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Events for February – June 2019

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Feb. 04, 2019 3pm (Math 350)	Ulam Mitsuru Wilson (IMPAN) X In the seminal work of M. Ruzhansky et al., a very suitable pseudo-differential calculus on compact Lie groups was constructed using the Peter-Weyl decomposition of L^2 functions. I will explain this formalism, then present my generalization of it to compact quantum groups, which come from the theta-deformation of compact Lie groups. Pseudo-differential operators and their symbols on theta-deformation of compact quantum groups
Feb. 05, 2019 1pm (MATH 220)	Athena Sparks (CU Boulder) X Recently, Bulin, Krokhin, and Oprsal have found sufficient conditions for NP-hardness of a promise constraint satisfaction problem (PCSP). Their method involves defining a new class of problems, the problems of deciding triviality of a set of minor identities, which can been viewed as an algebraic version of Label Cover problems. Every PCSP with a fixed constraint language is log-space equivalent to a promise version of these problems. We will discuss this equivalence as well as how these new problems provide sufficient conditions for NP-hardness of PCSPs. NP-hardness of PCSPs (part 1)
Feb. 05, 2019 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X This semester the ALT seminar will give an introduction to Hopf algebras. We will begin with some survey talks, and also have some research talks later in the semester. This talk is the first such survey talk. Hopf algebras crop up naturally throughout mathematics, and require numerous natural operations to harmonize. This introductory talk focuses on two strands of Hopf theory: Hopf algebras as a direct generalization of group theory and Hopf algebras as a repository for combinatorial gizmos. We will motivate the study Hopf algebras with the first approach and then proceed to show how the second perspective completely alters the paradigm. Throughout we will try to highlight some of the types of questions that arise and key examples. Hopf structures in algebra and combinatorics
Feb. 05, 2019 2pm (MATH 220)	Michael Wheeler (CU Boulder) p = t via Nonstandard Models of Set Theory, Part 2
Feb. 05, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) Classifying spaces and tangential structures
Feb. 06, 2019 3pm (Math350)	Math Phys Markus J. Pflaum (University of Colorado, Math Department) The spectral theorem for operators on Hilbert spaces and its application in quantum mechanics.
Feb. 06, 2019 4pm (MATH 350)	Grad Student Seminar Mariusz Tobolski (IMPAN) X I will discuss main ideas of noncommutative topology and use the definition of the local-triviality dimension to illustrate how they work in practice. The local-triviality dimension was introduced in 2017 and it generalizes the concept of a locally trivial principal bundle, which is pivotal in algebraic topology, and fundamental in gauge field theories in physics. A (locally) trivial talk
Feb. 07, 2019 3pm (MATH 350)	Probability Isaac Sonin (UNC at Charlotte) X We present a Defense/Attack resource allocation model, where Defenders have some number of "locks" to protect n vulnerable boxes (sites), and attackers are trying to destroy these boxes, having m "bombs" that can be placed into boxes. Similar models were studied in game theory - (Colonel) Blotto games, but our model has a feature absent in previous literature. Attackers test the vulnerability of all sites before allocating their resources, and these tests are not perfect, i.e., a test can give plus for a box without a lock and minus for a box with a lock. We describe the optimal strategies in two versions of this Locks-Bombs-Testing (LBT) model. The inspiration for this model was the paper by K. Sonin, J. Wilson, A. Wright: Rebel Capacity, Intelligence Gathering, and the Timing of Combat Operations (manuscript, 2018). A small part of my talk will be devoted to a seemingly simple question: what does it mean that two events are independent ? The last time I gave a talk at your University, I mentioned that if we flip a fair coin and a fair die, then the total number of pairs of independent events is 888, 888. Locks, Bombs and Testing
Feb. 11, 2019 3pm (Math 350)	Ulam Piotr M. Hajac (CU Boulder / IMPAN) X This is part two of my "Pullbacks of graph C*-algebras" talk. This time I plan to explain how everything works in the U(1)-equivariant setting of the gauge actions, prove the main result about pullbacks, and show more examples. From pushouts to pullbacks in the equivariant setting
Feb. 12, 2019 1pm (MATH 220)	Athena Sparks (CU Boulder) NP-hardness of PCSPs (part 2)
Feb. 12, 2019 2pm (MATH 220)	Michael Wheeler (CU Boulder) p = t via Nonstandard Models of Set Theory, Part 3
Feb. 12, 2019 5pm (ECCR 1B40)	DeLong Kristin Lauter (Microsoft Research) X As the world rushes towards the adoption of Artificial Intelligence (AI) in all aspects of our lives, the promise is great, but the risks are many. AI can help to improve and simplify our daily lives and introduce new types of safety measures, but at the cost of sharing and potentially leaking or misusing our private data. Cloud services built on AI algorithms need to make use of customer or enterprise data, to train models and make predictions. It is up to us to protect that data, in storage and in use. Private AI is a set of tools being developed by the Cryptography Group at Microsoft Research which can help protect data while in use by encrypting it. The leading edge of Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction. This talk will give a snapshot of the state of the art and show some compelling demos of HE in action. This lecture will be preceded by tea at 4 PM in MATH 350 Private AI (or: How to Keep your Location Data Secret)
Feb. 13, 2019 3pm (Math350)	Math Phys Mike Hermele (University of Colorado, Physics Department) Lattice models Part 1
Feb. 13, 2019 4pm (ECCR 265)	DeLong Kristin Lauter (Microsoft Research) X As we move towards a world where quantum computers can be built at scale, we are forced to consider the question of what hard problems in mathematics our next generation of cryptographic systems will be based on. Supersingular Isogeny Graphs were proposed for use in cryptography in 2006 by Charles, Goren, and Lauter. Supersingular Isogeny Graphs are examples of Ramanujan graphs, which are optimal expander graphs. These graphs have the property that relatively short walks on the graph approximate the uniform distribution, and for this reason, walks on expander graphs are often used as a good source of randomness in computer science. But the reason these graphs are important for cryptography is that finding paths in these graphs, i.e. routing, is hard: there are no known subexponential algorithms to solve this problem, either classically or on a quantum computer. For this reason, cryptosystems based on the hardness of problems on Supersingular Isogeny Graphs are currently under consideration for standardization in the NIST Post-Quantum Cryptography (PQC) Competition, and have advanced to the second round of the competition. This talk will introduce these graphs, the cryptographic applications, and the various algorithmic approaches which have been tried to attack these systems. This lecture will be preceded by tea at 3 PM in MATH 220 Reception to follow Lecture at the Koenig Alumni Center, 1202 University Avenue, 5:00 - 7:30pm Supersingular Isogeny Graphs in Cryptography (or: How to Keep your Secrets in a Post-Quantum World)
Feb. 14, 2019 2pm (MATH 350)	Functional Analysis Leonard Huang (University of Nevada) X One of the most important and celebrated mathematical results in quantum mechanics is the Stone-von Neumann Theorem. This theorem, in its modern formulation that is due to George Mackey and Marc Rieffel, says: If $G$ is a locally compact Hausdorff abelian group, and $R$ and $S$ are strongly-continuous unitary representations of $G$ and $\hat{G}$ respectively on a Hilbert space $ℋ$ that satisfy the Weyl Commutation Relation, i.e., $S() ∘ R(x) = (x) ⋅ [R(x) ∘ S()]$ for all $x ∈ G$ and $∈ G$ , then the triple $(H,R,S)$ must be unitarily equivalent to a direct sum of copies of the triple $(L2(G),U,V)$ , where $U$ is the unitary representation of $G$ on $L2(G)$ by left translations, and $V$ is the unitary representation of $\hat{G}$ on $L2(G)$ by phase modulations. In this talk, we will present a new generalization of the Stone-von Neumann Theorem that, for any Hilbert space $ℋ$ and any $C^{}$ -dynamical system of the form $(G,K(H),α)$ , classifies up to unitary equivalence all quadruples $(X,ρ,R,S)$ where $X$ is a Hilbert $K(H)$ -module, $ρ$ is a non-degenerate $$ -representation of $K(H)$ on $X$ by adjointable operators, and $R$ and $S$ are strongly-continuous unitary representations of $G$ and $\hat{G}$ respectively on $X$ that satisfy the Weyl Commutation Relation as well as the following two commutation relations: (i) $R(x) ∘ ρ(a) = ρ(αx(a)) ∘ R(x)$ for all $x ∈ G$ and $a ∈ A$ , and (ii) $S() ∘ ρ(a) = ρ(a) ∘ S()$ for all $∈ G$ and $a ∈ A$ . We will also present an infinitesimal (or unbounded) version of our result, which involves the Heisenberg Commutation Relation. We will conclude by elucidating a deep connection between our result and Takai-Takesaki Duality. This is joint work with Lara Ismert of the University of Nebraska-Lincoln. The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on C*-Algebras of Compact Operators Sponsored by the Meyer Fund
Feb. 18, 2019 3pm (Math 350)	Ulam Frederic Latremoliere (University of Denver) X We present our recent ideas on how to show that a family of metric spectral triples is continuous in the sense of the Gromov-Hausdorff propinquity. Convergence of spectral triples
Feb. 19, 2019 11am (MATH 350)	Number Theory Dang-Khoa Nguyen (University of Calgary) X For every complex number $x$ , let $∥ x∥:=\|x-m\|: m∈Z$ . Let $K$ be a number field, let $k∈N$ , and let $α1,…,αk$ be non-zero algebraic numbers. We solve the problem of the existence of $θ∈ (0,1)$ such that there are infinitely many tuples $n,q1,…,qk)$ satisfying $∥ q1α1n+…+qkαkn∥$ where $n∈N$ and $q1,…,qk∈ K$ having small logarithmic height compared to $n$ . In the special case when $q1,…,qk$ have the form $qi=qci$ for fixed $c1,…,ck$ , our work yields results on algebraic approximations of $c1α1n+…+ckαkn$ of the form $m/q$ with $m∈ Z$ and $q∈ K$ . Various results on linear recurrence sequences also follow as an immediate consequence. The case $k=1$ and $q1$ is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Schmidt's Subspace Theorem based on work of Corvaja-Zannier together with several modifications play an important role in the proof. This is joint work with Kulkarni and Mavraki. An Extension of Results by Mahler and Corvaja-Zannier
Feb. 19, 2019 12:10pm (MATH …	Kempner Brian Rider (Temple) X Random matrix theory grew out of fundamental considerations in mathematical statistics and theoretical physics, and now has applications to such disparate areas as number theory, wireless communications and bioinformatics. A perfect example of the field's reach is offered by the Tracy-Widom laws. First discovered as the fluctuation limit of the largest eigenvalue of certain symmetric random matrices at infinite dimension, these laws are now understood to serve the role of the "bell curve” for a wide area of nonlinear phenomena. I will discuss a relatively new description of the Tracy-Widom laws (along with a natural family of generalizations) in terms of random Schroedinger type operators. Operator limits of random matrices
Feb. 19, 2019 2pm (MATH 220)	Michael Wheeler (CU Boulder) p = t via Nonstandard Models of Set Theory, Part 4
Feb. 19, 2019 2pm (MATH 350)	Lie Theory Sarah Salmon (CU) X This talk will focus on previous work done in describing the center of the nil Temperley–Lieb algebra of type affine A. We will describe a faithful representation using particle configurations and give small hints toward future work. Exploring the nil Temperley–Lieb algebra of type affine A
Feb. 19, 2019 3pm (MATH 350)	Probability Brian Rider (Temple) X In random matrix theory there are three fundamental local limit regimes of interest: the spectral bulk and the so-called “hard” and “soft” edges. At this point all three regimes, for all values of the inverse temperature parameter beta, have descriptions in terms of random differential operators. On the other hand, the mathematical physics literature abounds with more exotic limits theorems arising from degenerate situations or various double scalings. With few exceptions, the latter have been restricted to beta = 2 on account of the presence of exact formulas in that setting. I’ll describe, via random operators, the first general beta result for a crossover (double scaling) ensemble. This is joint work with Jose Ramirez (Universidad de Costa Rica). A general beta crossover ensemble
Feb. 19, 2019 4pm (MATH 350)	Topology Jonathan Quartin (CU Boulder) Symmetric monoidal categories and other categorical preliminaries
Feb. 20, 2019 3pm (Math350)	Math Phys Mike Hermele (University of Colorado, Physics Department) Lattice models Part 2
Feb. 20, 2019 4pm (MATH 220)	Functional Analysis Ulrik Enstad (University of Oslo) X In the 80's, Rieffel gave a general procedure to construct projective modules over twisted group C*-algebras. As an application, he constructed and classified projective modules over noncommutative tori. The modules obtained from this general set-up have been termed Heisenberg modules, and have since seen several other applications, e.g. in the construction of projective modules over noncommutative solenoids due to Latremoliere and Packer. More recently, Luef and Jakobsen have shown that Heisenberg modules provide a natural framework for the theory of Gabor frames over locally compact abelian groups. In fact, the atoms of multi-window Gabor frames serve as generators of Heisenberg modules. I this talk I will give a survey of frames in Hilbert modules and of the connection between Heisenberg modules and Gabor frames. If time permits, I will show how Heisenberg modules can be used to shed light on the Balian-Low theorem, a classical theorem in Gabor analysis. Heisenberg modules and their connection to Gabor frames
Feb. 20, 2019 5pm (Math 350)	MathClub Peter Elliott (CU Boulder) X An overview of mathematics in which subjects are studied that do not, at first, seem relevant to the problem at hand. Sometimes it all comes together
Feb. 25, 2019 3pm (Math 350)	Ulam Alexandru Chirvasitu (SUNY Buffalo) X Compact quantum groups are non-commutative spaces with a sufficient amount of structure to resemble classical compact groups and recover much of their behavior. Their actions on classical or non-commutative spaces capture "quantum symmetries". I will discuss a curious phenomenon whereby "sufficiently regular" classical structures do not admit genuinely quantum symmetries: a compact quantum group acting in a structure-preserving fashion is automatically an ordinary compact group. This is the case, for instance, for compact quantum groups acting isometrically on the underlying geodesic metric space of a compact connected Riemannian manifold, with or without boundary. (Joint work with Debashish Goswami.) Quantum-rigid manifolds
Feb. 26, 2019 10am (Math 350)	Geometry/Analysis Nathan Glatt-Holtz (Tulane University) X We develop ‘scaling’ and ‘saturation’ methods on general state spaces which can be used to solve low mode control problems for certain nonlinear partial differential equations. These methods naturally generalize ideas of Jurdjevic and Kupka from the finite-dimensional setting of ordinary differential equations. Our approach is highlighted with applications to specific questions concerning the Korteweg-de-Vries (KdV), Euler and Boussinesq equations. In particular, we derive support properties and unique ergodicity results for certain randomly-forced versions of these equations. Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations Sponsored by the Meyer Fund
Feb. 26, 2019 11am (MATH 350)	Number Theory Damien Roy (University of Ottawa) X The theorem of Lindemann-Weierstrass asserts that the exponentials of distinct algebraic numbers are linearly independent over the field of rational numbers. The proof uses a construction of simultaneous rational approximations to such exponentials values, which goes back to Hermite. In this talk, we show that, from an adelic perspective, these approximations are essentially best possible. The point of view partly explains the nature of the algebraic numbers whose exponentials have a structured continuous fraction expansion. We conclude with a conjecture regarding simultaneous approximations to such values in adèle rings. Simultaneous rational approximation to exponentials of algebraic numbers
Feb. 26, 2019 2pm (MATH 350)	Lie Theory Bruce Sagan (Michigan State University) X The Lucas sequence is a sequence of polynomials in $s, t$ defined recursively by $0=0$ , $1=1$ , and $n=sn-1+tn-2$ for $n≥2$ . On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$ -integers $n]q$ . Given a quantity which is expressed in terms of products and quotients of positive integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with ${n}$ . It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give a combinatorial model for the Lucas analogue of binomial coefficients. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek. We then give an algebraic method for proving polynomiality using a connection with cyclotomic polynomials. This part of the talk is joint work with Jordan Tirrell and based on an idea of Richard Stanley. Finally we, also consider Catalan numbers and their relatives, such as those for finite Coxeter groups. Combinatorial and algebraic interpretations of Lucas analogues Sponsored by the Meyer Fund
Feb. 26, 2019 3pm (MATH 350)	Topology Amit Patel (CSU) X In this pretalk, I will give an introduction to classical persistent homology. I will highlight ideas MacPherson and I generalize to the setting of constructible maps to manifolds. Persistent Homology (Pretalk)
Feb. 26, 2019 4pm (MATH 350)	Topology Amit Patel (CSU) X We are interested in developing a theory of persistence for maps X —> M that are constructible with respect to a stratification of M. I will present ideas from the first paper in this direction. We give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions. Persistent Local Systems
Feb. 27, 2019 3pm (Math350)	Math Phys Markus J. Pflaum (University of Colorado, Math Department) Algebraic QFT a la Haag-Kastler Part 1
Feb. 27, 2019 5pm (Math 350)	MathClub Agnes Szendrei (CU Boulder) X Algebras are devices for computation. Computers are often used to search for examples of algebras with particular properties, or to experiment with algebras to form conjectures. In this talk I will discuss how likely it is that random search will produce an interesting example of an algebra that is general enough to serve as a test case in an experiment. Are Random Algebras Really `Random' ?
Mar. 04, 2019 3pm (Math 350)	Ulam Konrad Aguilar (Arizona State University, Tempe) X In this talk, we will place quantum metrics, in the sense of Rieffel, on certain unital inductive limits of C-algebras built from quantum metrics on the terms of the given inductive sequence with certain compatibility conditions. One of these conditions is that the inductive sequence forms a Cauchy sequence of quantum metric spaces in the dual Gromov-Hausdorff propinquity of Latremoliere. Since the dual propinquity is complete, this will produce a limit quantum metric space. Based on our assumptions, we then show that the C-algebra of this limit quantum metric space is isomorphic to the given inductive limit, which finally places a quantum metric on the inductive limit.This then immediately allows us to establish a metric convergence of the inductive sequence to the inductive limit. Another consequence to our construction is that we place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. Inductive limits of C*-algebras and compact quantum metric spaces
Mar. 05, 2019 10am (Math 350)	Noncomm Geometry John Lott (University of California, Berkeley) X In S.-Y. Cheng's Berkeley thesis from 1974, he gave upper bounds on the eigenvalues of the function Laplacian on a compact Riemannian manifold, in terms of geometric data. I will give an extension of Cheng's results to the differential form Laplacian. The proof uses Alexandrov spaces, i.e. metric spaces with curvature bounded below. I will also construct differential form Laplacians on Alexandrov spaces and prove a Hodge theorem. Eigenvalue estimates and differential form Laplacians on Alexandrov spaces Sponsored by the Meyer Fund
Mar. 05, 2019 10am (MATH 350)	Geometry/Analysis John Lott (UC Berkeley) X In S.-Y. Cheng's Berkeley thesis from 1974, he gave upper bounds on the eigenvalues of the function Laplacian on a compact Riemannian manifold, in terms of geometric data. I will give an extension of Cheng's results to the differential form Laplacian. The proof uses Alexandrov spaces, i.e. metric spaces with curvature bounded below. I will also construct differential form Laplacians on Alexandrov spaces and prove a Hodge theorem. Eigenvalue estimates and differential form Laplacians on Alexandrov spaces Sponsored by the Meyer Fund
Mar. 05, 2019 2pm (MATH 220)	Agnes Szendrei (CU Boulder) X This sequence of talks will focus on how to construct (regular) ultrafilters which have properties that are relevant to Keisler's order. The material is based on the paper `Constructing regular ultrafilters from a model-theoretic point of view' by M. Malliaris and S. Shelah (Trans. Amer. Math. Soc., 2015). Ultrafilters with Model-Theoretically Significant Properties
Mar. 05, 2019 2pm (MATH 350)	Lie Theory Stephanie van Willigenburg (University of British Columbia) X The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in his seminal 1995 paper. This function is currently experiencing a flourishing renaissance, in particular the study of the positivity of chromatic symmetric functions when expanded into the basis of elementary symmetric functions, that is, e-positivity. In this talk, we approach the question of e-positivity from various angles. Most pertinently we resolve the 1995 statement of Stanley that no known graph exists that is not contractible to the claw, and whose chromatic symmetric function is not e-positive. This is joint work with Soojin Cho, Samantha Dahlberg and Angele Foley, and no prior knowledge is assumed. The e-positivity of chromatic symmetric functions Sponsored by the Meyer Fund
Mar. 05, 2019 3pm (MATH 220)	Topology Anna Marie Bohmann (Vanderbilt University) X The speaker will discuss some background for the 4 pm seminar. Pretalk
Mar. 05, 2019 3pm (MATH 350)	Algebraic Geometry Damiano Fulghesu (Minnesota State Moorehead) X In this talk, we will show some general results about the integral intersection ring of the moduli stack of irreducible plane curves of fixed degree. We will also present a more detailed structure of the intersection ring in the case of cubics and quartics. This is joint work with Andrea Di Lorenzo and Angelo Vistoli. Intersection Theory of Irreducible Plane Curves
Mar. 05, 2019 4pm (MATH 350)	Topology Anna Marie Bohmann (Vanderbilt University) X Freyd's generating hypothesis is a long-standing conjecture in stable homotopy theory. The conjecture says that if a stable map between finite CW complexes induces the zero map on homotopy groups, then it must actually be nullhomotopic. I will formulate the appropriate generalization of this conjecture in the case where a group G acts on the complexes and give some results about this setting. In particular, I will show that the rational version of this conjecture holds when G is finite, but fails when the group is S^1. The Generating Hypothesis with G-actions
Mar. 06, 2019 3pm (Math350)	Math Phys Markus J. Pflaum (University of Colorado, Math Department) Algebraic QFT a la Haag-Kastler Part 2
Mar. 06, 2019 4pm (Math 35)	Thesis Defenses Paul Lessard (CU Boulder) X Synthesizing the theory of sketches for algebraic theories, Cisinski model category theory, and Kan's combinatorial presentation of spectra by way of stable cells we develop a new presentation of the (\infty,1)-category of spectra as locally finite \Z-groupoids and thereby put fort a new naive stable homotopy hypothesis. Spectra as Locally Finite \Z-Groupoids
Mar. 07, 2019 12:10pm (MATH …	Kempner Laura Scull (Fort Lewis College) X Homotopy theory studies deformations of spaces and maps between them, an inherently continuous concept. In this talk, we will explore how this translates to the discrete category of graphs. We will see how to define homotopy for graphs, and show that we can break down a deformation of graph maps into moves of one vertex at a time. We also discuss how to recognize when two graphs are the same up to deformation (‘homotopy equivalent’), and look at developing homotopy invariants for graphs. No prior knowledge of either homotopy theory or graph theory will be needed, and many examples and pictures will be given. Parts of this work are joint with Dr. Tien Chih at MSU Billings, and parts were developed in collaboration with Fort Lewis College undergraduate students Coleman Kane, Diego Novoa and Jonathon Thompson. Homotopy theory for graphs
Mar. 07, 2019 2pm (Math 350)	Functional Analysis Laura Scull (Lewis College) X The equivariant fundamental groupoid was originally defined by Tom Dieck for spaces with group actions, as a way to get an invariant that captured more of the structure of the fixed sets of the action than the traditional fundamental group. This construction was later reinterpreted as a Grothendieck category. I will give the definition and examples illustrating the structure of this category. Then I will talk about how the categorical interpretation leads to a potential generalization to groupoids, and give an outline of how such an argument might go. This is work in progress, so we have not yet filled in all the steps necessary to get a well-defined orbifold invariant. I will give the results we have and explain what still needs to be done. This is joint work with Dorette Pronk at Dalhousie university. An equivariant fundamental group for orbifolds Sponsored by the Meyer Fund
Mar. 08, 2019 4pm (MATH 350)	Grad Student Seminar Sebastian Bozlee (CU Boulder) X Finite fields are finite collections of things which it makes sense to add, subtract, multiply, and divide. They are of interest in Galois theory, algebraic number theory, and other areas of pure mathematics. In this talk, we will define finite fields, construct examples, and present a real-life application of finite fields to the recovery of data on computers. An Application of Finite Fields
Mar. 11, 2019 3pm (Math 350)	Ulam Albert J. L. Sheu (University of Kansas, Lawrence) X The notion of the quantum space of all maps between quantum spaces was invented by Piotr M. Soltan. His pioneering work was mainly focused on finite-dimensional C-algebras, which are matrix-algebra bundles over a finite set S. We propose a generalization of this concept that includes arbitrary compact Hausdorff spaces X (instead of finite sets S) and takes into account the topology of X. In this context, the notion of the free product of copies of a unital C-algebra topologically indexed by a compact Hausdorff space arises naturally, and enjoys some desired functoriality. Based on joint work with Thomas Timmermann. The Quantum family of maps
Mar. 12, 2019 11am (Math 350)	Thesis Defenses John Willis (University of Colorado, Boulder) Topological foundations of tropical geometry
Mar. 12, 2019 1pm (MATH 220)	Nik Ruskuc (St Andrews) X I will compare and contrast the congruence lattices of the following: - the symmetric group (folklore); - the full transformation monoid (an old result of Malcev); - the partition monoid and its significant submonoids (diagram monoids) - recent results with East, Mitchell and Torpey and with East. In Part 1 I will talk about the finite case, and in part 2 the infinite. Congruence lattices of diagram monoids, 1
Mar. 12, 2019 2pm (MATH 220)	Peter Mayr (CU Boulder) X I will discuss several results that were presented at the Workshop on General Algebra 97 at Vienna last weekend. Results from AAA 97
Mar. 12, 2019 2pm (MATH 350)	Lie Theory Richard Green (CU Boulder) X The Totally Asymmetric Simple Exclusion Process (TASEP) is a dynamical model that has applications in statistical physics and molecular biology. I will describe this model and its relationship with the nil Temperley--Lieb algebra of type affine C. Applications of the nil Temperley--Lieb algebra of type affine C
Mar. 12, 2019 4pm (MATH 350)	Topology Robin Deeley (CU Boulder) Bordism and classical invariants
Mar. 13, 2019 5pm (Math 350)	MathClub Piotr M. Hajac (ULAM visitor) Game of Theorems
Mar. 14, 2019 10am (Math 350)	Thesis Defenses Kevin M. Berg X We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, $ℒ$ , and take as input an algebra homomorphism $f : X \to Z$ between two finite $ℒ$ -algebras $X$ and $Z$ , along with an intermediate finite $ℒ$ -algebra $Y$ . The decision problem asks whether there are homomorphisms $g : X \to Y$ and $h : Y \to Z$ such that $f=hg$ . We show that this problem is NP-complete for most languages. We also investigate special cases where homomorphism factorization can be performed in polynomial time. The Complexity of Homomorphism Factorization
Mar. 18, 2019 3pm (ECCR 155)	Functional Analysis Alex Kumjian (University of Nevada) X We review the definition and elementary properties of the Crainic-Moedijk homology in the setting of ample groupoids. We show that there is a natural homomorphism from the homology of a higher-rank graph to that of its path groupoid. Given an ample groupoid G with compact unit space which is minimal and effective, Matui conjectured that the K-theory of the reduced C*-algebra of G is isomorphic to the homology, and verified his conjecture in a number of important cases. In joint work my coauthors, Carla Farsi, David Pask and Aidan Sims, and I have shown that the isomorphism in the conjecture holds for the path groupoids of higher-rank graphs of ranks one or two even when the unit space is no longer compact. The homology of ample groupoids and the Matui conjecture (Joint with the Ulam Seminar) Sponsored by the Meyer Fund
Mar. 18, 2019 3pm (Math 350)	Ulam Alexander Kumjian (University of Nevada, Reno) X We review the definition and elementary properties of the Crainic-Moerdijk homology in the setting of ample groupoids. We show that there is a natural homomorphism from the homology of a higher rank graph to that of its path groupoid. Given an ample groupoid G with compact unit space which is minimal and effective, Matui conjectured that the K-theory of the reduced C*-algebra of G is isomorphic to the homology and verified his conjecture in a number of important cases. In joint work my coauthors, Carla Farsi, David Pask and Aidan Sims, and I have shown that the isomorphism in the conjecture holds for the path groupoids of higher rank graphs of ranks one or two even when the unit space is no longer compact. The homology of ample groupoids and the Matui conjecture
Mar. 19, 2019 12:10pm (MATH …	Kempner Alex Kumjian (University of Nevada/Reno) X Higher rank graphs were introduced nearly twenty years ago as a combinatorial model for the construction of C^-algebras generalizing that of the C^algebra of a directed graph. We review some basic definitions and examples and survey some recent results. Higher Rank Graphs and the Associated C^*-algebras
Mar. 19, 2019 1pm (MATH 220)	Nik Ruskuc (St Andrews) Congruence lattices of diagram monoids, 2
Mar. 19, 2019 2pm (MATH 220)	Peter Mayr (CU Boulder) More results from AAA 97
Mar. 19, 2019 4pm (MATH 350)	Topology Markus Pflaum (CU Boulder) Topological field theories
Mar. 20, 2019 3pm (Math350)	Math Phys Daniel Spiegel (University of Colorado, Physics Department) Quantum Field Theory from a physicists point of view
Mar. 20, 2019 4pm (MATH 350)	Grad Student Seminar Lucas Laird (CU Boulder) X Biological sequences are one of the primary types of data used in computational biology. Such sequences can be represented as $k$ -mers, strings of length $k$ with symbols chosen from a reference alphabet. In contrast, powerful machine learning algorithms often require numeric vector representations of the data that are low-dimensional and minimally distorted/biased. Hamming graphs and their so-called "resolving sets" have been shown to produce efficient vector embeddings of $k$ -mers. Here, we reduce the question of whether a set of nodes in a Hamming graph is resolving to solving a constrained linear system and show that such a system can be solved by finding the intersection of a matrix null space with the roots of a polynomial system. Gröbner bases then provide a highly efficient characterization of resolvability on Hamming graphs with the goal of improving $k$ -mer vector representations. Resolvability of Hamming Graphs for Applications in Computational Biology
Mar. 21, 2019 12:10pm (MATH …	Kempner Boris Hanin (Texas A&M) X Deep learning is the study and use of artificial neural networks, which are finite dimensional spaces of non-linear functions. Although neural nets have been around since the 1940’s, they have recently achieved state-of-the-art in a variety of machine learning tasks, ranging from machine vision to natural language processing, and reinforcement learning. The fact that they work at all is a bit surprising and leads to a number of interesting mathematical and statistical questions. I will begin by defining neural networks and giving a sense of how they are used in practice. I will then focus on what I think are several important problems related to approximation theory, random matrix theory, and optimization. In each case, I will explain a bit of what is known. Deep Learning for Mathematicians
Mar. 21, 2019 3pm (MATH 350)	Thesis Defenses Noah Williams (CU Boulder) TBA
Apr. 01, 2019 3pm (Math 350)	Ulam Sarah Reznikoff (Kansas State University, Manhattan) X Abelian operator algebras are understood. General operator algebras can be analyzed by examination of large abelian subalgebras. Renault's seminal result relates Cartan subalgebras, which are abelian and particularly nice, to algebraic structures known as groupoids. The talk will define all relevant notions and proceed to explain an extension of Renault's result to the situation where a compact abelian group acts in a convenient way on a C-algebra. This is joint work with Jonathan Brown, Adam Fuller, and David Pitts. A Renault-type correspondence between groupoid twists and C-algebras acted upon by abelian groups
Apr. 02, 2019 10am (MATH 350)	Geometry/Analysis Peter Olver (University of Minnesota) X I will survey the equivariant method of moving frames and how it is used to construct differential invariant signatures for solving equivalence and symmetry problems. I will then present recent applications in image processing, in particular the reassembly of 2D and 3D jigsaw puzzles and the geometric classification and reassembly of broken bones in anthropology. Moving frames, differential invariants, and the reassembly of broken objects Sponsored by the Meyer Fund
Apr. 02, 2019 1pm (MATH 220)	Agnes Szendrei (CU Boulder) X This is a continuation of the talk ``Ultrafilters with Model-Theoretically Significant Properties''. Flexible Ultrafilters and the Theory of the Random Graph
Apr. 02, 2019 2pm (MATH 350)	Lie Theory John Shareshian (Washington University in St. Louis) X Given a finite graph $G=(V,E)$ , a proper coloring of $G$ is a function $k$ from $V$ to the set of positive integers such that $k(i)$ and $k(j)$ are distinct whenever ${i, j}$ is in $E$ . Associated to each such $k$ is the monomial $x(k)$ in variables $x1,x2,...$ in which the exponent of $x_{i}$ is the number of vertices $v$ satisfying $k(v)=i$ . The sum of such monomials over all proper colorings $k$ is the chromatic symmetric function $X_{G}$ , introduced by Richard Stanley. If $V=1,...,n$ , one may multiply introduce another variable $t$ and multiply each $x(k)$ by $t asc(k)$ , where $asc(k)$ is the number of edges ${i, j}$ in $E$ such that $i < j$ and $k(i)$ . The sum of $t asc(k)xk$ is called the chromatic quasisymmetric function of $G$ . These quasisymemtric functions were introduced in joint work with Michelle Wachs. When $G$ is a unit interval graph, labeled appropriately, it turns out that this quasisymmetric function is in fact a symmetric function. Moreover, this symmetric function is Schur positive and therefore is the Frobenius characteristic of some representation of the symmetric group $S_{n}$ . Patrick Brosnan and Tim Chow proved a conjecture of ours, that the representation in question is, up to a sign twist, the "dot action" (first investigated by Julianna Tymoczko) of $S_{n}$ on the cohomology of a variety naturally associated to the unit interval graph $G$ . This variety is a subvariety of the flag variety that admits a torus action, thus allowing one to study its cohomology by considering a moment graph. (I will explain all of these terms, it is not necessary for the audience to be familiar with them already.) The Brosnan-Chow Theorem allows a geometric attack on a longstanding conjecture of Stanley and John Stembridge, but this conjecture remains open. Chromatic quasisymmetric functions and regular semisimple Hessenberg varieties Sponsored by the Meyer Fund
Apr. 02, 2019 4pm (MATH 350)	Topology Paul Lessard (CU Boulder) Higher categories and the Grothendieck Homotopy Hypothesis
Apr. 03, 2019 3pm (Math350)	Math Phys Sheng-Jie Huang (University of Colorado, Physics Department) Chern-Simons Theory
Apr. 03, 2019 5pm (Math 350)	MathClub Jeanne Clelland (CU Boulder) X We tend to think of elections in simple terms: Everyone votes for their favorite candidate, and the candidate with the most votes wins. For elections with only two candidates, this works perfectly well. But as soon as there are three or more candidates, it gets complicated: What if no candidate receives a majority of the votes? In this talk I will describe several possible approaches to this problem, along with some of their advantages and disadvantages, including some political implications of various options. The Will of the People: How we vote and why it matters
Apr. 04, 2019 10am (MATH 220)	Thesis Defenses Robert Hines X We investigate continued fractions and some aspects of Diophantine approximation in the following settings: - approximation of vectors in the product of the Archimedean completions of a number field by elements of the field (embedded diagonally), - approximation of vectors in Euclidean space by elements of definite Clifford algebras. In the first setting, when the number field is one of the Euclidean imaginary quadratic fields, we more thoroughly investigate two algorithms: nearest integer continued fractions (introduced by A. Hurwitz) and continued fraction algorithms associated to a class of right-angled hyperbolic Coxeter groups (variations on the continued fraction algorithms of A. L. Schmidt). In the second setting, we go into slightly more detail concerning approximation in three-dimensional Euclidean space by quotients of Lipschitz quaternions. In both settings we give explicit examples of badly approximable numbers via: the Dani correspondence and bounded geodesic trajectories in the appropriate locally symmetric spaces associated to the special linear group of degree two, boundedness of partial quotients in various continued fraction algorithms, and elementary arguments in the vein of Liouville's theorem concerning rational approximation to algebraic numbers. In addition, some properties of the discrete Markoff spectrum are explored, namely we demonstrate the transcendence of limits of roots of Markoff forms and evaluate a sums over Markoff numbers. The former result follows from work of Bugeaud and Adamczewski (an application of the Subspace Theorem) and the latter is a special case of Mcshane's identity (for a sum over lengths of simple closed geodesics on a once-punctured hyperbolic torus). Applications of hyperbolic geometry to continued fractions and Diophantine approximation
Apr. 04, 2019 11am (Math 350)	Thesis Defenses Paul Lessard (CU Boulder) X TBA Spectra as Locally Finite Z-groupoids
Apr. 04, 2019 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X I will discuss index theory in the context of both manifolds with Baas-Sulivan singularities and their generalization to fibered singularities. A basic but informative class of examples are Z/k-manifolds. As will be developed in detail in the talk, a Z/k-manifold is obtained from a manifold with boundary whose boundary decomposes into k-copies of a single closed manifold. The fibered generalization in this special case is obtained from a manifold with boundary whose boundary is the total space of a k-fold covering space. Much of what will be discussed is based on "Positive scalar curvature on manifolds with fibered singularities" by Botvinnik and Rosenberg (https://arxiv.org/abs/1808.06007). Index theory for manifolds with fibered Baas-Sulivan singularities
Apr. 04, 2019 3pm (MATH 350)	Probability Zhenhua Wang (CU Boulder) X Let S be the random walk obtained from "coin turning" with some sequence {p_n}_{n>=1}. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const n^{-1} (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk. Sketch of proof will be given if time is allowed. The ‘coin-turning walk’ and its scaling limit
Apr. 08, 2019 3pm (Math 350)	Ulam Alexander Gorokhovsky (CU Boulder) X Using a relative form of the Chern character in cyclic homology, we give a uniform construction of the higher indices of elliptic operators associated to Alexander-Spanier cocycles of either parity in terms of a pairing a la Connes between the K-theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators. While the formula for the lowest index of an elliptic operator D on a closed manifold M (which coincides with its Fredholm index) reproduces the Atiyah-Singer index theorem, our formula for the highest index of D (associated to a volume cocycle) yields an extension to arbitrary manifolds of any dimension of the Helton-Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres. In fact, the totality of higher analytic indices for an elliptic operator D amounts to a representation of the Connes-Chern character of the K-homology cycle determined by D in terms of expressions which extrapolate the Helton-Howe formula below the dimension of M. Based on joint work with H. Moscovici. Index pairing with Alexander-Spanier cocycles
Apr. 09, 2019 12:10pm (MATH …	Kempner Adam Sikora (State University of New York at Buffalo) X The Jones polynomial is simple to define, yet a mysterious invariant of links in R^3. It extends to links in thickened surfaces, leading to the notion of the skein algebra of a surface, which turns out to be a quantization of the Teichmuller space. We will study some fundamental properties of these algebras using the theory of measured foliations and pseudo-Anosov diffeomorphisms of surfaces. From the Jones polynomial to Quantum Teichmuller Theory
Apr. 09, 2019 1pm (Math 245)	Thesis Defenses Jon Belcher (CU Boulder) X The connection between Hochschild and cyclic cohomologies with generalized De Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al. Here we generalize these cohomology theories even further, essentially creating a theory that establishes a step-wise bridge between the two. Motivation for this construction comes from trying to generalize the Hochschild-Kostant-Rosenberg-Connes theorem to manifolds with boundary, and applications in tracial constructions in certain classes of pseudo-differential operators. Further geometric and topological interests of this theory include extending Chern-Weil theory to manifolds with boundary via pairings between bridge cohomology and higher K-theories. Bridge Cohomology
Apr. 09, 2019 1pm (MATH 220)	Thesis Defenses Steven Weinell (CU Boulder) X This thesis characterizes the potential behaviors of higher commutators in a simple algebra. On the Descending Central Series of Higher Commutators
Apr. 09, 2019 1pm (MATH 220)	Steven Weinell (CU Boulder) X This thesis characterizes the potential behaviors of higher commutators in a simple algebra. On the Descending Central Series of Higher Commutators (Thesis defense)
Apr. 09, 2019 2pm (MATH 220)	Steven Weinell (CU Boulder) On the Descending Central Series of Higher Commutators (Thesis defense)
Apr. 09, 2019 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (University of Colorado Boulder) X The theory of $P$ -partitions was introduced by Stanley, and a $P$ -partition can be viewed as a generalization of the tableaux. Stembridge introduced enriched $P$ -partitions, and these combinatorial objects are a generalization of the shifted Young tableaux. The polynomials corresponding to the enriched $P$ -partitions produce the peak algebra. We introduce and develop a theory of colored $P$ -partitions and colored enriched $P$ -partitions, and this theory give us tools to define the peak algebra in non-commutating variables. In the end, we generalize the peak algebra to an arbitrary combinatorial Hopf algebra. This is joint work with Shu Xiao Li. Hopf structures in algebra and combinatorics: The peak algebra
Apr. 09, 2019 4pm (MATH 350)	Topology Michael Hermele (CU Boulder) Extended Topological Field Theories and the Cobordism Hypothesis
Apr. 10, 2019 3pm (Math350)	Math Phys Markus J. Pflaum (University of Colorado, Math Department) Algebraic QFT a la Haag-Kastler Part 3: The Doplicher-Haag-Roberts theory
Apr. 10, 2019 4pm (MATH 350)	Grad Student Seminar Jun Hong (CU Boulder) X In physics, the divergent sums $∑n≥ 0 n$ and $∑n≥ 0 n3$ appear in the study of the Casimir effect, which is the existence of an attractive force between two parallel conducting plates in vacuum. Curiously, the experimental results agree with the ad-hoc assumption that these divergent series converge to the Riemann-zeta function evaluated at $s=-1$ and $s=-3$ , respectively. In this talk, we will explore some summation methods that assign values to divergent series, and introduce a general algebraic framework that unifies the various summation methods. Summation of Divergent Series
CANCELED Apr. 11, 2019	Functional Analysis Hao Guo (Texas A&M) X It is a classical result of Lichnerowicz that the index of the Dirac operator on a closed, spin manifold vanishes in the presence of positive scalar curvature. When the manifold is non-compact but has bounded geometry, one can define a refinement of this index that lives in the K-theory of the maximal Roe algebra. The aim of this talk is to outline a proof of the fact that, if the manifold has uniformly positive scalar curvature, then this index vanishes. The argument relies on a subtle distinction between the usual Roe algebra and a uniform version of it; the latter allows one to avoid some of the analytical difficulties encountered when working with the former, in particular when trying to establish a version of the functional calculus in the maximal setting. This work is joint with Zhizhang Xie and Guoliang Yu. A Lichnerowicz vanishing theorem for the maximal Roe algebra Sponsored by the Meyer Fund
Apr. 15, 2019 3pm (Math 350)	Ulam Kenny De Commer (Vrije Universiteit Brussel) X Let G be a semisimple real Lie group with simply connected complexification G_c. Let U be the maximal compact subgroup of G_c. Then U is a compact semisimple Lie group which admits a q-deformation into a compact quantum group U_q. In this talk, we will explain how also the quotient space X = G\G_c can be q-deformed into a U_q-equivariant quantum space X_q with a classical space of orbits, denoted X_q/U_q. We further show how one particular orbit in X_q/U_q gives rise to a quantum symmetric space (in the sense of G. Letzter) through an adaptation of the universal K-matrix formalism (due to M. Balagovic and S. Kolb). This is joint work with Marco Matassa. Quantum symmetric spaces and quantizations of semisimple real Lie groups
Apr. 16, 2019 10am (MATH 350)	Geometry/Analysis Taylor Klotz (CU Boulder) X A control system with symmetry is cascade linearizable if its solutions can be determined by “composing” two associated linear systems. One first uses a linearizable quotient of the system by some symmetries to reconstruct the control system as a linear part plus an equation of lie type. One then restricts the reconstructed system along so-called partial contact curves so that the restricted equations are now themselves linearizable as well. In this talk we present new results concerning existence of linearizable partial contact curve reductions. The Partial Contact Curve Reduction Step in Cascade Linearization
Apr. 16, 2019 11am (MATH 350)	Number Theory Chong Gyu Lee (Soongsil University) X Spiro showed that multiplicative function which is additive on prime numbers should be the identity function. After Spiro's work, there are many variations. One of such result is work of Chung and Phong; multiplicative function which is additive on triangular numbers should be the identity. But there is non-identity functions which is multiplicative and is additive on square numbers. In this talk, we check the difference of triangular and square numbers and expand it the other polygonal numbers, pentagonal and hexagonal numbers. Multiplicative functions, additive on polygonal numbers
Apr. 16, 2019 12pm (MATH 350)	Kempner Satyan Devadoss (University of San Diego) X A 2-ton interactive sculpture came to life at Burning Man 2018, the world’s most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron was illuminated by 16,000 LEDs, requiring 6500 person hours and $40,000 in funds. Its interior, large enough to hold 15 people, was fully lined with massive mirrors, alluding to a possible shape of our universe. The unfolding exterior points to the 500-year-old work of Albrecht Durer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron. We discuss the state-of-the-art, and consider higher-dimensional unfolding analogs, with elegant geometric and combinatorial relationships to integer partitions and chord diagrams. Unfolding Mathematics at Burning Man Sponsored by the Meyer Fund
Apr. 16, 2019 12pm (MATH 350)	Topology Satyan Devadoss (University of San Diego) X A 2-ton interactive sculpture came to life at Burning Man 2018, the world’s most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron was illuminated by 16,000 LEDs, requiring 6500 person hours and $40,000 in funds. Its interior, large enough to hold 15 people, was fully lined with massive mirrors, alluding to a possible shape of our universe. The unfolding exterior points to the 500-year-old work of Albrecht Durer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron. We discuss the state-of-the-art, and consider higher-dimensional unfolding analogs, with elegant geometric and combinatorial relationships to integer partitions and chord diagrams. Unfolding Mathematics at Burning Man Sponsored by the Meyer Fund
Apr. 16, 2019 1pm (MATH 220)	Keith Kearnes (CU) Congruence n-permutability is not join prime
Apr. 16, 2019 2pm (MATH 220)	Jordan DuBeau (CU) An uncountable Jonsson Jonsson-Tarski Algebra
Apr. 16, 2019 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (University of Colorado Boulder) X The theory of $P$ -partitions was introduced by Stanley, and a $P$ -partition can be viewed as a generalization of the tableaux. Stembridge introduced enriched $P$ -partitions, and these combinatorial objects are a generalization of the shifted Young tableaux. The polynomials corresponding to the enriched $P$ -partitions produce the peak algebra. We introduce and develop a theory of colored $P$ -partitions and colored enriched $P$ -partitions, and this theory give us tools to define the peak algebra in non-commutating variables. In the end, we generalize the peak algebra to an arbitrary combinatorial Hopf algebra. This is joint work with Shu Xiao Li. Hopf structures in algebra and combinatorics: The peak algebra in non-commuting variables
Apr. 16, 2019 3pm (MATH 350)	Topology Ezra Getzler (Northwestern University) X The moduli space of vector bundles on a closed oriented surface is a symplectic manifold. More recently, this result has been generalized by Toën and Vezzosi and their collaborators in several directions, in the framework of derived stacks: there is a Chern character in the de Rham complex of differential forms on the moduli space (or derived stack) of finite dimensional complexes on an oriented d-dimensional manifold, and the second Chern character class in this differential form is a shifted symplectic form (of total degree 2-d). In my work with Kai Behrend, I have sought more a constructive approach to this result: the original approach made heavy use of closed model categories and Lurie’s proof of the cobordism hypothesis (for the circle). The goal of these two talks is to explain what a derived stack is, construct some examples explicitly, and explain how to define the Chern character in this setting. To give a small flavor of the subject, here is a typical result from our work: if A is a finite-dimensional differential graded algebra over the complex numbers vanishing in degree -k and below, then its quasi-invertible elements (invertible up to a homotopy) are the objects of a derived k-groupoid. Derived stacks and symplectic forms Sponsored by the Meyer Fund
Apr. 17, 2019 3pm (Math350)	Math Phys Victor Gurarie (University of Colorado, Physics Department) Conformal Field Theory Part 1
Apr. 17, 2019 4pm (MATH 350)	Grad Student Seminar Hunter Davenport (CU Boulder) X The Robertson-Seymour Theorem (aka the Graph Minor Theorem) states that given any infinite family of finite graphs, at least one graph in the family appears as a “minor” in one of the other graphs, or that finite graphs are “well-quasi-ordered” by the minor relation. The proof spans about 500 pages of research beginning in 1983 and concluding in 2004. One of the more accessible steps involves showing that trees are well-quasi-ordered by the graph minor relation and then classifying all finite graphs by their tree-width, a measurement of how tree-like they are. In this talk, we will define tree-decompositions and identify the canonical obstruction to a graph having small tree-width. This will lead into a discussion of the characterization of graphs by forbidden minors. We will conclude with some surprising consequences and, time permitting, an outline of the role of the bramble in the remainder of the proof. Tree-Decompositions and the Graph Minor Theorem
Apr. 17, 2019 5pm (Math 350)	MathClub Panel: Faculty and Graduate Students (CU Boulder) X The CU Faculty and Graduate Students will offer advice on how to best prepare for graduate school in mathematics. Panel on Graduate School
Apr. 18, 2019 2pm (MATH 350)	Functional Analysis Hao Guo (Texas A&M) X It is a classical result of Lichnerowicz that the index of the Dirac operator on a closed, spin manifold vanishes in the presence of positive scalar curvature. When the manifold is non-compact but has bounded geometry, one can define a refinement of this index that lives in the K-theory of the maximal Roe algebra. The aim of this talk is to outline a proof of the fact that, if the manifold has uniformly positive scalar curvature, then this index vanishes. The argument relies on a subtle distinction between the usual Roe algebra and a uniform version of it; the latter allows one to avoid some of the analytical difficulties encountered when working with the former, in particular when trying to establish a version of the functional calculus in the maximal setting. This work is joint with Zhizhang Xie and Guoliang Yu. A Lichnerowicz vanishing theorem for the maximal Roe algebra Sponsored by the Meyer Fund
Apr. 18, 2019 3pm (MATH 350)	Probability Hoi Nguyen (Ohio State University) X We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve in flat tori are universal under various families of randomness. Based on joint work with M-C. Chang, O. Nguyen and V. Vu. Random eigenfunctions on flat tori: universality for the number of intersections
Apr. 22, 2019 3pm (Math 350)	Ulam Masoud Khalkhali (University of Western Ontario, London) X Our understanding of the idea of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative differential geometry started when a Gauss-Bonnet-type theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansion suggest a general way of defining local curvature invariants for noncommutative Riemannian-type spaces where the metric structure is encoded by a Dirac-type operator. To carry out explicit computations, one needs a quite intriguing new set of ideas that are completely absent in the classical setting. In this talk, I shall try to highlight what has happened in the subject by giving an account of the most recent developments. For more details, I refer to https://arxiv.org/abs/1901.07438 and references therein. Curvature in noncommutative geometry
Apr. 23, 2019 10am (MATH 350)	Geometry/Analysis Markus Pflaum (University of Colorado, Boulder) Introduction to Lie groupoids from a geometric point of view
Apr. 23, 2019 1pm (MATH 220)	Keith Kearnes (CU) Congruence n-permutability is not join prime, 2
Apr. 23, 2019 2pm (MATH 220)	Agnes Szendrei (CU Boulder) Flexible Ultrafilters and the Theory of the Random Graph, Part 2
Apr. 23, 2019 2pm (MATH 350)	Lie Theory Nathan Lindzey (CU) X We discuss a Hecke algebra associated with the permutation representation of the symmetric group acting on the set of matchings of the complete graph. This permutation representation is reminiscent of a more complicated representation that arises when deciding the semi-simplicity of Brauer algebras (see Hanlon & Wales), but our motivation comes from optimization, namely, symmetric semidefinite relaxations of the perfect matching problem. Realizing this Hecke algebra as a non-commutative matrix algebra, we give combinatorial formulas for many of the eigenvalues of operators in this algebra. This is joint work in progress with Gary Au and Levent Tunçel. Harmonic Analysis on Matchings
Apr. 24, 2019 3pm (Math350)	Math Phys Victor Gurarie (University of Colorado, Physics Department) Conformal Field Theory Part 2
Apr. 24, 2019 4pm (MATH 350)	Grad Student Seminar Taylor Klotz (TK) (CU Boulder) X You likely know of probability distributions and you may even know of distributions from analysis (or generalized functions) but did you know differential geometry also has distributions? Distributions are subbundles of the tangent bundle to a manifold and such objects encode all kinds of interesting geometry. We’ll introduce the notion of distributions, their counterpart exterior differential systems, and the Frobenius theorem (also of the third kind?). From there we’ll explore ways in which the Frobenius theorem fails and what kinds of geometry this puts us in contact with. Close Encounters of the Third Kind of Distribution
Apr. 25, 2019 2pm (MATH 350)	Functional Analysis Mahya Ghandehari (University of Delaware) X Beurling-Fourier algebras are analogues of the Beurling algebra in the non-commutative setting.These algebras for general locally compact groups were defined by Lee and Samei as the predual of certain weighted von Neumann algebras, where a weight on $\hat{G}$ is defined to be a suitable unbounded operator affiliated with the group von Neumann algebra. In this talk, we present the general definition of a Beurling-Fourier algebra, and discuss how their spectra can be identified. In particular, we determine the Gelfand spectrum of Beurling-Fourier algebras for some representative examples of Lie groups, emphasizing the connection of spectra to the complexification of underlying Lie groups. This talk is based on joint work with Lee, Ludwig, Spronk, and Turowska. On non-commutative weighted Fourier algebras Sponsored by the Meyer Fund
Apr. 29, 2019 3pm (Math 350)	Ulam Paul F. Baum (Penn State University, State College) X On a closed (i.e. compact, no boundary) odd-dimensional manifold, the index of any elliptic differential operator is zero. Thus, on odd-dimensional manifolds, interesting examples can only be obtained by dropping either "elliptic" or "differential". This talk will explain the two cases. If "differential" is dropped, then the examples are Toeplitz operators (which are elliptic pseudo-differential operators of order zero). A corollary is the result of Louis Boutet de Monvel about Toeplitz operators associated to the boundary of a strictly pseudo-convex domain. If "elliptic" is dropped, then examples are the differential operators on contact manifolds studied by E. van Erp, Baum-van Erp, and Gorokhovsky-van Erp. Index theory on odd-dimensional manifolds
Apr. 30, 2019 10am (MATH 350)	Geometry/Analysis Elaine Cozzi (Oregon State University) X Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. One can easily show that this is the case for sufficiently smooth solutions having spatial decay. In this talk, we consider a broader class of weak solutions with vorticity lacking spatial decay, and we show that such stability still holds. This is based on joint work with James Kelliher. Incompressible Euler equations and the effect of changes at a distance Sponsored by the Meyer Fund
Apr. 30, 2019 11am (MATH 350)	Number Theory Clayton Petsche (Oregon State University) X I will describe a certain two-parameter family of automorphisms of the p-adic affine plane. Each of these automorphisms admits a chaotic attractor, with non-integral Hausdorff dimension, on which it is topologically conjugate to a full two-sided shift map. The attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction; this measure is a p-adic analogue of the SRB measures in smooth dynamics. Attractors associated to a family of hyperbolic p-adic plane automorphisms
Apr. 30, 2019 12:10pm (MATH …	Kempner Anton Dzhamay (University of Northern Colorado) X Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study these systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. We also will describe some examples of applications. Geometry of Discrete Integrable Systems: QRT Maps and Discrete Painlevé Equations
Apr. 30, 2019 2pm (MATH 220)	Agnes Szendrei (CU Boulder) Flexible Ultrafilters and the Theory of the Random Graph, Part 3
Apr. 30, 2019 2pm (MATH 350)	Lie Theory Shawn Burkett (Kent State) X Given a normal subgroup $N$ of a finite group $G$ , we define the subgroup $V(G∣ N)$ to be the product of the vanishing-off subgroups of all irreducible characters of $G$ not containing $N$ in their kernel. The map $N↦V(G∣ N)$ gives an order-preserving, join-preserving map from the lattice of normal subgroups of $G$ to itself. Thus there is right adjoint $N↦U(G∣ N)$ so that these two maps give a Galois connection on the lattice of normal subgroups of $G$ . In this talk we will explore the properties of the subgroups $U(G∣ N)$ , with special attention given to the subgroup $U(G)=U(G∣ Z(G))$ , which is the largest normal subgroup $U$ of $G$ for which every character in $Irr(G∣ U)$ vanishes off of $Z(G)$ . We will discuss groups satisfying $U(G)>1$ , and see that these are essentially $p$ -groups. Of particular interest will be nested generalized vanishing center (nested GVZ) groups --- finite groups satisfying the following two properties: $•$ (nested) The centers of the irreducible characters form a chain; $•$ (GVZ) Every irreducible character vanishes off of its center. We will use the vanishing-off subgroups and their adjoints to construct central series that will be used to give new characterizations of nested GVZ groups. This is joint work with Mark L. Lewis (Kent State University). Characterizations of nested GVZ groups in terms of vanishing-off subgroups of characters Sponsored by the Meyer Fund
Apr. 30, 2019 3pm (MATH 350)	Algebraic Geometry Yuchen Liu (Yale University) X We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is joint work with Chenyang Xu. K-stability of cubic threefolds Sponsored by the Meyer Fund
Apr. 30, 2019 4pm (MATH 350)	Topology Katharine Adamyk (CU Boulder) Madsen Tillman Spectra
May. 01, 2019 3pm (Math350)	Math Phys Ravishankar Sundararaman (Rensselaer Polytechnic Institute) X Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy. In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT. Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
May. 01, 2019 3pm (MATH 350)	Topology Ravishankar Sundararaman (Rensselaer Polytechnic Institute) X Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy. In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT. Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
May. 01, 2019 4pm (MATH 350)	Grad Student Seminar Yu Wang (CU Boulder) X Symplectic manifolds are a special kind of manifolds which have rich applications in physics and mathematics. Roughly speaking, a symplectic manifold is a smooth manifold equipped with a symplectic form, which makes mathematicians only need to study global properties of the manifold. Combining with a compatible almost complex structure, it will give a math version of mirror symmetry conjecture from string theory. Most modern geometry are built up on Kahler manifolds and Calabi-Yau manifolds that can be gotten from giving more analytic structures to a symplectic manifold. In the talk, I am going to roughly talk about basic properties of symplectic manifolds, complex manifolds and Kahler manifolds. Eventually, I am gonna mention a very famous theorem called Hodge decomposition of Kahler manifolds and a famous conjecture named “Homological Mirror Symmetry”. From symplectic manifolds to Kahler manifolds
May. 02, 2019 3pm (MATH 350)	Probability Steven N. Evans (Departments of Statistics and Mathematics, University of California at Be…) X The $α$ -Lipschitz minorant of a function is the greatest $α$ -Lipschitz function dominated pointwise by the function, should such a function exist. I will discuss this construction when the function is a sample path of a (2-sided) L\'evy process. The contact set is the random set of times when the sample path touches the minorant. This is a stationary, regenerative set. I will provide a description of the excursions of the sample path away from the contact set that is analogous to It\^o's theory for the excursions of a Markov process away from some point in the state space. In particular, I will elucidate the probabilistic structure of both a ``generic'' excursion and the special excursion that straddles the time zero. Lipschitz minorants of L\'evy processes
May. 06, 2019 2pm (MATH 350)	Lie Theory Julianne Rainbolt (Saint Louis University) X A possible Supercharacter Theory based on the degrees of the irreducible characters of a finite group will be discussed. A proof that this Supercharacter Theory applies for certain families of finite groups but does not apply for other families will be outlined. Supercharacter Theory and Degrees of the Irreducible Characters of a Group
May. 06, 2019 3:30pm (MATH 3…	Topology Jonathan Wise (CU Boulder) Ansatz
Jun. 03, 2019 1pm (350)	Ulam Jonathan Belcher (CU Boulder) X The connection between Hochschild and cyclic cohomologies with generalized De Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al. Here we generalize these cohomology theories even further, essentially creating a theory that establishes a step-wise bridge between the two. Motivation for this construction comes from trying to generalize the Hochschild-Kostant-Rosenberg-Connes theorem to manifolds with boundary, and applications in tracial constructions in certain classes of pseudodifferential operators. Further geometric and topological interests of this theory include extending Chern-Weil theory to manifolds with boundary via pairings between bridge cohomology and higher K-theories. Bridge cohomology: a generalization of Hochschild and cyclic cohomologies with applications to Chern-Weil theory
Jun. 20, 2019 2pm (MATH 350)	Topology Henry Adams (Colorado State University) X This talk is an introduction to computational topology, as applied to data analysis and to sensor networks. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets. As a second application, I will describe how topology has been applied to coverage problems in mobile sensor networks. An introduction to applied topology

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