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Events for March – July 2023

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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
Apr. 04, 2023 1:25pm (MATH 2…	Algebra/Logic Wentao Yang (Carnegie Mellon University) View Online:https://cuboulder.zoom.us/j/96896272260 X We propose an analogue definition of No Independence Property (NIP) for abstract elementary classes (AECs) that coincides with NIP when the class is elementary. We construct a forking-like relation on AECs with NIP, and show that its negation leads to being able to encode subsets. An NIP-like notion in abstract elementary classes
Apr. 04, 2023 2:30pm (Math 3…	Lie Theory Aparna Upadhyay (University of Arizona) X The core of a finite dimensional kG-module M is obtained by deleting all the summands of M that are projective. Benson and Symonds defined an invariant for M, called the gamma invariant. This invariant is a result of studying the asymptotics of the core of tensor powers of M. In this talk, we will see some interesting properties of this invariant. We will explore the sequence of dimensions of the core of $M⊗ n$ , and see that it has algebraic Hilbert series when M is Omega-algebraic. When M is $Ω +$ -algebraic then these dimension sequences are eventually linearly recursive. This partially answers a conjecture by Benson and Symonds. The asymptotics of the decomposition of tensor powers of modules Sponsored by the Meyer Fund
Apr. 04, 2023 3:30pm (MATH 3…	Topology Emily Montelius (CU Boulder) Segal's Gamma-Spaces and the Barratt-Priddy-Quillen Theorem
Apr. 05, 2023 3:30pm (Math350)	Math Phys Rebecah Storms (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 Profinite dimensional manifolds
Apr. 06, 2023 9:05am (RAMY N…	Thesis Defenses Michael Wheeler (CU) View Online:https://cuboulder.zoom.us/j/96126023746 X We use the methods of model-theoretic nonstandard analysis, in particular enlargements, to study the properties of regular and good ultrafilters and their role within Keisler's order on countable complete first-order theories. This understanding is used to produce alternative proofs of several key theorems in the study of Keisler's order, such as the well-definedness of Keisler's order and the maximality of theories with SOP_2 (Strong Order Property 2). Furthermore, we provide an analysis of the logical structure of the types used in the proof that theories with SOP_2 are Keisler maximal and use this analysis to give an easy-to-state set-theoretic characterization of ultrafilters that are both regular and good. A Nonstandard Approach to Keisler's Order
Apr. 06, 2023 3:35pm (MATH 3…	Probability Hoi Nguyen (Ohio State University) X In this talk we discuss the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we will show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are different. We will also discuss partial progress toward the case of non-gaussian coefficients. Central Limit Theorem for the number of real roots of gaussian random polynomials
Apr. 06, 2023 4pm (MATH 220)	Init Conditions Colin Jackson Limits and colimits in set
Apr. 11, 2023 10am (MATH 350)	Geometry/Analysis Thomas Ivey (College of Charleston) X In a 2009 paper, Anderson, Fels and Vassiliou showed that, for a class of Darboux-integrable (DI) hyperbolic systems of PDE, a canonical integrable extension exists and is constructed using the action of a novel invariant called the Vessiot group; moreover, the extension splits as the product of two simpler differential systems both of Lie type. Each solution of the DI system thus arises as a `superposition' of a pair of solutions to the simpler systems. In this preliminary report on joint work with Mark Fels, we outline a construction of a canonical integrable extension for elliptic DI systems. By contrast, in this case the extension doesn't split but is a holomorphic Lie equation involving the complexified Vessiot group. In several examples the extension is contact-equivalent to a prolongation of the Cauchy-Riemann equations, leading to solution formulas for the PDE in terms of holomorphic data. Geometry of Elliptic Darboux-Integrable Systems Sponsored by the Meyer Fund
Apr. 11, 2023 12:20pm (MATH …	Kempner Andrew Linshaw (Denver University) X W-algebras are an important class of vertex algebras associated to a Lie (super)algebra g and an even nilpotent element in g. Trialities of W-algebras are isomorphisms between the affine cosets of three different W-algebras. A large family of such isomorphisms were conjectured in physics by Gaiotto and Rapcak in 2017, and were recently proven in my joint work with Thomas Creutzig. This result vastly generalizes both Feigin-Frenkel duality and the coset realization of principal W-algebras of classical types. In this talk, I will give an overview of vertex algebras and discuss the Gaiotto-Rapcak trialities. Time permitting, I will describe some possible generalizations of this result to a larger class of W-algebras. This is a joint work with Thomas Creutzig (University of Alberta) and Vladimir Kovalchuk (University of Denver). Trialities of W-algebras
Apr. 11, 2023 1:25pm (Online)	Algebra/Logic Guram Bezhanishvili (New Mexico State University) View Online:https://cuboulder.zoom.us/j/96896272260 X I’ll survey our current knowledge of the lattice of varieties of Heyting algebras. Towards the end of the talk, I’ll discuss some recent developments, including some open problems that continue to fight back. Varieties of Heyting algebras: what we (still don’t) know
Apr. 11, 2023 3:30pm (MATH 3…	Topology Sarah Petersen (CU Boulder) X In this largely expository talk, we will overview homotopical and geometric motivations for constructing Brown-Gitler spectra, as well as describe a number of different constructions of these spectra due to a range of authors. We will discuss how Brown-Gitler spectra are used in homotopy computations and indicate areas of current research interest. Come to this talk to find out what abelian groups ordinary homology spits out when it eats Brown-Gitler spectra! Brown-Gitler Spectra: An introduction & current directions
Apr. 12, 2023 3:30pm (Math350)	Math Phys Connor Mccranie (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 Holomorphic functions in infinite dimensional vector spaces
Apr. 12, 2023 4:40pm (MATH 3…	Grad Student Seminar Calum Shearer (CU Boulder) X Sometimes a topological space looks locally like some kind of product space, but globally there may be some kind of 'twist' which prevents us from viewing it globally as a product. Such spaces are called fiber bundles, and arise frequently in topology, geometry and algebraic geometry (or so I've heard). In this talk I will introduce fiber bundles in a fairly informal way, explaining what kinds of structure and features such spaces have, how we generalize graphs of functions to these spaces, and what sorts of questions are natural to ask about them. I will focus on the two examples of the cylinder and the Mobius strip, and explain ways in which we can show that these two spaces are different. I have even constructed some fiber bundles out of paper to use as props, which is the closest thing I've done in a while to anything practical as a math student! A friendly introduction to fiber bundles
Apr. 13, 2023 2:30pm (MATH 3…	Functional Analysis Matt Lorentz (Michigan State University) X Many times in analysis we focus on the “small scale” structure of a metric space, e.g. continuity, derivations, etc. However, to examine the “large scale” structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X,dX) is coarsely equivalent to (Y, dY ) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely bornological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules. The Hochschild Cohomology of Roe Type Algebras
Apr. 13, 2023 3:35pm (MATH 3…	Probability Tom Trogdon (University of Washington) X Many iterative algorithms from numerical linear algebra (Krylov subspace methods, in particular) connect directly to polynomials orthogonal with respect to a measure generated by a matrix and a vector, the so-called eigenvector empirical spectral distribution (VESD). For many random matrix distributions, the limiting VESD is known (or at least characterized) yet the ill-conditioned nature of the mapping from measure to orthogonal polynomials appears to limit the conclusions one can make. To partially overcome this difficulty, we develop a new Riemann--Hilbert-based perturbation theory for orthogonal polynomials that takes the local law as input. This allows one to make precise statements about the random polynomials as their degree diverges, and therefore, about algorithms applied to random matrices as the number of iterations diverges. This is joint work with Percy Deift (NYU), Xiucai Ding (UC Davis), and Elliot Paquette (McGill). Local laws for random matrices, random orthogonal polynomials and algorithms
Apr. 18, 2023 11am (Online)	Algebra/Logic Peter Jipsen (Chapman University) View Online:https://cuboulder.zoom.us/j/96896272260 X In this talk I will report on some joint research in the past year with J.B. Nation and Ralph Freese on ortholattices and with Melissa Sugimoto, José Gil-Férez and Sid Lodhia on involutive partially ordered monoids. Rather than just present the results I will attempt to describe how the automated theorem prover Prover9/Mace4 was used in some aspects of this research. For ortholattices we prove that the variety $ℋ$ generated by the 6-element hexagonal ortholattice has an equational basis given by any equational basis of the pentagon lattice variety. We find a list of 9 finite ortholattices that generate varieties covering $ℋ$ and present partial results towards showing that this list of finitely generated covers is exhaustive. For locally integral involutive po-monoids we prove that they are Plonka sums of integral involutive po-monoids and we give a more compact dual description of these sums. These results generalize an earlier structural description of finite commutative idempotent involutive residuated lattices by replacing the lattice order by a partial order, idempotence by local integrality and removing finiteness and commutativity. In the idempotent lattice-ordered case we give equational bases for finitely generated varieties by relating them to varieties of Brouwerian algebras. Using Prover9 for research on ortholattices and locally integral involutive po-monoids
Apr. 18, 2023 11:15am (MATH …	Diversity Facilitators: Howy Jordan and Maya Ornstein CHAT+ Disabilities in academia
Apr. 18, 2023 2:30pm (MATH 3…	Lie Theory ALT Faculty X Richard Green, Flor Orosz Hunziker and Nat Thiem will give brief introductions to their research and their mentoring style. ALT Open House
Apr. 18, 2023 3:30pm (MATH 3…	Topology Guchuan Li (University of Michigan) X Chromatic homotopy theory is a powerful tool to study periodic phenomena in stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories E_h. The homotopy groups of these fixed points are periodic and computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights h and all finite subgroups G of the Morava stabilizer group, the G-homotopy fixed point spectral sequence of E_h collapses after the N(h,G)-page and admits a horizontal vanishing line of filtration N(h,G). Our proof uses new equivariant techniques developed by Hill-Hopkins-Ravenel in their solution of the Kervaire invariant one problem. As an application, we give a computation of E_2^{hG_{48}} based on this vanishing result. This is joint work with Zhipeng Duan and XiaoLin Danny Shi. Vanishing results in Chromatic homotopy theory at the prime 2 Sponsored by the Meyer Fund
Apr. 18, 2023 3:35pm (MATH 2…	Probability Todd Kemp (University of California San Diego) X The Lie group ${GL}_{N}$ of invertible (complex) $N \times N$ matrices is a Riemannian manifold (with its tangent spaces equipped with the metric inherited from the usual Hilbert--Schmidt inner product on matrices), and therefore has a standard Brownian motion $Gt = G(N)t$ . The process $G_{t}$ can also be thought of as a (complex) matrix-valued geometric Brownian motion. It is, with probability $1$ , never a normal matrix, and so the study of the evolution of its eigenvalues is quite complicated. It is yet unknown whether the empirical measure of its eigenvalues (at a fixed time) converges (as $N\to\ifnty$) to some measure on the complex plane. A putative limit measure, called the Brown measure (yes, the Brownian Brown measure) was computed by Driver, Hall, and me in 2022. As a partial result towards the desired limit theorem, in this talk I will present a tightness result: for each fixed time $t>0$ , there is a certain annulus that, with probability $1$ , contains all the eigenvalues of $G(N)t$ for all large $N$ . This shows that the sequence of empirical laws of eigenvalues for $G(N)t$ forms a tight sequence of random measures. Along the way to the proof, I will also show a ``Dyson Brownian motion'' style interacting particle system that describes the dynamic evolution of the eigenvalues of $G_{t}$ . This is joint work with Bruce Driver and Vaki Nikitopoulos. Tightness for eigenvalues of ${GL}_{N}$ Brownian Motion
Apr. 19, 2023 5pm (MATH 350)	Grad Student Seminar Michael Wheeler X There are now several ways of formalizing the use of real infinitesimals in the study of calculus. These ways of adding in infinitesimals result in various extensions of the real numbers that can be interesting in their own right. In this talk, we will explore some of the fun properties of one such extension resulting from an enlargement of R and discuss how these properties allow us to gain insight into more than just calculus. A Glimpse of the Nonstandard Landscape
Apr. 20, 2023 2:30pm (Math350)	Math Phys Bruno Nachtergaele (UC Davis) View Online:https://cuboulder.zoom.us/j/95206919281 X I will discuss recent work proving that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system spectral gaps uniform in the system size. To obtain this result, we extend the Bravyi-Hastings-Michalakis strategy so it can be applied to perturbations of the GNS Hamiltonian of the infinite-system ground state. Stability of the bulk gap for quantum lattice systems Sponsored by the Meyer Fund
Apr. 20, 2023 3:35pm (MATH 3…	Probability Bojan Basrak (University of Zagreb) X Using the example of stationary subcritical branching processes with immigration, we discuss two essentially different ways in which power law tails can arise. For instance, if the immigrant distribution is regularly varying, then the same property holds for the stationary distribution. In this talk, we explain how the assumption that the branching takes place in a random environment also induces power law tail behavior of the stationary distribution. We explore probabilistic properties of the resulting stationary process. The original motivation comes from the seminal 1975 paper by Kesten, Kozlov, and Spitzer, which relates random walks in random environment models to a special Galton-Watson process with immigration in random environment. We show how our approach leads to new results even in this well-studied setting. This talk is based on joint work with P. Kevei (University of Szeged). On stationary branching models and random walks in random environment
Apr. 20, 2023 4pm (MATH 220)	Init Conditions Howy Jordan Adjunctions
Apr. 25, 2023 1:25pm (Online)	Algebra/Logic Ralph Freese (University of Hawaii) View Online:https://cuboulder.zoom.us/j/96896272260 X In the first part we discuss Bjarni Jónsson's arguesian law for lattices, and its implications for lattice theory and for universal algebra. The geometric diagram in ALV2, discussed in December's talk, is explained (as was promised). We continue with some higher order arguesian identities first shown to us by Bill Lampe. The second part presents some new and old results about congruence varieties and congruence prevarieties. This part includes a discussion of commutator properties and Mal'tsev conditions. In the third part we discuss the importance of some unpublished notes of Alan Day on higher order Polin varieties and their applications to solving some problems on the collection of congruence varieties. New old results
Apr. 25, 2023 2:30pm (MATH 3…	Lie Theory Angela Carnevale (University of Galway) X This talk will be devoted to certain generating functions associated with spaces of matrices. First, I will discuss how imposing linear relations subject to mild conditions ("playing board games") on modules of matrices does not affect associated generating functions. Secondly, we will relate Hadamard products of such rational generating functions to shuffle products of permutations. This is based on joint work with T. Rossmann, and work in progress with T. Rossmann and V. Moustakas. Board and card games... and matrices Sponsored by the Meyer Fund
Apr. 25, 2023 3:30pm (MATH 3…	Topology Rachel Chaiser (CU Boulder) X Matui's HK-conjecture proposes an in-principle 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 the following phenomenon in algebraic topology. For a CW-complex of dimensiom at most 3, the Atiyah--Hirzebruch spectral sequence computes the K-theory explicitly as a direct sum of cohomology groups. However, this explicit computation fails to hold in general 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
Apr. 26, 2023 3:30pm (Math350)	Math Phys Howy Jordan (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 Infinite dimensional stratifications
Apr. 27, 2023 9am (MATH 350)	Algebraic Geometry Bob Kuo X A well known result in algebraic geometry is that smooth cubic surfaces contain exactly 27 lines over an algebraically closed field. This is no longer true over arbitrary fields. For example, real cubic surfaces contain either 3, 7, 15, or 27 lines. Nevertheless, these lines fall into the class of elliptic lines or hyperbolic lines, and (the number of hyperbolic lines) - (the number of elliptic lines) = 3. In their paper "An arithmetic count of the lines on a smooth cubic surface", Jesse Leo Kass and Kirsten Wickelgren extend this count to smooth cubic surfaces over arbitrary fields. This talk will be a summary of Kass and Wickelgren's results. An arithmetic count of the lines on a smooth cubic surface
Apr. 27, 2023 4pm (MATH 220)	Init Conditions Henry Maher X The Yoneda lemma tells us a great deal about the structure of presheaves, but there is more to be said; I will discuss a pair of theorems that provide even more insight into the structure of presheaf categories. The density theorem and free cocompletion
May. 02, 2023 11am (MATH 350)	Topology Hiro Lee Tanaka (Texas State University) X If you pour water on a statue, the water will descend in predictable ways. Amazingly, if you know exactly how streams of water on a statue can break, you can reconstruct the statue's shape. This fact is a toy example of Morse theory, a field of topology that tries to understand shapes of things in terms of streams of least resistance (these streams are sometimes called gradient flows). I want to introduce this theory in some fun, concrete examples, and as time allows, talk about a mysterious, recent discovery that connects two disparate branches of math: The geometry of Morse theory encodes exactly the algebra of associativity (yes, like associativity of multiplication). Topology Day: Streams of water, the shapes of things, and the geometry of associativity Sponsored by the Meyer Fund
May. 02, 2023 1:25pm (online)	Algebra/Logic Will Boney (Texas State University) View Online:https://cuboulder.zoom.us/j/96896272260 X Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernible. Building generalized indiscernibles in AECs with set theory
May. 02, 2023 2:30pm (MATH 3…	Topology Teena Gerhardt (Michigan State University) X The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are very difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. Equivariant homotopy theory, a branch of algebraic topology which studies topological objects with a group action, has been particularly important in the study of algebraic K-theory. In this talk I will give a gentle introduction to algebraic K-theory and its applications, and explain the interesting role of equivariant homotopy theory in this story. Topology Day: Algebra in topology and topology in algebra: an introduction to algebraic K-theory Sponsored by the Meyer Fund
May. 02, 2023 3:30pm (MATH 3…	Topology Yuriy Mileyko (University of Hawaii) X The last couple of decades saw a considerable increase in applications of topological techniques within other disciplines, particularly in data analysis, leading to a creation of a nascent field of topological data analysis. The premise of the latter is that data have shape, and that shape matters. Mathematically, this typically translates into an assumption that the data under consideration represent a sample from some topological space whose homotopy type is of interest. For computational reasons, the focus shifts to recovering homology groups of the underlying space, and several important results have been obtained to address this important task. In this talk I will cover some of the key results which allow one to compute homology groups of a subspace of a metric space given only a finite (possibly noisy) sample from it. Not surprisingly, such results rely on certain assumptions about the global geometry/topology of the subspace. I will then show that these assumptions (and results) become more subtle if the goal is to recover local rather than global homology groups of the subspace. Topology: Day Recovering homology from samples: from global to local Sponsored by the Meyer Fund
May. 03, 2023 4:40pm (MATH 3…	Grad Student Seminar Maya Ornstein (CU Boulder) TBA
May. 04, 2023 9am (MATH 350)	Algebraic Geometry Jonathan and Yano X Jonathan and Yano will talk a bit about what it's like to do research in algebraic geometry. Younger grad students are especially encouraged to come. Algebraic geometry open house
May. 04, 2023 3:35pm (MATH 3…	Probability Oanh Nguyen (Brown University) X The Kac polynomial is one of the most studied models of random polynomials. It has the form $fn(x) = ∑i=0n xii xi.$ It is known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. Disproving a natural conjecture for these holes, we will show that all the hole radii are of the same order. We will also discuss open problems and recent fascinating conjectures concerning the flow of roots. Joint work with Hoi Nguyen. Hole radii for the Kac polynomials
May. 04, 2023 4pm (MATH 220)	Init Conditions Howy Jordan X A category of sets is foundational to mathematics, and a category of complete inner product spaces is foundational to modern physics. The former has been captured by categorical language through the work of topos theory. As for the latter, the culmination of reconstruction programs led to a categorical characterization in 2022 by Heunen and Kornell. This could be summarized by saying that the category of Hilbert spaces and continuous linear maps is the dagger-abelian dagger monoidal category, with a simple monoidal separating unit and with directed colimits of dagger monics. In the first part of this talk, we'll consider the axioms of elementary topoi, which capture Set-like categories, and see how to distinguish Set among these. We'll introduce the idea of subobjects in a category, and see that their classifying objects correspond to truth values giving us classifying functions of subsets. As time permits, we'll encounter related objects, like power objects and exponentials. In the second part, we'll consider axioms for the category of Hilbert spaces. In particular, we'll introduce dagger and monoidal structures on categories, overview how a monoidal unit can obtain the structure of a field, and see how the axioms impose an orthomodular structure on the subobjects. The Category of Sets, The Category of Hilbert Spaces
May. 08, 2023 10am (Math 350)	Thesis Defenses Connor Meredith (CU) View Online:https://cuboulder.zoom.us/j/96126023746 X We determine the possible sizes of algebras of finite type that are nilpotent, nonabelian, dualizable, and generate a congruence modular variety. We also show that localization preserves dualizability, then use this result to construct a finite nilpotent algebra A that is nondualizable, yet such that no algebra in ISP(A) has a supernilpotent nonabelian congruence. Finally, we show that if A is a Mal’cev algebra, then higher commutators in any localization of A are closely controlled by higher commutators in subalgebras of A. Nilpotence and Dualizability of Algebras of Finite Type

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