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Events for November 2018 – March 2019

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Nov. 06, 2018 11am (MATH 350)	Number Theory Christelle Vincent (University of Vermont) X The shape of a number field is an invariant which refines the discriminant. In this talk we present Piper H's PhD thesis on the equidistribution of shapes of S_3-, S_4-, and S_5-number fields when they are ordered by discriminant. We then discuss joint work that is in progress on the joint distribution of the shapes of a field and its resolvent. This talk will be accessible to anyone who knows some Galois theory as well as the definition of a number field and its ring of integers. On the equidistribution of the shape of certain number fields
Nov. 06, 2018 1pm (MATH 220)	Grad Algebra/Logic Election Day! Vote!
Nov. 06, 2018 2pm (MATH 220)	Election Day! Vote!
Nov. 06, 2018 2pm (MATH 350)	Lie Theory Henry Kvinge (CSU) X First formally defined by Borodin and Olshanski, a coherent system on a graded graph is a sequence of probability measures which respect the action of certain down/up transition functions between graded components. In one common example of such a construction, each measure is the Plancherel measure for the symmetric group and the down transition function is induced from the restriction functor. In this talk we will show how to generalize the above framework to the case of any free Frobenius tower where we no longer assume that our algebras are semisimple. In particular, we describe two coherent systems on graded graphs defined via representation theory of a tower of algebras and connect one of these systems to a family of central elements of the tower. When the algebras are not semisimple, the resulting coherent systems reflect the duality between simple modules and indecomposable projective modules. Coherent systems of probability measures on graphs for representations of free Frobenius towers Sponsored by the Meyer Fund
Nov. 06, 2018 3pm (MATH 350)	Algebraic Geometry Jeff Achter (CSU) X Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients. I'll use recent advances in the theory of K3 surfaces to construct arithmetic period maps which explain these uniformizations. Arithmetic moduli for lattice-polarized K3 surfaces, or, Three views of six points Sponsored by the Meyer Fund
Nov. 06, 2018 4pm (MATH 350)	Topology Alex Mundey Dirac operators
Nov. 07, 2018 4pm (MATH 350)	Grad Student Seminar Jonathan Quartin (CU Boulder) X The Inverse Galois Problem is easily stated— “does every finite group occur as the automorphism group of some finite Galois extension of ” While this is still unsolved, if we replace by , the statement is true. The proof of this fact highlights an interesting boundary between Algebra and Geometry. In this talk, I will discuss Complex manifolds and Covering spaces—two of the key ingredients that go into the proof. A geometric approach to a 200 year old problem in Algebra
Nov. 08, 2018 3pm (MATH 350)	Probability Indrajit Jana (Temple) X In this talk, we discuss about a CLT for the linear eigenvalue statistics of non-Hermitian random band matrices. Let is a random matrix with iid entries, and be it's eigenvalue. Define the linear eigenvalue statistics . Rider, and Silverstein '06 had shown that the unscaled and centered with analytic converges to a Gaussian distribution. The variance of the limiting distribution is completely determined by the test function only. We extend that result for non-Hermitian random band matrices. In case of band matrices, the linear eigenvalue statistics needs to be scaled properly. In addition, the variance of the limiting Gaussian distribution depends on the test function, as well as the growth rate of the bandwidth. Linear eigenvalue statistics of non Hermitian random band matrices
Nov. 13, 2018 1pm (MATH 350)	Ulam Markus Pflaum (CU Boulder) X The inertia space of a compact Lie group action or, more generally, of a proper Lie groupoid, has an interesting singularity structure. Unlike the quotient space of the group action, respectively the groupoid, the inertia space cannot be stratified by orbit types, in general. In this talk, we explain this phenomenon. We also indicate a connection between the inertia space and the non-commutative geometry of the underlying groupoid in terms of basic relative Grauert-Grothendieck forms on the inertia space. We finally pose a conjecture about the Hochschild homology of the convolution algebra of the transformation groupoid of a compact Lie group action. The talk is based on joint work with with C. Farsi and Ch. Seaton (singularity theory of inertia groupoids) and with H. Posthuma and X. Tang (Grauert-Grothendieck forms and Hochschild homology of convolution algebras). Inertia groupoids and a conjecture on the Hochschild homology of the convolution algebra of a compact Lie group action
Nov. 13, 2018 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr (CU) X We discuss Rosenberg's classification of maximal clones on finite sets and its consequences for the structure of finite algebras with WNU. CSP maximal clones
Nov. 13, 2018 2pm (MATH 220)	Agnes Szendrei (CU Boulder) Cofinality Spectrum Problems, Part 1: A motivating example by Malliaris and Shelah
Nov. 13, 2018 3pm (MATH 350)	Algebraic Geometry Gregory Pearlstein (Texas A&M University) X The theory of normal functions and the Hodge conjecture have their origin in the study of algebraic cycles by Lefschetz and Poincare. I will sketch the history of the subject and discuss some of my recent work on singularities of normal functions to the Hodge conjecture and the zero locus of a normal function to a conjectural filtration of Bloch and Beilinson. Normal Functions and the Hodge Conjecture Sponsored by the Meyer Fund
Nov. 14, 2018 3pm (Math350)	Math Phys Magdalena Lottner On causal fermionic systems, continued
Nov. 14, 2018 4pm (MATH 350)	Grad Student Seminar André Davis (CU Boulder) X Did you know that one can deduce the irrationality of nth roots of 2 from Fermat's last theorem? Or that Cayley's theorem (every group is isomorphic to a subgroup of a symmetric group) follows as a special case of the Yoneda lemma? In this lighthearted talk we will prove a multitude of theorems using unconventional techniques. The only criteria is that the proof must be short, beautiful, and unexpected. We will end with a proof from THE BOOK. Swatting Flies With a Sledgehammer
Nov. 15, 2018 2pm (MATH 350)	Functional Analysis Ulrik Enstad (University of Oslo) X In time-frequency analysis, some of the central objects of study are Gabor frames. A sequence consisting of time-frequency translates of a single function in is called a Gabor system. The Balian-Low theorem, a fundamental theorem about Gabor systems, implies that for sufficiently nice , Gabor systems can never be orthonormal bases. A solution to this is to settle with (Gabor) frames, which can be overcomplete in general. The theory of Gabor frames is available in for any locally compact abelian group , but has so far been investigated mostly for ``elementary'' groups such as , , , and products of these. As first discovered by F. Luef, Gabor frames in are related to M. Rieffel's Heisenberg modules over twisted group -algebras. In this talk I will show how the Balian-Low theorem, which concerns , might or might not generalize to , and that the answer depends on the structure of certain Heisenberg modules, or equivalently, certain line bundles associated to . I will also present joint work with Luef and M. Jakobsen on the existence of Gabor frames in , with being either the adeles over the rationals or the group .? Gabor frames, Heisenberg modules and vector bundles
Nov. 15, 2018 3pm (MATH 350)	Probability Lu Wei (University of Michigan-Dearborn) X Quantum bipartite model is a standard model in describing the interaction of a physical object and its environment for different quantum systems. We wish to understand the degree of entanglement of such systems as measured by von Neumann entropy via its exact moments. The problem of computing the moments of von Neumann entropy was first considered by Don Page in 1994, where a formula for the first moment was conjectured. Page’s conjecture was proved independently by a number of works in the years followed. Recently, an expression for the second moment was conjectured by Vivo, Pato, and Oshanin, which was subsequently proved by us last year. In this talk, we will first review the computations for the first two moments. We will then show how our approach can be extended to obtain the third moment and beyond. Moments of von Neumann Entropy of Quantum Bipartite Systems: Results in Progress and in Perspectives
Nov. 27, 2018 11am (MATH 350)	Number Theory Özlem Ejder (CSU) X The points on the modular curve X_1(n) (roughly) classifies the pairs (E,P) up to isomorphism where E is an elliptic curve and P is a point of order n on E. We call a closed point x on X_1(n) sporadic if there are only finitely many closed points of degree at most deg(x); hence classifying sporadic points on X_1(n) is closely related to determining the torsion subgroups of elliptic curves over a degree d field. When d=1 or 2, Mazur and Kamienny's work show that there are no sporadic points of degree d on X_1(n). In this talk, I will discuss that the sporadic j-invariants of bounded degree is finite. This is joint with A. Bourdon, Y. Liu, F. Odumudu and B. Viray. Sporadic points on modular curves
Nov. 27, 2018 12:10pm (MATH …	Kempner Xudong Chen (CU Boulder) X Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of individual control systems. As a dual, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the (continuum) ensemble. We introduce in the talk a novel class of ensembles of nonlinear control systems, termed distinguished ensemble systems. Every such system has two key components, namely a set of finely structured control vector fields and a set of co-structured observation functions. In the first half of the talk, we demonstrate that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a principle for ensemble system design. In the second half of the talk, we address the issue about existence of a distinguished ensemble system for a given manifold. We will focus on the case where the state space of every individual control system is an arbitrary semi-simple Lie group or its homogeneous space. Structure Theory for Ensemble Control and Estimation of Nonholonomic Systems
Nov. 27, 2018 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr X We discuss the final steps in Zhuk's algorithm: reductions for absorbing subuniverses, centers, polynomially complete quotients and linear quotients. CSP linear quotients 1
CANCELED Nov. 27, 2018	Ulam Erik van Erp (Dartmouth College) X This is joint work with Robert Yuncken. We developed a coordinate-free geometric definition and a unified approach to various pseudodifferential calculi. The starting point of the construction is the identification of an appropriate groupoid, which is a purely geometric object. These groupoids are suitable adaptations of Connes' tangent groupoid. We show how, once the appropriate groupoid has been constructed, the details of the calculus are developed on its foundation. Our construction recovers the classical pseudodifferential calculus as well as, for example, the Heisenberg calculus (on contact manifolds) and its generalizations (to filtered manifolds). A groupoid approach to Pseudodifferential calculi
Nov. 27, 2018 2pm (MATH 220)	Agnes Szendrei (CU Boulder) Cofinality Spectrum Problems, Part 2: A motivating example by Malliaris and Shelah
Nov. 27, 2018 3pm (MATH 350)	Topology Paul Lessard X The simplex category, the category of finite totally ordered sets, is a strict test category, meaning that there exists a product preserving Quillen equivalence between simplicial sets, presheaves on the simplex category, and "nice topological spaces". It is not clear however how the combinatorics of the simplex category relate to spectra, the objects which represent cohomology theories for spaces. All the well known models of spectra, Adams spectra, symmetric spectra, L-spectra, etc. may or do use simplicial sets as their presentation for spaces, but only one almost forgotten model, Kan's semisimplicial spectra of 1963 leverage the combinatorics of the simplex category for elegance and insight. We will begin by asking why the simplex category is a strict test category and why the the category of globes is not. We will then then develop a strict test category, the theta category of Joyal, and construct an elegant suspension functor for cellular sets, presheaves on the category theta. We will then present a conjectured new model of spectra based on this category which recovers the best aspects of Kan's semisimplicial spectra. We will also comment on how this model of spectra may provide a universal algebraic lens to the distinction between stable and unstable homotopy theory. A Cellular Suspension Functor for a New Presentation of Spectra
Nov. 27, 2018 4pm (MATH 350)	Topology Andrew Stocker Analytic K-homology
Nov. 28, 2018 3pm (Math350)	Math Phys Daniel Spiegel Symmetries in QFT and Bargmann's theorem
Nov. 28, 2018 4pm (MATH 350)	Grad Student Seminar Mark Pullins (CU Boulder) X Groups of finite Morley rank can be viewed as infinite groups equipped with a notion of dimension. In this talk, we’ll enter the subject rapidly via the axiomatic definition introduced by Borovik, discuss some of the major tools used to study these groups (using only very basic model theory), and present current avenues of research along with major obstructions. Recent study has involved tools inspired by finite group theory, algebraic geometry, Lie groups, and model theory - so there should be something of interest to (almost) everyone! A Brief Survey of Groups of Finite Morley Rank
Nov. 29, 2018 12pm (Math 350)	Jun Hong X As we all know, the secret to giving a Good Talk is to not give a Bad Talk. Jun Hong has volunteered to give an example of what not to do. While the mathematics will be correct, in 20 minutes Jun will strive to leave us bored, annoyed, and more confused on the subject than before we came in. We will then discuss what happened and workshop ideas for avoiding some of the most common pitfalls. Improve your presentation skills: Workshop on how to avoid giving a bad seminar talk
Dec. 04, 2018 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr CSP linear quotients 2
Dec. 04, 2018 2pm (MATH 220)	Agnes Szendrei (CU Boulder) Cofinality Spectrum Problems, Part 3
Dec. 04, 2018 2pm (MATH 350)	Lie Theory Spencer Gerhardt (CSU) X Let be conjugacy classes of a simple algebraic group defined over a sufficiently large field . We consider the following question: what are necessary and sufficient conditions for the existence of a tuple ? A solution to this problem has several applications, including generic stabilizers in linear representations of algebraic groups, and random generation of finite groups of Lie type. Results and outstanding cases are discussed. Some work in this talk is joint with Robert Guralnick and Tim Burness. Topological generation of algebraic groups and applications
Dec. 04, 2018 4pm (MATH 350)	Topology Robin Deeley The Atiyah-Singer index theorem a la Paul Baum
Dec. 05, 2018 4pm (MATH 350)	Grad Student Seminar Sangman Lee (CU Boulder) X Cantor's pairing function is a well-known bijection used in his proof that rational numbers are denumerable. We will have a brief survey on Cantor's pairing function and discuss open problems on this topic (just not to look too simple). During the talk, we will have some fun with a math puzzle regarding paring functions and, more importantly, a cat. We will also explore what happens if we generalize the puzzle: Cantor's cat will dance on a donut, a ball, and a Möbius strip. Playing with Cantor's Cat
Dec. 06, 2018 3pm (MATH 220)	Topology Prasit Bhattacharya (The University of Virginia) X Connective topological modular form has led to a lot of important calculations in chromatic homotopy theory. One of the key features of is that its cohomology as a module over Steenrod algebra is isomorphic to , where is a subalgebra of A generated by . On the other hand, is an exterior element of . For any A-module one can define a chain complex where the action of acts as a differential and its homology is the Margolis homology . The calculation Margolis homology is not straightforward as the action do not follow Leibniz rule. In this talk, we will develope a new technique that will allow us to calculate Margolis homology. In particular we will give a complete answer for Margolis homology of . The P_2^1 Margolis homology of the connective topological modular form.
Dec. 11, 2018 10am (Math 350)	PDE/Analysis Taylor Klotz (CU Boulder) X Given a nonlinear control system with symmetry one might want to ask if there is a related system that is linearizable and whose solutions can be used to construct solutions to the original nonlinear system. One ad hoc technique is to attempt to add additional differential equations to the system and hope for a new linearizable system. This is currently known as Dynamic Feedback Linearization. In a different direction, if we assume the nonlinear system has symmetries then one can form a smaller system via a quotient by the symmetries. If this system is linearizable then its solutions can be used to construct solutions to the original system. This talk will explore recent work on the connection between these two solution methods and how the latter sheds light on a potential theory of the former. Cascade Linearization and Dynamic Feedback Linearization of Control Systems with Symmetry
Dec. 11, 2018 11am (MATH 350)	Number Theory Dan Stroock (MIT and CU) Continued Fractions and Pell's Equation
Dec. 11, 2018 12:10pm (MATH …	Kempner Florian Pfender (CU Denver) X The flag algebra method is a technique recently developed by Razborov, and it has revolutionized parts of extremal combinatorics. The method draws from the theory of graph limits, analysis, probability and optimization. We will quickly present the fundamental ideas and then show a number of diverse applications. Somewhat counter intuitively, one can even use it to show statements about relatively small graphs, and we will show how it can be used to find new bounds on graph Ramsey numbers. joint work with B. Lidicky Semidefinite programming in extremal graph theory -- advances on the flag algebra method
Dec. 11, 2018 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr CSP linear quotients 3
Dec. 11, 2018 2pm (MATH 220)	Trevor Jack (CU Boulder) X Given generators for finite transformation semigroups and finite matrix semigroups, we will give explicit polynomial time algorithms for finding all of the left and right identities for the generated semigroups. Computing Left and Right Identities in a Semigroup
Dec. 13, 2018 3pm (MATH 350)	Probability Tulasi Ram Reddy (New York University, Abu Dhabi) X We consider the primary Brownian loop soup (BLS) layering vertex fields and show the existence of the fields in smooth bounded domains for a suitable range of parameters 's. To show this at a fixed cutoff, we use Kahane's theory of Gaussian multiplicative chaos. On the other hand, when the cutoff is removed, we use Weiner-Ito chaos expansion to establish that the limit as the intensity of the BLS diverges and goes to 0 such that is constant, is a complex Gaussian multiplicative Chaos. ( Based on joint work with F. Camia, A. Gandolfi and G. Peccati) Gaussian multiplicative chaos limit of the Brownian loop soup Poisson layering fields.
Jan. 22, 2019 12:10pm (MATH …	Kempner Zhouli Xu (MIT) X Based on work of Kervaire-Milnor, Browder, and Hill-Hopkins-Ravenel, Guozhen Wang and I recently proved that the 61-sphere has a unique smooth structure, and it is the last odd dimensional case: The only odd dimensional spheres that have a unique smooth structure are the ones in dimensions 1, 3, 5 and 61. The proof is a computation of stable homotopy groups of spheres. In recent work with Dan Isaksen and Guozhen Wang, we developed a new method that can be used to compute stable homotopy groups of spheres in a much more effective way than known ones. This new method uses motivic homotopy theory, which was introduced by Morel and Voevodsky to study algebraic geometry. Smooth structures on spheres, stable homotopy groups of spheres, and motivic homotopy theory
Jan. 22, 2019 3pm (MATH 350)	Topology Danny XiaoLin Shi (Harvard University) X We show that Lubin-Tate spectra at the prime are Real oriented and Real Landweber exact. The proof is an application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height , we compute the entire homotopy fixed point spectral sequence for with its -action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these -fixed points. Then, I will talk about the slice spectral sequence of a -equivariant spectrum. This spectrum is a variant of the detection spectrum of Hill--Hopkins--Ravenel and is very closely related to the height 4 Lubin--Tate theory. Real Orientations of Lubin-Tate Spectra
Jan. 22, 2019 4pm (MATH 350)	Topology Zhouli Xu (MIT) X A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi. The geography problem on 4-manifolds: 10/8 + 4
Jan. 23, 2019 4pm (MATH 350)	Grad Student Seminar Sebastian Bozlee (CU Boulder) X Injective modules are important objects in homological algebra and algebraic geometry. We will give the standard definition, then we will show that there is an alternative, more concrete characterization resembling the definition of an algebraically closed field. Injective modules are algebraically closed
Jan. 24, 2019 12:10pm (MATH …	Kempner Sooran Kang (Chung-Ang University) X Quantization is known as a way of constructing (noncommutative) quantum mechanical system from (commutative) classical mechanical system, mathematically forming operators from ordinary functions. The word `` quantize'' means ``discretize'' and we discuss where it came from by reviewing the motivation and background of quantum mechanics. Then we discuss mathematical formulation of (deformation) quantizations and introduce two -algebras, noncommutative two tori and quantum Heisenberg manifolds, which are main examples of Rieffel's strict deformation quantizations. We also introduce noncommutative (or quantum) Yang-Mills theory developed by Connes and Rieffel on noncommutative tori, and then review some of the results of Yang-Mills connections on noncommutative two tori and quantum Heisenberg manifolds if we have time. Introduction to deformation quantizations and noncommutative Yang-Mills theory
Jan. 24, 2019 2pm (MATH 350)	Functional Analysis Palle Jorgensen (University of Iowa) X The study of representations of higher rank-graph algebras, and related non-commutative structures, is of interest in its own right. It is also part of wider themes of analysis of non-commuting geometries. The talk will focus on systems that arise in such applications as representation theory, symbolic dynamics, sub-band filters from signal processing and wavelet analysis. We shall give an account of joint results, with many co-authors, on the use of representations of the Cuntz relations O_N, and their generalizations, Cuntz-Krieger, higher-rank, etc. Related applications include a class of filter problems used in the analysis of fractals, and in geometric measure theory. By their nature, these representations reflect intrinsic self-similarity; and thus serve ideally to encode sub-bands, and more generally iterated function systems (IFSs), their dynamics, and their associated measures. At the same time, these representations offer a new harmonic analysis of signals. A key analytic tool will be classes of “transfer operators,” arising in such diverse areas as symbolic dynamics, analysis of Markov processes, and their associated path-space measures. The role of path-space and path-space measures in symbolic dynamics, in multi-resolutions (such as generalized wavelets), and in representation theory Sponsored by the Meyer Fund
Jan. 28, 2019 3pm (Math 350)	Ulam Piotr M. Hajac (CU Boulder / IMPAN) X We define an admissible decomposition of a graph E into subgraphs E_1 and E_2, and consider the intersection graph Fas a subgraph of both E_1 and E_2. We prove that, if the graph E is row finite and its decomposition into the subgraphs E_1 and E_2 is admissible, then the graph C-algebra C(E) of E is the pullback C-algebra of the canonical surjections from C(E_1) and C(E_2) onto C(F). Pullbacks of graph C*-algebras from admissible pushouts of graphs
Jan. 29, 2019 11am (MATH 350)	Number Theory Antoine Marmat and Nikolay Moshchevitin (Univ. Graz and Moscow State Univ.) X We will discuss the problem of optimal inequalities between the ordinary and the uniform Diophantine exponents, for simultaneous approximation as well as for approximation with one linear form. The first result, dealing with approximation to two numbers, was obtained by V. Jarnik in 1940. Quite recently W,M. Schmidt and L. Summerer introduced a new method of Parametric Geometry of Numbers which gave new understanding of related problems. In particular they formulated a conjecture about the optimal bounds between exponents.We will speak about a proof of this conjecture and related topics Uniform and ordinary Diophantine exponents
CANCELED Jan. 29, 2019	Kempner Catherine Sulem (University of Toronto) X The Derivative Nonlinear Schrodinger (DNLS) equation is a long-wave, weakly non-linear model that arises in the context of the Hall magnetohydrodynamic equations for conducting fluids. It was known since the work of Hayashi (1992) that solutions exist globally in time for small initial data but the case of large data remained open. Exploiting the complete integrability of the DNLS equation discovered by Kaup and Newell (1978), we prove global wellposedness for general initial conditions in weighted Sobolev spaces. Furthermore, for initial conditions that support bright solitons (but exclude spectral singularities), we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. This property is usually referred to in the literature as the soliton resolution conjecture. This is a joint work with Robert Jenkins (Colorado State University), Jiaqi Liu (University of Toronto) and Peter Perry (University of Kentucky). Global wellposedness and soliton resolution for the derivative nonlinear Schrodinger equation
Jan. 29, 2019 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr (CU Boulder) X We show that solvable quotients of subdirect products of perfect groups are nilpotent of arbitrary degree. This answers a question by Derek Holt. Joint work with Keith Kearnes and Nik Ruskuc. Subdirect products of perfect groups
Jan. 29, 2019 2pm (MATH 220)	Michael Wheeler (CU Boulder) X Until recently, the question of whether p < t is consistent with ZFC had been the longest open problem regarding cardinal invariants of the continuum. Malliaris and Shelah proved that p < t is inconsistent and so p = t for all models of ZFC. In these talks we will present a simplified version of Malliaris and Shelah's argument due to Douglas Ulrich. In this first week, we define omega-nonstandard models of ZFC, give some sense of what these models look like, and give a brief outline of the proof that p = t. We will present the basic definitions needed for the argument, and, time permitting, will present the result that "small" types of omega-nonstandard models that are consistently realized in a given pseudofinite set are always realized. p = t via Nonstandard Models of Set Theory, Part 1
Jan. 29, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) X I will give some background on topological field theories with a view towards the Freed-Hopkins-Teleman results on invertible field theories. This will partly be an organizational meeting and talks will be assigned for the rest of the semester. Topological field theories - Intro
Jan. 30, 2019 3pm (Math350)	Math Phys Mike Hermele (University of Colorado, Physics Department) Basics of quantum mechanics from a physicist’s point of view
Jan. 30, 2019 4pm (MATH 350)	Grad Student Seminar Piotr M. Hajac (CU Boulder / IMPAN) X Operator algebras are the language of quantum mechanics just as much as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. One of these endeavours is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C-algebras) and locally compact Hausdorff spaces. Thus noncommutative C-algebras are thought of as quantum topological spaces, and are researched from this perspective. However, such C-algebras can enjoy features impossible for commutative C-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new ("quantum") intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces, and how it explains its K-theory. Operator Algebras that One Can See
Jan. 30, 2019 5pm (Math 350)	Math Club Danny Moritz (CU Boulder) X A web through time and space exploring the Geodesic Dome’s historical origins, structural facets, a middle schooler’s construction using Euler Paths, Euler’s formula for polyhedra with two interesting proofs, and extensions into Fullerenes reconsidered with signatures and adjacency matrices. Geodesic Domes, Graph Theory, and Beyond
Jan. 31, 2019 3pm (MATH 350)	Probability Zhen-Qing Chen (University of Washington) X Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. I will then present some recent results in this area, in particular on the probabilistic representation to the solutions of time fractional equations with source terms. Two-sided estimates on the fundamental solutions will also be given if time permits. Anomalous diffusions and fractional order differential equations
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)	Grad Algebra/Logic 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 220)	Michael Wheeler (CU Boulder) p = t via Nonstandard Models of Set Theory, Part 2
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 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)	Grad Algebra/Logic 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 is a locally compact Hausdorff abelian group, and and are strongly-continuous unitary representations of and respectively on a Hilbert space that satisfy the Weyl Commutation Relation, i.e., for all and , then the triple must be unitarily equivalent to a direct sum of copies of the triple , where is the unitary representation of on by left translations, and is the unitary representation of on by phase modulations. In this talk, we will present a new generalization of the Stone-von Neumann Theorem that, for any Hilbert space and any -dynamical system of the form , classifies up to unitary equivalence all quadruples where is a Hilbert -module, is a non-degenerate -representation of on by adjointable operators, and and are strongly-continuous unitary representations of and respectively on that satisfy the Weyl Commutation Relation as well as the following two commutation relations: (i) for all and , and (ii) for all and . 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 , let . Let be a number field, let , and let be non-zero algebraic numbers. We solve the problem of the existence of such that there are infinitely many tuples satisfying where and having small logarithmic height compared to . In the special case when have the form for fixed , our work yields results on algebraic approximations of of the form with and . Various results on linear recurrence sequences also follow as an immediate consequence. The case and 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)	Math Club 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)	PDE/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 defined recursively by , , and for . On specialization of and one can recover the Fibonacci numbers, the nonnegative integers, and the -integers . 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 in the expression with . It is then natural to ask if the resulting rational function is actually a polynomial in and 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)	Math Club 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)	PDE/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)	Grad Algebra/Logic 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)	Math Club 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, , and take as input an algebra homomorphism between two finite -algebras and , along with an intermediate finite -algebra . The decision problem asks whether there are homomorphisms and such that . 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)	Grad Algebra/Logic 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 -mers, strings of length 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 -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 -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

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