will touch on Sections 11.3 and 11.4 of Nielsen and Chuang’s quantum information theory book. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Introduction to von Neumann entropy
Nov. 16, 2022 4:40pm (MATH 3…	Grad Student Seminar Jon Kim (CU Boulder) X This GSS talk will focus on introducing and motivating the basic tools of algebraic geometry in affine space (and maybe projective space) using commutative rings and fields. We will introduce rational normal curves in affine space and show how we can use these tools to study them. This talk is aimed at people who are curious about what algebraic geometry is about and only knowledge of algebra will be assumed. A Friendly Introduction to Algebraic Geometry with Rational Normal Curves
Nov. 17, 2022 9am (MATH 350)	Algebraic Geometry Jonathan Quartin (Boulder) X In recent years, there have been attempts to use Algebraic Geometry to model routing problems in space communications. In this talk, I’ll introduce sheaves on graphs, and demonstrate their ability to capture the local nature of routing problems. Time permitting, I’ll discuss some work from my NASA summer internship, and NASA’s plans for future research into the applicability of Algebraic Geometry to routing. Sheaf-Theoretic Models for Communications
Nov. 29, 2022 11am (Math 350)	Number Theory Collective X We'll use the regular Number Theory seminar this week to plan a graduate-student reading group in number theory for next semester. The goal is to organize a low-commitment, accessible reading group for graduate students at all career stages. Everyone is welcome to join, and we are in the market for suggested topics and sources! Reading group organizational meeting
Nov. 29, 2022 1:25pm (Zoom)	Reinhard Pöschel (TU Dresden, Germany) X It is well-known that a function $f$ (of arity $n≥ 2$ ) preserves an equivalence or a quasiorder relation $ρ$ if (and only if) each unary polynomial function preserves $ρ$ which can be obtained from $f$ by substituting constants at all but one arguments. Thus the polymorphism clone Pol $ρ$ is completely determined by the endomorphism monoid End $ρ$ . We ask which other relations and which monoids of unary mappings may have the same property. We introduce so-called generalized quasiorders which play a central role for answering these questions (moreover, they lead to new algebraic notions generalizing, e.g., affine completeness). Join work with Danica Jakubikova-Studenovska (Kosice, Slovakia) and Sandor Radeleczki (Miskolc, Hungary) Generalized quasiorders
Nov. 30, 2022 3:30pm (Math350)	Math Phys Rebecah Storms (CU Boulder) Einstein Field Equations
Nov. 30, 2022 4:40pm (MATH 3…	Grad Student Seminar Ezzeddine El Sai (CU Boulder) X In this talk, we explore how similar ideas appear in different areas of mathematics. One of those recurring themes is the notions of convolution and the Fourier transform which originated in the context of analysis. We will see how these ideas appear in computing, representation theory, algebraic geometry, and category theory. Since these ideas appear at different levels of abstraction, we will be able to examine the power and limitations of abstraction in mathematics. Fourier analysis without the analysis
Dec. 01, 2022 9am (MATH 350)	Algebraic Geometry Sebastian Bozlee X The moduli space M_{g,n} of smooth algebraic curves of genus g with n distinct marked points is not compact. However, it admits many compactifications that are themselves moduli spaces, and it remains an outstanding problem in algebraic geometry to classify these modular compactifications. An important family of examples is that of the moduli spaces of "weighted pointed" curves constructed by Hassett, in which a vector of real numbers determines which of the n marked points are permitted to come together. In this talk, I will present joint work with Vance Blankers that constructs modular compactifications of M_{g,n} using a simplicial complex rather than vector of weights as an input. Not only do the resulting "simplicial" moduli spaces generalize Hassett's, but they also classify the modular compactifications coming from colliding markings. On modular compactifications of M_g,n with colliding points
Dec. 01, 2022 2:30pm (MATH 3…	Functional Analysis Andrew Stocker (CU Boulder) X This talk will for the most part be an expository talk about the relationships between inverse semigroups, groupoids, and dynamical systems. Inverse semigroups are semigroups with the property that for every element x there is a unique element x* such that xxx = x and xxx* = x*. I will present a construction of a shift space, given a finite graph with edges labeled in an inverse semigroup. I will discuss how one can construct, from the labeled graph, a groupoid which contains the stable synchronizing algebra of the shift space as a sub-groupoid. Dynamical Systems from Inverse Semigroups
Dec. 01, 2022 3:35pm (Online)	Probability Jiaming Chen (ETH Zurich) X How do we simulate a singular stochastic diffusion with high efficiency? In the context of Schramm- Loewner evolution, we study the approximation of its traces via the Ninomiya-Victoir Splitting Scheme. We prove a strong convergence in probability with respect to the sup-norm to the distance between the SLE trace and the output of the Ninomiya-Victoir Splitting Scheme when applied in the context of the Loewner differential equation. Randomized algorithm to Schramm-Loewner evolution and its convergence
Dec. 06, 2022 1:25pm (Zoom)	Alex Kruckman (Wesleyan University) X Since the birth of modern model theory in the work of Morley and Shelah, notions of independence have been an important tool for studying complete first-order theories. Assumptions on the complexity of the theory often imply, and can in fact be characterized by, useful properties of these independence relations. For example, forking independence is a generalization of linear independence in vector spaces, which is defined in arbitrary theories; but a theory T is stable if and only if forking independence is stationary, and T is simple if and only if forking independence is symmetric. In this talk, I will describe a duality between certain notions of independence and certain ideals in the Boolean algebra of formulas, and I will show how constructions with ideals can be used to produce new notions of independence. As an application, I will prove a type amalgamation result that holds (surprisingly, at least to me) in arbitrary first-order theories. Ideallic independence relations
Dec. 08, 2022 2:30pm (MATH 3…	Functional Analysis Judith Packer (University of Colorado Boulder) X Let $𝕃$ be a length function on a discrete abelian group $Γ .$ Let $σ$ be a multiplier on $Γ$ (two cocycle taking values in $𝕋 .$ Rieffel and collaborators, following work of Connes, have used $𝕃$ to construct a ``Dirac" operator on $ℓ2(Γ),$ giving spectral triples with for the twisted group $C$ -algebra $C(Γ,σ),$ mostly when $Γ$ is finitely generated Following Rieffel's work and recent work of Long and Wu, we construct odd spectral triples for noncommutative solenoids $C( Z[1/p]× Z[1/p],σ)$ first studied by the author and Latremoliere. The Dirac operator involved can be used to construct a metric on the state space of the twisted group $C$ -algebra, which induces a topology coinciding with the weak- $*$ topology. This talk is based on joint work with C. Farsi, T. Landry and N. Larsen. Spectral triples for noncommutative solenoids
Dec. 13, 2022 1:25pm (Zoom)	Freese, McKenzie, McNulty, Taylor X Volume I of Algebras, Lattices, Varieties appeared in 1987. Volumes II and III have just now appeared. We will touch on what's in the them and what went into writing them, including some of the history of both the volumes and the field. Algebras, Lattices, Varieties Vol 1-N (ALVIN)
Jan. 19, 2023 9am (MATH 350)	Algebraic Geometry Fumiaki Suzuki (UCLA) X The Tate conjecture is a long-standing problem in arithmetic geometry, describing algebraic cycles on an algebraic variety in terms of Galois representation on étale cohomology. In contrast, an integral analogue of the Tate conjecture is known to fail in general, and a natural question is whether the failure is always caused by geometry. Over a finite field, we construct the first counterexamples to this question: in codimension 2 on our examples, a geometric cycle map is surjective but an arithmetic cycle map is not. We also show positive results toward a conjecture of Colliot-Thélène and Kahn on the third unramified cohomology group for threefolds over a finite field. This is a joint work with Federico Scavia. Non-algebraic geometrically trivial cohomology classes over finite fields Sponsored by the Simons Foundation
Jan. 25, 2023 3:30pm (Math350)	Math Phys Markus J. Pflaum (Department of Mathematics, CU Boulder) Classical variational calculus
Jan. 25, 2023 4:40pm (MATH 3…	Grad Student Seminar Dan Lyness (CU Boulder) X Galois Connections are something that arise in many different areas of Maths. We normally see them for the first time in Galois Theory, while studying the connection between field extensions and automorphism groups. In this talk I will consider this example, and some more examples of Galois Connections including the connection between prime ideals and varieties in Algebraic Geometry, and between kernels and cokernels in Category Theory. We will see how Galois Connections lead to a closure operator, that in some cases can be used to define a topology on our spaces. I intend to keep the talk accessible, and hopefully approach some familiar concepts from a new perspective. Galois Connections and the Zariski Topology
Jan. 31, 2023 1:25pm (Online)	Algebra/Logic Žaneta Semanišinová (Technische Universität Dresden) View Online:https://cuboulder.zoom.us/j/96896272260 X For a given relational structure $m a t h f r a k B$ , the constraint satisfaction problem CSP( $m a t h f r a k B$ ) asks whether a given finite input structure maps homomorphically to $m a t h f r a k B$ . We study the complexity of CSP( $m a t h f r a k B$ ), where the domain of $m a t h f r a k B$ is infinite. Our basic setting is that if we have a complexity classification for the CSPs of first-order expansions of a structure $m a t h f r a k B$ , then it can be transferred to obtain a classification of the CSPs of first-order expansions of another structure $m a t h f r a k C$ . We exploit the algebraic product of structures that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of finite algebraic powers of $m a t h b b Q; <)$ . By combining our classification result with classification transfer techniques, we answer several open questions in spatial reasoning from (Balbiani, Condotta, 2002), (Balbiani, Condotta, del Cerro, 1999) and (Balbiani, Condotta, del Cerro, 2002). This is joint work with Manuel Bodirsky, Peter Jonsson, Barnaby Martin and Antoine Mottet. Complexity Classification Transfer for CSPs via Algebraic Products
Feb. 01, 2023 3:30pm (Math350)	Math Phys Howy Jordan (CU Boulder) X Infinite dimensional manifolds arise naturally even from considerations of finite dimensional spaces. But in general, infinite dimensional topological vector spaces lose many of the nice properties from the finite case that we casually take for granted in differential geometry. In this talk, we consider the nicer cases of Banach and Hilbert spaces and manifolds modeled on these. The basic building blocks of differential geometry (like charts, smooth structures, and tangent vectors) can be easily defined, and we will observe some differences and similarities between the finite theories of submanifolds and vector bundles and their respective general Banach/Hilbert counterparts. Banach and Hilbert Manifolds
Feb. 02, 2023 5pm (MATH 350)	Grad Student Seminar Chase Meadors (CU Boulder) X In this talk, we will start by finding out where the arrows are in algebra so we can turn them all around, yielding a subject called coalgebra. Coalgebra turns out to model various forms of automata and dynamical systems. A "coequation", then, is something that controls, specifies, or constrains the behavior of such systems. We will explore the analogy of "equational logic" for algebras with "modal logic" for coalgebras. Finally, we will proceed to co-solve co-systems of co-linear co-equations and end by learning the co-quadratic formula. Coalgebra 1
Feb. 07, 2023 3:30pm (MATH 3…	Topology Organizational Meeting Organizational Meeting
Feb. 08, 2023 3:30pm (Math350)	Math Phys Rebecah Storms (CU Boulder) Jets and jet bundles
Feb. 08, 2023 4:40pm (MATH 3…	Grad Student Seminar Alex LaJeunesse (CU Boulder) X In this talk we’ll be looking at how symmetric monoidal categories encode the many instances of monoid-like structures that appear across mathematics, and exploring different ways of universally completing them to abelian groups (in the case of honest monoids) or grouplike spaces (in the case of categories). In particular we will focus on two main examples of symmetric monoidal categories: finite sets with disjoint union, and finitely-generated projective modules over a ring with direct sum. By the end, we will see how group-completing these two categories realizes a natural relationship between stable homotopy groups and algebraic K-theory. Group Completion of Monoids and Monoidal Categories
Feb. 09, 2023 4pm (MATH 220)	Init Conditions Adrian Neff Naturality and Equivalence of Categories
Feb. 14, 2023 12:20pm (MATH …	Kempner Mandi Schaeffer Fry (Metropolitan State University of Denver) X Widely known as the founder of modular representation theory, Richard Brauer set the stage for the study of the so-called local-global conjectures in the character theory of finite groups. These are conjectures that relate the representations and structure of a finite group to that of certain proper (“local”) subgroups. In this talk, I'll discuss some of these conjectures and their implications for other character-theoretic properties and lingering questions of Brauer. Character Tables and Brauer’s Legacy
Feb. 14, 2023 1:25pm (Online)	Algebra/Logic Will Brian (University of North Carolina at Charlotte) View Online:https://cuboulder.zoom.us/j/96896272260 X I will sketch a proof that, assuming 0^† does not exist, if there is a partition of R into ℵ_ω Borel sets, then there is also a partition of R into ℵ_ω+1 Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of ℵ_ω.) This contrasts starkly with the situation for cardinals with uncountable cofinality and their successors, where the spectrum of possible sizes of partitions of R into Borel sets can be made completely arbitrary via forcing. Partitions of the real line into Borel sets
Feb. 14, 2023 3:30pm (MATH 3…	Topology Agnes Beaudry, Robin Deeley and Markus Pflaum X In three 20 minute talks, each of us will introduce our research and what it is like to work in our research group. An Introduction to the CU Boulder Topology Group
Feb. 15, 2023 3:30pm (Math350)	Math Phys Connor Mccranie (CU Boulder) Locally convex topological vector spaces
Feb. 15, 2023 4:40pm (MATH 3…	Grad Student Seminar Evan Young (CU Boulder) X Fourier transforms. Laplace transforms. Mellin transforms. Discrete Fourier transforms. Hadamard transforms. Oh my! In this presentation, we will discuss how all of the above examples of integral transforms are actually particular instances of a very general association of commutative Banach algebras known as the Gelfand representation. This talk will serve as an introduction to the use of algebraic techniques in analysis and conversely, to the use of analytic techniques in algebra. The One Transform to Rule Them All
Feb. 21, 2023 12:10pm (MATH …	Kempner Klaus Hulek (Leibniz University Hannover) X The classification of mathematical objects is an important problem in many branches of mathematics, notably also in algebraic geometry. Typically, these classification cannot be achieved by (finite) lists. Instead one builds a new algebraic variety whose points correspond to then isomorphism class of the objects to be classified — a moduli space. This leads to the questions: how does one construct such classifying spaces and what can one say about the geometry of these moduli spaces? I will discuss these question in exemplary cases (old and new). Classification in algebraic geometry and the geometry of moduli spaces
Feb. 21, 2023 3:30pm (MATH 3…	Topology Daniel Spiegel (CU Boulder) X The study of parametrized many-body gapped Hamiltonians and their topological phases represents a fast-growing frontier at the intersection of mathematical physics and condensed matter physics. In the limit of infinite system size, observable quantities are represented by elements of a uniformly hyperfinite C-algebra A and the object of study is a continuous function into the space P(A) of pure states of A. With a continuous family of pure states, questions arise of how to do classical C-algebraic maneuvers like the GNS construction and Kadison transitivity theorem in a way that depends continuously on the input data. I will present new theorems pertaining to both the norm and weak* topologies on P(A) to address these questions and others that have arisen in the study of parametrized many-body quantum systems. Topological Results on Pure State Space Inspired by Parametrized Quantum Phases
Feb. 21, 2023 5pm (Online)	Algebra/Logic Zarathustra Brady (MIT) View Online:https://cuboulder.zoom.us/j/96896272260 X If you have a non-invertible unary operation $f$ on a finite set, a natural thing to do with it is to create a compositional power $f^{n}$ which satisfies $fn(fn(x)) approx fn(x)$ . Using this compositional power of $f$ , you can produce a simpler clone on a smaller set which satisfies every height-one identity which is satisfied by the original clone. By repeating this process, we can eventually reduce questions about height-one identities in finite algebras to the case of idempotent algebras. It turns out that one can do something similar with two-variable operations: starting from any idempotent binary operation $t$ on a finite set, there is a (nontrivial) functorial construction of a two-variable operation $s$ in the clone generated by $t$ which satisfies the identities $s(x,s(x,y)) approx s(s(x,y),x) approx s(x,y)$ . Such operations were used extensively by Andrei Bulatov in his proof of the CSP Dichotomy Conjecture, and I call them ``partial semilattice operations''. I will describe this functorial construction and show how we can make use of such operations to simplify the behavior of binary absorption in finite algebras while preserving many types of height-one identities. Simplifying clones with partial semilattice operations
Feb. 22, 2023 3:30pm (Math350)	Math Phys Calum Shearer (CU Boulder) Differentiability notions in infinite dimensional vector spaces
Feb. 23, 2023 9am (MATH 350)	Algebraic Geometry Klaus Hulek (Leibniz University Hannover) X Although K3 surfaces and Enriques surfaces are closely related, their moduli spaces behave very differently. As the degree increases, one one obtains an infinite series of different moduli spaces of polarized K3 surfaces., In contrast, there are only finitely many different moduli spaces of (numerically) polarized Enriques surfaces. In fact there are exactly 87 orthogonal modular varieties which appear in context with moduli of polarized Enriques surfaces. Together with Dutour-Sikiric we classified the corresponding arithmetic groups. I will discuss this classification and related questions. Moduli of polarized Enriques surfaces Sponsored by the Simons Foundation
Feb. 23, 2023 2:30pm (MATH 3…	Functional Analysis Angel Roman (University of Washington at St. Louis) X In the 1970's, George Mackey proposed an analogy between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group, known as the Cartan motion group. Recently, Nigel Higson and Alexandre Afgoustidis made this analogy precise between equivalence classes of irreducible tempered representations of a reductive groups and equivalence classes of irreducible unitary representations of the associated Cartan group. First, I will introduce tempered unitary representations of a reductive Lie group to develop the Mackey analogy. The goal of this talk is to show that the reduced group C-algebra of the Cartan motion group can be embedded into the reduced group C-algebra of the reductive group. To accomplish this, I will show the construction of a continuous fields of C-algebras generated from a Lie groupoid known as deformation to the normal cone. I will also discuss the Fourier transform picture. C-Algebraic Mackey Analogy of Reductive Groups
Feb. 23, 2023 4pm (MATH 220)	Init Conditions Bob Kuo Examples & the art of the diagram chase
Feb. 28, 2023 1:25pm (Online)	Algebra/Logic Albert Vucaj (Technical University Vienna, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. Over the years, it has been shown that such a goal seems arduously reachable even if we only focus on clones over three-element sets. In a recent turn of events, the minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form $f(x1,dots,xn)approx g(y1,dots,ym)$ , also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure $m a t h b b A$ only depends on the set of minor-identities satisfied by the polymorphism clone of $m a t h b b A$ . In this talk, I will present a general overview of the poset arising by considering clones over some finite set with the following order: we write $m a t h c a l C l e q m a t h c a l D$ if there exist a minor-preserving map from $m a t h c a l C$ to $m a t h c a l D$ . In particular, I will focus on rather natural questions such as: is this poset a lattice? Is it countable? Does it have atoms or coatoms? Joint work with Manuel Bodirsky and Dmitriy Zhuk. Clones over finite sets up to minor-equivalence
Feb. 28, 2023 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X A k-root is an easily defined type of polynomial that can be expressed as an integer linear combination of squarefree monomials of degree k in n indeterminates. The case k=1 corresponds to a root system of type D. As is the case for root systems, k-roots can be either positive or negative, and there is a basis for their span analogous to a set of simple roots. This talk will describe some interesting constructions that can be done by summing all the positive k-roots in certain judiciously chosen subsets. A generalization of the positive roots of type D_n
Feb. 28, 2023 3:30pm (MATH 3…	Topology Juan Moreno (CU Boulder) Duality for Morava E-Theory
Mar. 01, 2023 3:30pm (Zoom)	Math Phys John Baez (University of California at Riverside) View Online:https://cuboulder.zoom.us/j/94002553301 X Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 The importance of the tenfold way in physics was only recognized in this century. Simply put, it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when the topologist C. T. C. Wall classified the associative real super division algebras and found ten of them. The three 'purely even' examples were already familiar: the real numbers, complex numbers and quaternions. The rest become important when we classify representations of groups or supergroups on Z/2-graded vector spaces. We explain this classification, its connection to Clifford algebras, and some of its implications. The Tenfold Way
Mar. 02, 2023 2:30pm (MATH 3…	Functional Analysis Robin Deeley (CU Boulder) X Given a map $g : Y \to Y$ that is continuous and onto, one can construct a solenoid, which is the stationary inverse limit associated to g. This process leads to a space X and a homeomorphism $v a r p h i$ . The dynamics of g and $v a r p h i$ are very much related. I will discuss various examples of this process and the C-algebras associated with solenoids. In particular, we will see examples where $Y$ is non-Hausdorff, but the associated solenoid $X$ is Hausdorff. Solenoids and their C-algebras
Mar. 02, 2023 3:35pm (MATH 3…	Probability Manuel Lladser (CU Boulder) X Resolvability is a generalization of the notion of trilateration of the plane to an arbitrary metric space. Given a finite metric space $X,d)$ , a subset $R=r1,…,rk⊂ X$ is said to resolve $X,d)$ when the transformation $x→(d(x,r1),…,d(x,rk))$ , with $x∈ X$ , is one-to-one. In particular, this transformation can be used to represent points in $X$ as Euclidean vectors and, due to the triangular inequality, points close by in $X$ are represented as vectors with similar coordinates in $ℝ^{k}$ . Resolving sets can therefore produce feature vectors for points in abstract metric spaces, which may find applications in classification problems of symbolic objects under suitably chosen metrics. Unfortunately, the problem of resolving optimally large metric spaces (i.e., minimizing the cardinality of $R$ ) is NP-complete. In fact, the state of the art algorithm, the Information Content Heuristic (ICH), has complexity $O(\|X\|3)$ and is only guaranteed to find a resolving set of cardinality at most $1+(1+o(1))⋅\|X\|$ times the optimal one. In this talk, I will present work in progress---which uses probabilistic arguments---regarding the multilateration of Jaccard spaces, i.e., metric spaces of the form $2X,d)$ , where $2X$ is the power set of a finite set $X$ , and $d$ is the Jaccard distance. Namely, for all $A,B⊂ X$ , $d(A,B):=\|AΔ B\|/\|A⊃ B\|$ , where $\|⋅\|$ denotes cardinality and $Δ$ the symmetric difference of sets. This work is partially funded by the NSF grant No. 1836914. Resolvability of Jaccard Metric Spaces
Mar. 02, 2023 4pm (MATH 220)	Init Conditions Stephanie Oh Representable functors and the Yoneda Lemma
Mar. 07, 2023 1:25pm (Online)	Algebra/Logic Michael Kinyon (University of Denver) View Online:https://cuboulder.zoom.us/j/96896272260 X The conjugacy equivalence relation in groups can be generalized to semigroups in various distinct ways. For example, in an inverse monoid $S$ , one might define $a,b∈ S$ to be conjugate if there exists $g∈ S$ such that $g-1ag = b$ and $gbg-1 = a$ . There are other semigroup generalizations of conjugacy which do not involve inverses at all. In this talk I will survey the various generalizations, focusing particularly on a notion known as \emph{natural conjugacy}, which, in inverse semigroups, coincides with the one defined above. Along the way, I will discuss connections with partial inner automorphisms and with the (universal algebra) center of an inverse semigroup. The work described herein is joint with J. Ara\'{u}jo, W. Bentz, J. Konieczny, A. Malheiro, V. Mercier and D. Stanovsk\'{y}. Conjugacy in semigroups
Mar. 07, 2023 2:30pm (MATH 3…	Lie Theory Nat Thiem (CU) X Symmetric functions appear throughout much of Stanley’s school of algebraic combinatorics. Algebraically, it is a combinatorial Hopf algebra indexed by integer partitions, and it is in fact a terminal object in the category of co-commutative combinatorial Hopf algebras. This talk explores a number of generalizations or variations on symmetric functions and how algebraic methods inform or confuse how these combinatorial Hopf algebras relate. This is ongoing work with F. Aliniaeifard. The problems with canonical maps
Mar. 07, 2023 3:30pm (MATH 3…	Topology Sheagan John (CU Boulder) X For f and g continuous and commuting self-maps of a Hausdorff space X, we investigate various conditions on X and on the pair (f , g) which provide existence of a coincidence value. We introduce generalized notions of the coincidence value property and use this added flexibility to determine how various coincidence properties of X are related to coincidence properties of associated fibre bundles and adjunction spaces. These investigations are motivated by the longstanding open question concerning whether Brouwer's classical fixed-point theorem has a coincidence value analog. "Lifts and Extensions" of Generalized Coincidence Value Properties
Mar. 08, 2023 3:30pm (Math350)	Math Phys Ezzedine El Sai (CU Mathematics) X PDEs are very familiar objects and have been studied for centuries. This is especially because they describe classical mechanical systems in physics. But in order to study the transition from classical to quantum mechanics, it's imperative to generalize our notion of what a PDE really is. In this talk, we will discuss some of these general notions, particularly Jets, D-modules, and connections with homological algebra. The geometry (and algebra) of PDEs
Mar. 09, 2023 2:30pm (MATH 3…	Functional Analysis Levi Lorenzo (CU Boulder) X K-theory, and its dual K-homology, are the generalized homology and cohomology theories on the category of C-algebras that undergird noncommutative algebraic topology. Further, the two functors admit a bilinear pairing to the integers, the Index Pairing. For a smooth manifold, M, the K-homology of C_0(M) has the additional property of being uniformly p-summable on the subalgebra C_0^\infty (M), for each p larger than the dimension of M. This is desirable because finitely summable (p < \infty) K-homology cycles allow one to compute the Index Pairing in a more straightforward formula, via cyclic cohomology. Given a C-algebra, we can ask whether it has a dense subalgebra on which all of its K-homology classes have p-summable representatives for some p < infinity. In general the answer is no, but there are few known counterexamples and not so many positive results either. In this talk, we will define K-homology, discuss what it means for K-homology to be finitely summable, argue why we should care, and discuss what we know about the answer to this question and directions of where we might find further answers. Finitely Summable K-homology
Mar. 09, 2023 3:35pm (Online)	Probability Ewain Gwynne (University of Chicago) View Online:https://cuboulder.zoom.us/j/96074949744 X A permuton is a probability measure on $0,1]2$ whose two coordinate marginals are Lebesgue measure. Permutons describe the large-scale geometry of random permutations. I will discuss a geometric construction of a certain class of random permutons using a pair of random space-filling curves called Schramm-Loewner evolution (SLE) and a random measure arising from Liouville quantum gravity (LQG). This class includes the limits of various types of random pattern-avoiding permutations as well as the conjectural limit of meandric permutations (permutations arising from a simple loop which crosses a line a specified number of times). I will then discuss some results about random permutations which can be proven using SLE and LQG, concerning, e.g., the length of the longest increasing subsequence and the fractal dimension of the support. I will not assume any prior knowledge about permutons, SLE, or LQG. Based on joint work with Jacopo Borga and Xin Sun. Permutons, meanders, and SLE-decorated Liouville quantum gravity
Mar. 09, 2023 4pm (MATH 220)	Init Conditions Henry Maher Universal Properties
Mar. 14, 2023 12:10pm (MATH …	Kempner David Renfrew (Binghamton University) X I will describe the limiting behavior of the eigenvalues of minors of large biunitarily random matrices and the roots of derivatives of polynomials with independent, random coefficients, by giving a convolution semi-group which relates the two processes together. Eigenvalues of minors of random matrices and roots of derivatives of random polynomials
Mar. 14, 2023 1:25pm (Online)	Algebra/Logic Dmitriy Zhuk (Charles University, Prague, Czechia) View Online:https://cuboulder.zoom.us/j/96896272260 X In 2017 Moiseev found 2 079 040 clones on 3 elements definable by binary relations using a computer. It is clear that only computer can deal with so many clones but another result from 2019 showed that even computers have their limitations. Moore proved that they cannot check whether a clone given by generating operations is finitely related (definable by a relation). These results made us believe that even a computer description of all the clones on 3 elements would never appear. However, not all clones are essentially different, and one might try to characterise clones modulo clone homomorphisms or minor preserving maps (only h1-identities are preserved). In 2022 we showed that there are only countably many clones of self-dual operations on 3-elements modulo a minor preserving map. Then using a computer we collapsed 2 079 040 into a very small number and now we hope that a complete characterization of all clones on 3 elements modulo minor preserving maps is possible. Joint work with Libor Barto, Jan Adam Zah\'alka, Albert Vucaj, and Florian Starke. Clones on 3 Elements: A New Hope
Mar. 14, 2023 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X When first encountering manifolds, there are basic examples of non-manifolds that should be met. In particular, simple constructions like the wedge product of two manifolds immediately give spaces with points which fail to have appropriate neighborhoods. These spaces fail to have nice properties of manifolds such as Poincare duality. But these spaces are mostly manifolds after all. Much of the theory of manifolds can be recovered by observing how the space is naturally decomposable into manifolds and respecting this decompostion. This is the starting point for the theory of stratified spaces, spaces equipped with decompositions into manifolds. We will see that the tools familiar from the theory of manifolds, like tangent bundles and vector fields, allow one to refine the theory of stratified spaces further. Thom's first isotopy lemma for instance gives sufficient conditions for when a stratified space is locally a product of a manifold and a lower dimensional stratified space. If time permits, we will also consider the basics of the theories of intersection homology and perverse sheaves on stratified spaces, which allow even Poincare duality to be recovered. Stratified Spaces
Mar. 15, 2023 3:30pm (Math350)	Math Phys Calum Shearer (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X Lie groups arise in mathematics and physics as groups of continuous symmetries. The theory of finite-dimensional Lie groups and their representations has a deep and rich theory, and plays an important role in modern physics, for example in quantum mechanics and in gauge theories. However, sometimes a group of symmetries depends on too many parameters to be modelled on a finite-dimensional space! Such spaces are also objects of interest: for example, loop spaces, diffeomorphism groups and groups of gauge transformations. In this talk, we introduce the notion of infinite-dimensional (locally convex) Lie groups, as treated by John Milnor and discuss what familiar features carry over from the finite-dimensional setting. In finite dimensions, there are many powerful tools for obtaining local or global information about a Lie group from its Lie algebra, for example via the exponential map. We discuss the limitations of this approach in infinite dimensions, and describe what partial results can still be obtained in this setting Infinite dimensional Lie groups a la Milnor
Mar. 16, 2023 2:30pm (MATH 3…	Functional Analysis Rachel Chaiser (CU Boulder) X Matui's HK-conjecture proposes an in-principal computation of the K-theory of the reduced C-algebra of a (nice enough) groupoid in terms of the homology of the groupoid. This can be viewed as a C-algebraic analogue of a technique in algebraic topology where one can compute the K-theory of a CW-complex of dimension at most 3 explicitly from its cohomology using the Atiyah--Hirzebruch spectral sequence. However, this technique may fail for spaces of dimension 4 and greater. Given a flat manifold and an expanding self-cover, one can construct a groupoid where the invariants in the HK-conjecture are computed from the corresponding invariants of the manifold. Using this construction, a flat manifold for which the Atiyah--Hirzebruch spectral sequence fails to compute K-theory from cohomology ought to give a counterexample to the HK-conjecture. This method was introduced by Deeley to construct a counterexample given a flat manifold of dimension at least 9. This naturally leads to the question: is there a flat manifold of dimension 4 giving a counterexample? Flat manifolds and the HK-conjecture
Mar. 16, 2023 3:35pm (Online)	Probability Sung-Soo Byun (Korea Institute for Advanced Study) View Online:https://cuboulder.zoom.us/j/92389457191 X In this talk, I will discuss real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the maximally non-Hermitian regime, which corresponds to the matrix model with real i.i.d. Gaussian entries, it was pioneered by Edelman, Kostlan, and Shub that the number of real eigenvalues is of order $\sqrt{N}$ , where $N$ is the size of the matrix. Moreover, it can be heuristically conjectured that as a real random matrix becomes more symmetric, it gets more real eigenvalues. I will demonstrate that such a statement can be made rigorous by presenting the large- $N$ expansion of the mean and variance of the number of real eigenvalues in the almost-Hermitian regime, where one can observe a non-trivial transition between real i.i.d. and real symmetric random matrices. Furthermore, I will explain the limiting empirical distributions of the real eigenvalues which interpolate the Wigner semicircle law and the uniform distribution. Real eigenvalues of elliptic random matrices
Mar. 16, 2023 4pm (MATH 220)	Init Conditions Jon Kim Limits and Colimits
Mar. 17, 2023 3:30pm (MATH 3…	Grad Student Seminar Chase Meadors (CU Boulder) X The pantheon of "controversial" propositions only logicians and set theorists care about includes things such as the Law of Excluded Middle, the Axiom of Choice, and the Continuum Hypothesis (CH). It makes sense to take any of these propositions as true or false, and either decision leads to very interesting phenomena in the set-theoretic universe. In this talk, we will explore consequences of CH and ¬CH in the format of a game show: For each wacky proposition, participants can vote on whether it implies or is implied by the continuum hypothesis or its negation. No serious set theory background is required to participate, and some very basic background will be given in the talk. CH or ¬CH? That is the question (among others)
Mar. 21, 2023 12:20pm (Math …	Diversity Teresa Wroe (Office of Institutional Equity and Compliance) X As classroom leaders it’s critical to recognize that unconscious bias can show up in lots of ways and is multi-directional when we are teaching, grading, evaluating, and providing feedback. This session will be an opportunity to examine bias and discuss strategies that can help mitigate its effects in the classroom and in our everyday interactions with students. Mitigating Unconscious Bias in the Classroom
Mar. 21, 2023 1:25pm (Online)	Algebra/Logic Pawel Idziak (Jagiellonian University, Krakow, Poland) View Online:https://cuboulder.zoom.us/j/96896272260 X The talk reports how the project of characterizing finite algebras with tractable equation solving problem led us to the concepts of circuits and non-uniform automata over arbitrary algebras. As a byproduct we get (under some complexity hypothesis, like Exponential Time Hypothesis and Constant Degree Hypothesis) a full characterization of finite algebras from congruence modular varieties with tractable checking if two circuits compute the same function. A similar (but not the same one) characterization is also provided for finite groups with respect to checking if two polynomials define the same function. Circuits over finite algebras
Mar. 22, 2023 4:40pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X The Riemann mapping theorem states that any two simple connected proper domains in the complex plane are conformally isomorphic, i.e. there exists an angle preserving bijection between them. In general, there is no explicit description of the conformal isomorphism in terms of the conformal domains. Despite this, Loewner developed a way to connect the dynamics of a family continually evolving simply connected domains to the dynamics of their conformal isomorphisms to the the upper half plane. We will do a brief review/introduction on conformal maps before diving deeper into Loewner evolution. Time permitting, we will discuss Schramm's groundbreaking idea connecting Loewner evolution to statistical mechanics. Conformal maps, the Riemann mapping theorem, and (Schramm--)Loewner evolution
Mar. 23, 2023 9:05am (Math 3…	Algebraic Geometry Jarod Alper (University of Washington) X We will survey how recent advances in moduli theory over the last five years have provided a new technique to construct projective moduli spaces. Advances in moduli theory Sponsored by the Simons Foundation

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