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Events for October 2022 – February 2023

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Oct. 04, 2022 11am (Math 350)	Number Theory YouTube (Castryck/Decru) X We will continue what we were up to last week, since we didn't even finish the first video. :) Attacks on Isogeny based cryptography continued
Oct. 04, 2022 2:30pm (MATH 3…	Lie Theory Justin Willson (CU) X Green and Xu defined the 2-root lattice of a simply laced Weyl group by looking at pairs of orthogonal roots in the root systems of simply laced Weyl groups with Y-shaped Dynkin diagrams. In this talk we will discuss the existence of elements of squared length 2 in the 2-root lattice in hyperbolic root systems as well as progress made on the conjecture that there are no such elements in the affine and finite cases. Lattice roots arising from 2-root lattices of type ADE
Oct. 05, 2022 4:40pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X You may be surprised to learn that differentiation of polynomials is still an active and even growing area of research. We will look at some recent work of Steinerberger answering the title question for polynomials with real roots, and the surprising connection to eigenvalues of matrices. Time permitting we may discuss the challenges of the complex root case. While there are interesting connections to probability, combinatorics, and operator algebras we will try to keep the vast majority of the talk at the level of calculus, linear algebra, and undergraduate analysis. How do polynomial roots move under repeated differentiation?
Oct. 06, 2022 9am (Math 220)	Algebraic Geometry Bob Kuo (CU Boulder) Modular compactifications of the space of genus 1 curves (part 1)
Oct. 06, 2022 3:35pm (MATH 3…	Probability Leonid Petrov (University of Virginia) X Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems. This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which "rewrite the history" of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap. Based on a joint work with Axel Saenz. Rewriting History in Integrable Stochastic Particle Systems
Oct. 08, 2022 9:30am (Math 3…	Number Theory Front Range Number Theory Day X We are excited to announce that we will be hosting Front Range Number Theory Day on October 8th, 2022, here at CU Boulder! The goal of the FRNTD is to provide a venue for faculty, graduate students, and undergraduates on the Front Range who are interested in number theory to meet, learn, and collaborate. For more information and to register, please visit our website: https://sites.google.com/colorado.edu/front-range-number-theory-day/home. If you would like to give a five-minute talk, please include your title on the registration form. We hope to see you there! Front Range Number Theory Day
Oct. 11, 2022 10am (MATH 350)	Number Theory Matthew de Courcy-Ireland (EPFL, Lausanne) X We discuss a family of cubic surfaces defined by x^2+y^2+z^2=xyz+k modulo prime numbers. The solutions form a graph, where each vertex (x,y,z) is joined to the other solution of the same quadratic in any of the three variables. These moves are related to a nonlinear action of the modular group PGL(2,Z) on the surface. We outline some ways these equations arise in geometry, and how we became interested in showing that the associated graphs cannot be embedded in the plane. We sketch a proof that 2, 3, and 7 are the only primes for which the graph is planar. For larger primes, we give an estimate of the Euler characteristic of a surface in which the graph can be embedded. Non-planarity of Markoff graphs mod p
Oct. 11, 2022 11am (MATH 350)	Number Theory Matthew Litman (University of California Davis) X Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran. Markoff-type K3 Surfaces: Local and Global Finite Orbits
Oct. 11, 2022 1:25pm (MATH 2…	Marcos Mazari-Armida (CU Boulder) X The classical model-theoretic notion of superstability was introduced by Shelah in the late sixties for first-order theories. In this talk we will show that superstability is a natural algebraic property by characterizing some classical classes of rings, such as noetherian rings and perfect rings, via superstability of certain classes of modules. Characterizing some classical rings via superstability
Oct. 11, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X Kazhdan and Lusztig defined partitions of an arbitrary Coxeter group W into left, right, and two-sided cells. Lusztig defined an integer a(w) for each element w of W, and proved that the a-function takes constant values on cells. In general, the problem of determining these cells, and the function a, is difficult. Elements of a-value 0 or 1 are well understood, and this talk will describe the structure of cells of elements of a-value 2 in the cases where the number of such elements is finite. This is joint work with Tianyuan Xu. The structure of Kazhdan—Lusztig cells of a-value 2
Oct. 12, 2022 3:30pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) Whightman functions, the Borchers–Uhlmann algebra and GNS type reconstruction of quantum fields
Oct. 12, 2022 4:40pm (MATH 3…	Grad Student Seminar Juan Moreno (CU Boulder) X Homotopy groups are important invariants of both spaces and spectra, but they're not everything. In this talk, I'll introduce Postnikov invariants which contain more information about the space or spectrum. Then we'll determine some Postnikov invariants arising from the physics of topological phases. Postnikov Invariants & Topological Phases
Oct. 13, 2022 9am (Math 220)	Algebraic Geometry Adrian Neff (CU Boulder) Modular compactifications of the space of genus 1 curves (part 2)
Oct. 13, 2022 12:20pm (Math …	Diversity Office of Institutional Equity and Compliance X Attitudes and stereotypes are developed about people and groups of people that affect our understanding and actions without our awareness or intentional control. This workshop examines unconscious bias and identifies key areas where it affects our work, interactions, and decision-making processes. Participants explore personal and group practices that can help reduce bias and increase inclusion, engagement, and productivity. Recognizing and Mitigating Unconscious Bias
Oct. 13, 2022 3:35pm (MATH 3…	Probability Dan Stroock (MIT) Applications of the Gaussian Isoperimetric Inequality
Oct. 18, 2022 1:25pm (Zoom)	Samson Leung (Carnegie Mellon University) X Let K be an abstract elementary class and \lambda=LS(K). K can be axiomatized by a sentence in L_{(2^\lambda)^+,\lambda^+}, allowing game quantification. This is parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives K = PC(T, \Gamma, L(K)) where T is a first-order theory in an expansion of L(K) and \Gamma is a set of T-types. We provide a better bound of \|\Gamma\| in terms of I_2(\lambda,K). We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah. Axiomatizing AECs and applications
Oct. 19, 2022 3:30pm (Math350)	Math Phys Sheagan John (CU Boulder) X A pair of orthonormal bases of the d-dimensional complex space are mutually unbiased if the square of their standard hermitian inner product is equal to 1/d. We briefly motivate why this notion of mutually unbiased bases (MUBs) is of great importance in quantum information theory, then discuss the existing mathematical tools and difficulties in constructing MUBs, including the relationship to orthogonal pairs of special linear Lie algebras. Mutually Unbiased Bases and Lie Algebras
Oct. 19, 2022 4:40pm (MATH 3…	Grad Student Seminar Adrian Neff (CU Boulder) X Michael Artin supposedly said that you can understand most concepts in moduli through the moduli space of triangles (source: someone told me this once). With this advice in mind, this talk will do just that: we will go through multiple variations on the moduli space of triangles in order to see various behaviors of moduli problems, and hopefully by the end (unless Artin was a liar), we will all understand moduli spaces. An Introduction to Moduli Spaces
Oct. 20, 2022 11:15am (Math …	Ulam Boris Tsygan (Northwestern University) X We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung). A survey of noncommutative differential forms.
Oct. 20, 2022 11:15am (Math …	Noncomm Geometry Boris Tsygan (Northwestern University) X We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung). A survey of noncommutative differential forms.
Oct. 20, 2022 2:30pm (MATH 3…	Functional Analysis Frederic Latremoliere (University of Denver) X We introduced a distance between spectral triples, inspired from our earlier work on the quantum Gromov-Hausdorff distance, and we have established a few first examples of convergence for this distance. In this talk, after reviewing our distance, we will discuss properties which are continuous, in an appropriate sense, for our distance, which includes the spectra of the Dirac operators in spectral triples. Spectral properties of the convergence of spectral triples Sponsored by the Simons Foundation
Oct. 25, 2022 1:25pm (MATH 2…	Michael Wheeler (CU Boulder) X A non-standard framework over a base set A is an extension of a fragment of set theory containing A that realizes many types with parameters from A and preserves bounded quantifier formulas. In particular, if A contains an indexing set I and U is an ultrafilter over I, then a saturated enough non-standard framework over A will contain elements that appear to be principal generators for U. The properties of the ultrafilter are reflected in properties of their hyperprincipal generators in a non-standard framework. In this talk we will go over some background information about non-standard frameworks and then discuss how the properties of being Regular and Good (key properties related to Keisler's Order for countable first-order theories) are reflected in their hyperprincipal generators. Hyperprincipal generators for regular and good ultrafilters
Oct. 25, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X After reviewing the character theory of Coxeter groups of types A, B, and D, we will study a particular family of representations of Coxeter groups of type D. Each representation comes equipped with two very different bases, one of which can be described in terms of orthogonal roots. Orthogonal roots and representations of Coxeter groups
Oct. 25, 2022 3:30pm (MATH 3…	Topology Vasileios Maroulas (University of Tennessee) X Topological data analysis (TDA) studies the shape patterns of data. Persistent homology (PH) is a widely used method in TDA that summarizes homological features of data at multiple scales and stores them in persistence diagrams (PDs). In this talk we will discuss a random persistence diagram generation (RPDG) method that generates a sequence of random PDs from the ones produced by the data. RPDG is underpinned by (i) a model based on pairwise interacting point processes for inference of persistence diagrams, and (ii) by a reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm for generating samples of PDs. An example on a materials science problem will demonstrate the applicability of the RPDG method. Random persistence diagram generator and an application to materials Sponsored by the Meyer Fund
Oct. 26, 2022 4:40pm (MATH 3…	Grad Student Seminar Levi Lorenzo (CU Boulder) X To the outsider, the idea of “noncommutative geometry” is admittedly confusing. Commutativity is an algebraic property, so what about geometry makes it commutative (or not)? No, it doesn’t concern whether Sphere x Torus = Torus x Sphere. What it does concern is the algebras of (continuous, smooth, L^\infty, etc.) functions, which commute under pointwise multiplication. These algebras (along with certain operators) determine the (topological, smooth, Riemannian, spin, measure theoretic, etc.) structure on the manifold. This allows us to recover the geometric structure in algebraic language. Remove the commutativity requirement from those algebraic objects and you arrive at noncommutative geometry. These tools shed light on and can prove results in the classic arena, but they also give a way to understand the structures of many noncommutative spaces that arise, for example, as quotients of ordinary spaces, leaf spaces of foliations, dynamical systems, quantum groups, groupoids, and many examples from physics. In this talk, we introduce the primary objects at play and tools at hand in Connes’ program for noncommutative geometry and equip ourselves to journey away from the familiar and into the land of that which does not commute. Step into a (Noncommutative) World
Oct. 27, 2022 9am (MATH 350)	Algebraic Geometry Jonathan Wise TBA
Oct. 27, 2022 2:30pm (MATH 3…	Functional Analysis Christopher Schafhauser (University of Nebraska) X In the early '90s, Elliott conjectured that separable simple nuclear C-algebras are classified up to isomorphism by their K-theory groups and traces, in analogy with the Connes--Haagerup classification of separably acting injective factors by their type and flow of weights. In the last few years, Elliott's classification program has been completed under the two additional hypotheses: Z-stability and the UCT. In the von Neumann algebraic setting, the classification of injective factors was followed by a delicate analysis of the symmetries of such factors. For example, Oceanu proved every amenable group admits a outer action on the hyperfinite II_1 factor R, which is unique up to cocycle conjugacy, and Popa and Takesaki proved the automorphism group of R is contractable. In the spirit of the latter result, I will discuss recent joint work with Jamie Gabe on the homotopy type of automorphism groups of ``classifiable'' C-algebras. Homotopy groups of automorphism groups of classifiable C*-algebras. Sponsored by the Meyer Fund
Oct. 27, 2022 3:35pm (MATH 3…	Probability Benedek Valko (University of Wisconsin Madison) X If one removes the first row and column of an n by n random Haar unitary matrix then the eigenvalues of the resulting matrix are inside the unit disk. Zyczkowski and Sommers (2000) derived the joint pdf of these eigenvalues, and showed that they form a determinantal process. Killip and Kozhan (2017) extended this result to circular beta ensembles (with beta=2 corresponding to the unitary case), describing the eigenvalues of the truncated model via a random recursion. (For general beta the resulting ensemble is not determinantal.) We derive the point process limit of the truncated ensemble together with the scaling limit of the normalized characteristic polynomials. The limiting objects are closely connected to the random analytic function appearing as the limit of the normalized characteristic polynomials of the (full) circular beta ensemble. Joint with Yun Li (Tsinghua University, Beijing) Scaling limits of the truncated circular beta ensemble
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

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