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Events for October 2021 – February 2022

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Oct. 05, 2021 1pm (Zoom)	Philipp Rothmaler (CUNY) X A partition of the set of unary positive primitive (pp) formulas for modules over an associative ring into four regions will be presented. These four types of formula have a bearing on various structural properties of modules, a few instances of which will be discussed in the talk. Domains, specifically Ore domains, turn out to play a prominent role. One of the four types of formula are called high. These are used to define Ulm submodules and Ulm length of modules over an arbitrary associative ring. Pure injective modules turn out to have Ulm length at most 1 (just as in abelian groups). As a consequence, pure injective modules over RD domains (in particular, pure injective modules over the first Weyl algebra over a field of characteristic 0) decompose into a largest injective and a reduced submodule. High and low formulas
CANCELED Oct. 05, 2021	Math Edu Jess Ellis Hagman (Colorado State University) X In this talk I will share my current research that addresses student success in STEM from a critical perspective. A critical perspective on student success means locating student challenges in STEM not within the students' themselves but within the systems at play, and that these systems are failing students not because they are broken but because they are functioning perfectly well based on their design. I will share my research teams' current work focused on student experiences in college precalculus and calculus from a critical perspective, discussing both recent qualitative research and quantitative research studies. I care about these courses personally because I think the content is fun, but more importantly because of the role they play in STEM students' college experiences. This talk is intended for a broad audience to gain a new perspective on student success in STEM and learn a bit about research coming from this perspective. Speaker bio: Jess Ellis Hagman is an Associate Professor in the Department of Mathematics at CSU. She completed her PhD in Mathematics Education from the joint program between San Diego State University and the University of California, San Diego. Her area of research is undergraduate mathematics education. Her work is focused on dramatically increasing the number and diversity of people who succeed in undergraduate mathematics-especially introductory mathematics courses that often function as a roadblock for STEM intending students. Her current research includes studying characteristics of successful precalculus and calculus programs, focusing on investigating ways departments can create diverse, equitable, and inclusive introductory mathematics programs. Talking Critically about Student Success in STEM
Oct. 06, 2021 4pm (Math 350)	MathClub Markus Pflaum (CU-Boulder) X The question "Can one hear the shape of a drum?" was posed by Marc Kac in 1966 in an article in the American Mathematical Monthly. This question has become a central one in geometric spectral theory, a particular discipline within modern analysis, and could be solved only 30 years later by Carolyn Gordon and David Webb. In the talk the mathematics underlying this famous question will be explained together some modern aspects and developments. And not to forget, the answer will be given at the end of the talk together with a video of a short performance of a famous scientist and bongo player who lectured on the same question. Can one hear the shape of a drum?
Oct. 06, 2021 5pm (MATH 350)	Grad Student Seminar Chase Meadors (CU Boulder) X Some of you may know of Hilbert's famous hotel, but how do you get there from the airport? You take the transfinite subway, which has stations at every ordinal number up to the first uncountable cardinal \aleph_1, the stop at which Hilbert's hotel is located. At each station, only 1 passenger gets off the train, while a countable infinity of passengers get on. How many passengers will arrive at Hilbert's hotel? This talk will introduce basic set-theoretic concepts of ordinals and cardinals, their theory, and a key theorem that will help us get a handle on this situation, hopefully convincing you that \aleph_1 is much weirder than you may or may not have previously thought. Time permitting we will explore how this weirdness also leads to some counterexamples in other fields like topology. The transfinite subway to Hilbert's Hotel
Oct. 07, 2021 11am (Zoom)	Geometry/Analysis Padi Fuster Aguilera (CU Boulder) X We study a particular PDE model for chemotaxis with logarithmic sensitivity and logistic growth. We obtain existence and uniqueness of solutions as well as results for the zero-diffusion limit of the solutions with Neumann boundary conditions. We also show the formation of a boundary layer for the problem with dynamic boundary conditions, as well as some stability results. Numerical simulations will bring some light about the time-scale parameters as well as possible extensions on the problem. This is joint work with Kyle Zhao, Vincent Martinez and Ricardo Cortez. https://cuboulder.zoom.us/j/96510976623 A PDE model for chemotaxis with logarithmic sensitivity and logistic growth
Oct. 07, 2021 3pm (Zoom)	Probability Evgeni S. Dimitrov (Columbia University) X Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability. Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others. Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel. Gibbsian line ensembles and beta-corners processes
Oct. 12, 2021 1pm (Zoom)	Anna Zamojska-Dzienio (Warsaw University of Technology, Poland) X The Yang-Baxter equation is a fundamental equation occurring in integrable models in statistical mechanics and quantum field theory. Description of all possible solutions seems to be extremely difficult and therefore there were some simplifications introduced (V.G. Drinfeld 1992). Biracks are algebras studied in low-dimensional topology which are in a one-to-one correspondence with set-theoretical, non-degenerate solutions to the Yang-Baxter equation. The use of the language of biracks allows us to apply universal algebra tools. In this talk, we describe the generalized retraction relation on a birack which gives new classes of solutions. This is joint work with Premysl Jedlicka and Agata Pilitowska. Biracks and solutions of the Yang-Baxter equation
Oct. 13, 2021 5pm (MATH 350)	Grad Student Seminar Emily Montelius (CU Boulder) X Join us for an interactive talk exploring the wonderful world of tiling. Rather than spending an hour thinking about complex and abstract ideas, come and play with dominoes as we discover the fun properties they have. Together we will work to find or disprove the existence of fault free tilings on rectangles, cylinders, tori, and Möbius strips. Fault Free Tileability of Rectangles, Cylinders, Tori, and Möbius Strips with Dominoes
Oct. 14, 2021 2pm (MATH 350)	Functional Analysis Frederic Latremoliere (University of Denver) X Spectral triples, introduced by A. Connes in 1985, have emerged as the preferred generalization of Riemannian geometry to the noncommutative realm, and provide a new formalism for the study of problems form quantum physics to fractal geometry. Many problems in physics and geometry suggests that a quantification on "how close" spectral triples can be from each others would prove very helpful; however, only heuristics can be found in the literature. Examples of problems where such a notion could prove useful include the convergence of matrix models in mathematical physics, or the geometry of fractal sets constructed, in a natural manner, as limits of simpler spaces. In this talk, we will survey our construction of an actual metric, up to unitary equivalence, on a large class of spectral triples, which indeed provides a framework to discuss such problems . Our construction starts with our work in noncommutative metric geometry, spurred by the work of Rieffel, and our construction of a noncommutative analogue of the Gromov-Hausdorff distance on the class of quantum compact metric spaces. From this initial step, we will see how to enhance our construction to define convergence for Hilbert modules over quantum compact metric spaces, convergence of certain dynamics over quantum compact metric spaces and their modules, and, as a synthesis of these various tools, how to define our metric on spectral triples. A geometry for the space of metric spectral triples Sponsored by the Simons Foundation
Oct. 19, 2021 1pm (Zoom)	Marcel Jackson (La Trobe University, Melbourne) X It is well-known and easy to prove that the variety of groups abstractly captures algebras of permutations under composition and inverse, that the variety of inverse semigroups capture algebras of partial injective functions under composition and inverse, and that the variety of semigroups abstractly capture the algebras of any of total functions, partial functions or binary relations under the operation of composition. In contrast to this, a landmark result of Hirsch and Hodkinson showing the undecidability of determining when a finite algebra is isomorphic to an algebra of binary relations under Tarski’s signature: the usual set theoretic Boolean operations, composition, converse and identity. This is a very rich signature, and it has subsequently been discovered that undecidability of representability begins in weaker signatures. This talk will survey some of the very extensive literature in this area, and an overview of the approaches to undecidability, possibly touching on some new results for one of the weakest known algebraic signature to experience undecidability of representability as binary relations. Undecidability of representability as binary relations
Oct. 19, 2021 2pm (Math 350)	Lie Theory Aram Bingham (Centro de Ciencias Matemáticas, UNAM) X Borel subgroup orbits of classical symmetric spaces are parametrized by families of signed involutions called clans, which provide a combinatorial model for studying questions related to Schubert calculus on symmetric spaces. For symmetric spaces of Hermitian type, clans are grouped into “sects” corresponding to Schubert cells of an associated Grassmannian variety, yielding a cell decomposition of the symmetric space and facilitating a combinatorial description of the closure (Bruhat) order on the orbits. This decomposition reveals coincidences of clans in the largest sect with other well-studied posets of matrix Schubert varieties. We further describe explicit bijections between clans for Hermitian type symmetric spaces and several other combinatorial families of objects, including certain rook placements, set partitions, and weighted Delannoy paths. (Based on joint work with Mahir Can and Ozlem Ugurlu.) The Bruhat order on symmetric spaces
Oct. 19, 2021 4pm (Zoom)	Math Edu Jess Ellis Hagman (Colorado State University) X In this talk I will share my current research that addresses student success in STEM from a critical perspective. A critical perspective on student success means locating student challenges in STEM not within the students' themselves but within the systems at play, and that these systems are failing students not because they are broken but because they are functioning perfectly well based on their design. I will share my research teams' current work focused on student experiences in college precalculus and calculus from a critical perspective, discussing both recent qualitative research and quantitative research studies. I care about these courses personally because I think the content is fun, but more importantly because of the role they play in STEM students' college experiences. This talk is intended for a broad audience to gain a new perspective on student success in STEM and learn a bit about research coming from this perspective. Speaker bio: Jess Ellis Hagman is an Associate Professor in the Department of Mathematics at CSU. She completed her PhD in Mathematics Education from the joint program between San Diego State University and the University of California, San Diego. Her area of research is undergraduate mathematics education. Her work is focused on dramatically increasing the number and diversity of people who succeed in undergraduate mathematics-especially introductory mathematics courses that often function as a roadblock for STEM intending students. Her current research includes studying characteristics of successful precalculus and calculus programs, focusing on investigating ways departments can create diverse, equitable, and inclusive introductory mathematics programs. Talking Critically about Student Success in STEM
Oct. 20, 2021 4pm (Math 350)	MathClub Robin Deeley (CU-Boulder) X Informally, a dynamic system is any physical system that evolves with time (e.g., a pendulum, a planet orbiting the sun, the weather, etc). From a more mathematically precise perspective, one can consider a function mapping a space to itself. For example, f(x)=x^2 defined on the set of real numbers. Using this formulation, time is represented by iterating the function. In the example f(x)=x^2, if the initial value is 2, then after one unit of time, the value is f(2)=4, after two units of time, the value is f(f(2))=f(4)=16 and so on. We will discuss a number of explicit examples of dynamical systems and the precise mathematical formulation of chaos. We will also learn the answer to the question: why is my dad's favorite number 6174? Dynamical systems, chaos, and my dad's favorite number
Oct. 21, 2021 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X Scarparo has constructed counterexamples to Matui's HK-conjecture. These counterexamples are essentially principal but not principal. For group actions the difference between essentially principal and principal is related to the difference between the action being topologically free and free. I will discuss a counterexample to the HK-conjecture that is principal. Like Scarparo's original counterexample, this counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. A counterexample to the HK-conjecture that is principal
Oct. 26, 2021 1pm (Zoom)	Michael Pinsker (Technical University Vienna, Austria) X Many mathematical objects are naturally equipped with both an algebraic and a topological structure. For example, the automorphism group of any first-order structure is, of course, a group, and in fact a topological group when equipped with the topology of pointwise convergence. While in some cases, e.g. the additive group of the reals, the algebraic structure of the object alone carries strictly less information than together with the topological structure, in other cases its algebraic structure is so rich that it actually determines the topology (under some requirements for the topology): by a result of Kechris and Solecki, the pointwise convergence topology is the only compatible separable topology on the full symmetric group on a countable set. Which topologies are compatible with a given algebraic object has intrigued mathematicians for decades: for example, Ulam asked whether there exists a compatible locally compact Polish topology on the full symmetric group on a countable set (by the above, the answer is negative). In the case of automorphism groups of first-order structures, the question of the relationship between the algebraic and the topological structure has been pursued actively over the past 40 years, and numerous results have been obtained: many of the most popular automorphism groups, including that of the order of the rationals and of the random graph, do have unique Polish topologies. The endomorphism monoid of a first-order structure is algebraically not as rich as its automorphism group, and often allows many different compatible topologies. We show, however, that there is a unique compatible Polish topology on the endomorphism monoids of the random graph, the weak linear order of the rational numbers, the random poset, and many more. This is joint work with L. Elliott, J. Jonušas, J. D. Mitchell, Y. Péresse, and C. Schindler. Uniqueness of Polish topologies on endomorphism monoids of countable structures
Oct. 26, 2021 3pm (MATH 350)	Topology Ezzeddine El Sai (CU Boulder) X Machine learning methods aimed at learning the underlying geometry of data predominantly assume that the underlying geometry is a smooth embedded manifold. This assumption fails for many physical models where the underlying geometry contains singularities. In this talk, we discuss learning the underlying geometry of data within the scope of algebraic geometry and we use this to detect singularities in data. Detecting Singularities in Data
Oct. 28, 2021 2pm (MATH 350)	Functional Analysis Andrew Stocker (CU Boulder) X Expansive systems are a class of topological dynamical systems which exhibit sensitivity to initial conditions. Following work by Klaus Thomsen, we will construct a C-algebra from an asymptotic equivalence relation on the points in a given expansive system. This generalizes the C-algebraic construction made by Ian Putnam for Smale Spaces, which are themselves an important special case of expansive systems. We will then present some results obtained for a type of expansive system called synchronizing systems, and we will compute this C-algebra for some specific examples. No knowledge of dynamical systems will be assumed for this talk. Expansive Systems and their C-Algebras
Nov. 02, 2021 1pm (Zoom)	Laszlo Zadori (University of Szeged, Hungary) X In the talk, I sketch a semantical proof of the conjecture of Garcia and Taylor that congruence permutability is a prime Maltsev condition in the lattice of interpretability types of varieties. The proof was obtained jointly with Gyenizse and Maróti, and it is based on a combinatorial property of certain digraph powers. I also discuss how the present proof is related to the proof of our earlier result on the non-primeness of n-permutability when n>4 and some other result that we obtained for 3-permutability. On the primeness of 2-permutability
Nov. 02, 2021 1:50pm (Zoom)	Math Edu Karina Uhing (University of Nebraska at Omaha) X This talk will highlight students’ learning experiences with active learning in Precalculus, Calculus 1 and Calculus 2 courses using data from Phase 1 of the Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL) project. Specifically, I will share how students perceive five elements involved with active learning: interactions with their peers during group work, relationships with instructors, formats of courses, assessment, and their affective experiences while learning. Finally, we will discuss how investigating students’ experiences with active learning can help us better design our courses to engage and motivate students to learn. Speaker Bio: Dr. Karina Uhing is an Assistant Professor of Mathematics at the University of Nebraska at Omaha. She has worked on the SEMINAL project for the last 4 years. Currently, Dr. Uhing is part of a large-scale effort to improve Quantitative Reasoning courses at UNO. This semester, she is helping to investigate the transition from an emporium to an active learning model in College and Intermediate Algebra and students’ mathematical and emotional experiences in these courses. Student Experiences with Active Learning in Precalculus Through Calculus 2
Nov. 02, 2021 3pm (On Zoom)	Topology Cherry Ng (CU Boulder) X Bredon homology (and its expanded cousin RO(G)-graded homology) are homology theories suitable for computing homotopical invariants of $G$ -spaces. Such equivariant homology theories have proved useful and powerful, and they played a role in landmark results such has Hill, Hopkins, and Ravenel's work on manifolds of the Kervaire invariant. In this talk, we will discuss both geometric and algebraic methods used to compute Bredon homology, in particular when $G$ is the nonabelian group of order 21. Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 Bredon homology of the nonabelian group of order 21
Nov. 03, 2021 4pm (Zoom)	Grad Student Seminar Boris Hanin (Princeton) X Abstract: Deep neural networks are a centerpiece of modern machine learning. They are also fascinating mathematical objects, about which much remains unclear. In this talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical questions in mathematics, ranging from random matrix theory, to optimal transport, and combinatorics of hyperplane arrangements. Join Zoom Meeting https://cuboulder.zoom.us/j/92773933686?pwd=S1pSMWtORmRGUjNQTEtFbFdFRlFSUT09 Meeting ID: 927 7393 3686 Passcode: 146970 What is a Neural Network?
Nov. 03, 2021 4pm (Math 350)	MathClub Magdalena Czubak (CU-Boulder) X The nonlinear Schrodinger equation (NLS) appears in a wide range of physical phenomena. For example, it is a leading order approximation of the slowly varying envelope of waves in optics, and it also the mean field limit in many-body quantum systems. In this talk, we focus solely on the mathematical analysis of NLS, and discuss what it means to solve the equation, and how using simple Calculus, we can establish a formation of a singularity in finite time. The nonlinear Schrodinger equation: wellposedness and blowup
Nov. 04, 2021 11am (Zoom)	Noncomm Geometry Yining Zhang (CU Boulder) X In this talk, I will discuss the curved Koszul duality for associative algebras presented by quadratic-linear-constant (QLC) relations. Let A be a QLC algebra, the Koszul dual coalgebra of the associated quadratic algebra qA is a curved DG coalgebra. Moreover, this curved DG coalgebra gives rise to resolutions of A which can be used to compute the Hochschild and cyclic homology of A. I will describe the cyclic (co)homology of a QLC algebra and its Koszul dual curved DG algebra, and extend a result due to Feigin and Tsygan. Curved Koszul duality and cyclic (co)homology
Nov. 04, 2021 3pm (Zoom)	Probability Konstantin Matetski (Columbia University) X The KPZ universality class includes random growing interfaces, which, after rescaling, are conjectured to converge to the KPZ fixed point. The latter is a Markov process, which has been characterized through the exact solution of TASEP, a particular model in the class. The KPZ equation plays a special role and is conjectured to be the only model connecting the Edwards-Wilkinson (Gaussian) and the KPZ fixed points. In the talk, I will introduce the KPZ fixed point and review recent progress on the KPZ universality. KPZ universality of random growing interfaces
Nov. 05, 2021 3pm (350)	Math Phys Boris Hanin (Princeton University) X Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based in part on joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases (i.e. networks at the start of training) are solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has a number of consequences for networks during and after training, which I will discuss if time permits. Random Neural Networks at Finite Width and Large Depth Sponsored by the Meyer Fund
Nov. 09, 2021 1pm (MATH 220)	Miguel Couceiro (Universite de Lorraine, France) X In this presentation we consider aggregation procedures (consensus functions) over median algebras (ternary algebras that subsume several ordered structures such as distributive lattices as well as several combinatorial structures such as median graphs). Our starting point is a recent Arrow type impossibility result that states that any median preserving consensus function over linearly ordered sets is trivial in the sense that it only depends on a single argument. In view of this result, a natural problem is then to identify those median algebras that lead to such impossibility results. In particular, we will show that such impossibility results are inevitable when the codomain contains no cycle, i.e., it is a "tree", and we will provide a surprisingly simple condition that completely describes the latter as median algebras. To broaden the talk, we will also present some recent results that answer the parametrized version of this problem in which dependence is restricted to k arguments. We will conclude by observing that the underlying property to proving such results is that of congruence distributivity, which naturally raises the question whether these results extend to other varieties of algebras, e.g., congruence modular varieties. Impossibility theorems over median algebras and beyond
Nov. 09, 2021 2pm (MATH 350)	Lie Theory Jiayuan Wang (George Washington University) X Fully commutative elements in Coxeter groups $B_{n}$ and $D_{n}$ are completely characterized and counted by Stembridge. Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving an enumeration of such elements in complex reflection groups $G(m,1,n)$ . In this talk, we will present a combinatorial connection between fully commutative elements in $B_{n}$ and $G(m,1,n)$ , which allows us to characterize fully commutative elements in $G(m,1,n)$ by pattern avoidance and enumerate them. Further, we will show that fully commutative elements in $G(m,m,n)$ do not have the pattern avoidance property and we will explore full commutativity in $G(m,m,n)$ with different generating sets. Fully commutative elements in complex reflection groups Sponsored by the Meyer Fund
Nov. 09, 2021 3pm (MATH 350)	Topology André Davis (CU Boulder) X In a precise sense, the prime ideals of the stable homotopy category are given by a class of extraordinary co/homology theories called the Morava K-theories K(n). Much like we study rings by localizing at prime ideals, we can study the stable homotopy category by localizing at K(n). Associated to each K(n) is a certain formal group law whose automorphism group G appears on the E_2 page of a spectral sequence that computes the K(n)-local homotopy groups of spheres. In this talk, I hope to provide the necessary background on categorical localization, cohomology theories, and formal group laws to define G (called the Morava stabilizer group) and to describe some related original results. A computation of the action of the Morava stabilizer group on the Lubin-Tate deformation ring
Nov. 10, 2021 4pm (Zoom)	MathClub organized by Padi Fuster (Math For All in Boulder) X Are you interested on learning how to make a scientific poster? Not sure how to get started? Have questions? Join us for an online poster workshop on November 10th from 4pm-5pm organized by Math For All in Boulder! We will cover the following topics: 1. What is a poster presentation? 2. How to pick a topic and find a mentor 3. How to design a poster 4. How to create a poster in LaTeX 5. How to prepare an elevator pitch 6. What to expect at the poster session Registration is not required for this workshop. More info about Math For All in Boulder conference here: https://sites.google.com/view/mathforallnola/satellite-conference/boulder-co Learn how to make a scientific poster!
Nov. 11, 2021 2pm (MATH 350)	Functional Analysis Menevse Eryuzlu (CU Boulder) X We construct a functor that maps C-correspondences to their Cuntz-Pimsner algebras. We then use our functor to investigate the passage of certain C-correspondence relations to the associated Cuntz-Pimsner algebras. If the time permits, we shall discuss other "possible" applications of this functor. A categorical approach to C-algebras arising from C-correspondences
Nov. 11, 2021 3pm (Zoom)	Probability Brian Hall (University of Notre Dame) X The two most basic results in random matrix theory are the circular law and the semicircular law. The circular law says that if N is large and Z is an N x N random matrix with i.i.d. entries of mean zero and variance 1/N, then the eigenvalues of Z will be almost uniformly distributed on the unit disk. The semicircular law says that if N is large and X is a Hermitian random matrix with i.i.d. entries of mean zero and variance 1/N on and above the diagonal, then the (real) eigenvalues of X will have a nearly semicircular distribution on the interval [-2,2]. We now make a simple observation about the relationship between the circular and semicircular laws: twice the real part of the eigenvalues in the circular law has the same limiting distribution as the eigenvalues in the semicircular law. The observation is at one level trivial—if z is uniformly distributed over the unit disk, then 2Re(z) has a semicircular distribution on [-2,2]. But it is, at another level, mysterious: Why should the real parts of the eigenvalues in one random matrix model be related to the eigenvalues in a different random matrix model? In this talk, I will try to persuade you that the relationship between the circular and semicircular laws is not a coincidence but part of a general phenomenon relating the bulk eigenvalue distributions of different random matrix models. For example, if X0 is an arbitrary Hermitian random matrix independent of Z and X, then there is a formula relating the real parts of the eigenvalues of X0+Z to the eigenvalues of X0+X. The talk will be self-contained and have lots of pictures. Relating different random matrix models
Nov. 16, 2021 1:50pm (Zoom)	Math Edu Nancy Kress (CU Boulder) X In this talk I will share results from my dissertation that demonstrate how undergraduate mathematics instructors who are committed to equitable and inclusive instruction can support students' success. I will share results from instructor and student interview data that describes specific instructional practices and the impacts these practices had on students' learning experiences. Areas of particular focus will include community building, use of flexible structures and resources, and student centered instructional strategies. Supporting Student Success in Early Undergraduate Mathematics Courses
Nov. 16, 2021 2pm (Zoom)	Peter Jipsen (Chapman University) X In June 2003 I gave a talk at the Annual Meeting of the Association for Symbolic Logic, University of Illinois at Chicago, on “An online database of classes of algebraic structures”. This list of mathematical structures is still on a website at http://math.chapman.edu/~jipsen/structures, but is mostly just an alphabetical list of links that point to (sometimes incomplete) axiomatic descriptions of about 300 categories of universal algebras. This past summer I started a project with Bianca Newell to recreate this list of (partially-ordered) algebraic structures as a computable LaTeX document that can be checked for consistency and updated more reliably than the previous collection of webpages. In this talk I will describe this project and recent joint work on partially ordered universal algebras with José Gil-Ferez. In this setting, a partially ordered universal algebra is a poset with finitary operations that are order-preserving or order-reversing in each argument, and congruences are replaced by compatible preorders. Our investigations are based on an unpublished paper from 2004 by Don Pigozzi: Partially ordered varieties and quasivarieties, available here [pdf]. A survey of partially ordered algebras
Nov. 16, 2021 2pm (Online)	Lie Theory Peter Tingley (Loyola University Chicago) X We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. This results is an expression for the multiplicity as counting words subject to some conditions, somewhat similar to the known Lyndon word combinatorics. As with that framework counting these words is hard, but in our setup error is concentrated, and we obtain a tractable over-count giving a good estimate. The framework is general, but we only work out specifics in rank two and certain rank 3 cases. This includes joint work with Colin Williams and with Patrick Chan. (https://cuboulder.zoom.us/j/94126918788) A quiver variety approach to root multiplicities
Nov. 16, 2021 3pm (zoom)	Topology Martina Rovelli (University of Massachusetts Amherst) X Power's 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (\infty,2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (\infty,2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus. https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 An (\infty,2)-categorical pasting theorem
Nov. 17, 2021 4pm (Math 350)	MathClub Menevse Eryuzlu (Postdoc CU-Boulder) X The theory of Hilbert spaces is the building block on which functional analysis is based, and is a crucial tool for quantum mechanics. This talk will provide an introduction to the theory, and a brief discussion on how Hilbert spaces are used in the development of one of the best mathematical formulations of quantum mechanics. Hilbert Space Theory and the Beauty of Infinite Dimensions
Nov. 30, 2021 11am (MATH 350)	Diversity CHAT+ (The Diversity Committee) X We kindly ask you to RSVP at the following link to get a headcount for food: https://forms.gle/DGkd4onizvu2Sby59 Overview What: CHAT+ Who: All math faculty, staff, and graduate students When: Tuesday November 30, 11:00AM-12:00PM + lunch provided after the discussion Where: MATH 350 Why: Promoting the success of first-generation students in our classrooms Moderators: Breeann Wilson and Jennifer Gensler Event description CHAT(+)s are discussion sessions for graduate students and faculty of the Department of Mathematics organized by the Diversity Committee. They are structured conversations about key issues, including discussions of issues relating to diversity, for graduate students (CHATs) and between graduate students and faculty (CHAT+s). The goal of the CHAT+ organized for November 30th is to address the challenges faced by first-generation students in higher education and discuss practices we can implement in our classrooms that promote the success of these students. The discussion will be guided by the following short readings: 1) https://all4ed.org/blog/the-challenges-and-privileges-of-being-a-first-generation-college-student/ 2) https://equityalliance.stanford.edu/content/equity-and-intersectionality-serving-first-generation-students-creating-inclusive 3) https://firstgen.naspa.org/files/dmfile/FactSheet-01.pdf (this is a graphic) Faculty and students are asked to take a look at the above readings before the CHAT+. If you are unable to do the readings, however, please do not be deterred from participating. If you can only attend for part of the discussion, please join us when you are able to. Here are some additional readings that may prove insightful: 1) https://serc.carleton.edu/sage2yc/firstgen/challenges.html 2) https://www.insidehighered.com/advice/2021/06/01/first-generation-grad-student-laments-lack-career-opportunities-academe-opinion If you have any questions, please reach out. We hope to see you there! Breeann and Jennifer, on behalf of the Diversity Committee Serving First-Generation Students in our Classrooms
Nov. 30, 2021 12pm (Zoom)	Michael Kompatscher (Charles University Prague) X Let $G$ be a permutation group on a $n$ -element set. We then say that an algebra $A$ has a $G$ -term $t(x1,…,xn)$ , if $t$ is invariant under permuting its variables according to $G$ , i.e. $A ⊨ t(x1,…,xn) ≈ t(xπ(1),…,xπ(n))$ for all $π ∈ G$ . Since $G$ -terms appear in the study of constraint satisfaction problems and elsewhere, it is natural to ask for their classification up to interpretability. In the first part of my talk I would like to share a few partial results on this problem. In the second part I am going to discuss the complexity of deciding whether a given finite algebra has a $G$ -term. The most commonly used strategy in showing that deciding a given Maltsev condition is in P, is to show that it suffices to check the condition locally (i.e. on subsets of bounded size). We show that this „local-global“ approach works for all $G$ -terms induced by regular permutation groups $G$ (and direct products of them), but fails for some other „rich enough" permutation groups, such as $Sym(n)$ for $n ≥ 3$ . This is joint work with Alexandr Kazda. $G$ -terms and the local-global property
Nov. 30, 2021 1pm (Zoom)	Jennifer Hyndman (University of Northern British Columbia, Canada) X Birkhoff and Mal'cev independently posed the problem: Describe all subquasivariety lattices. Nurakunov in 2009 showed that there are many unreasonable subquasivariety lattices where unreasonable means there is no algorithm to determine if a particular finite lattice is a sublattice. This sugests refinements of the original question are needed. A subquasvariety lattice has a natural equaclosure operator. Adaricheva and Gorbunov in 1989 defined an equaclosure operator abstractly as having the properties that are known to hold in a natural equaclosure operator. The soon-to-be-published book, A Primer of Quasivariety Lattices by Kira Adaricheva, Jennifer Hyndman, JB Nation, and Joy Nishida, refines the abstract definition of equaclosure operator and provides some answers to the refined question: When is a lattice with an equaclosure operator representable by a subquasivariety lattice and the natural equaclosure operator. This presentation explores some of this new approach. A reader's guide to A Primer of Subquasivariety Lattices
Nov. 30, 2021 3pm (Online)	Lie Theory Jieru Zhu (Okinawa Institute of Science and Technology) X In many settings, the Temperley-Lieb algebra is isomorphic to the endomorphism algebra on the tensor representation for $SL2$ . In characteristic zero, the Jones-Wenzl projectors give descriptions at the morphisms level for tilting modules in characteristic zero. Their characteristic p versions provided by Burrull-Libedinsky-Sentinelli, describe the tilting category for $SL2$ and admit a quiver representation given by Tubbenhauer-Wedrich. Our project extends this result to the mixed case, i.e. a field with characteristic p containing all ?’th roots of unity, gives fusion rules for tensoring with the natural module regarding the objects as well as the morphisms. This is joint work with L. Sutton, D. Tubbenhauer, and P. Wedrich. (Zoom Meeting ID: 941 2691 8788, passcode: algebra) The $SL2$ Tilting Category in Mixed Characteristic
Nov. 30, 2021 3pm (Zoom)	Topology Katharine Adamyk (University of Western Ontarion) X One common application of topological methods to data analysis is the clustering of data sets. Many of the methods for clustering depend on the value of certain parameters, and as these parameters vary, they index what is known as a hierarchical clustering. Jardine introduced the notion of layer points as a condensed description of hierarchical clusterings in 2020, in the context of the 2-parameter clustering defined by path components of degree-Rips complexes. In this talk I will present recent work that develops the theory of multiparameter layer points more generally, as well as a specific application to the original context. A main theme will be stability—under what conditions do similar inputs for a clustering algorithm guarantee similar resulting clusterings? I will focus particularly on the stability problem with respect to subsampling a data set. https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 Stability for Clustering Methods in Topological Data Analysis
Dec. 02, 2021 1pm (Zoom)	Diversity Teresa Wroe (OIEC) X Join Zoom Meeting https://cuboulder.zoom.us/j/95676446414 Meeting ID: 956 7644 6414 Managing Difficult Classroom Dynamics - This session focuses on how to integrate effective classroom norms to foster a respectful and inclusive environment and minimize challenging student behavior. Participants will discuss scenarios and practice responding effectively to problems in the classroom. Teresa Wroe from OIEC will be facilitating. Managing Difficult Classroom Dynamics
Dec. 07, 2021 12:40pm (Zoom)	Math Edu Shandy Hauk (San Francisco State University) X The session is a sampler across aspects of professional learning related to teaching, scholarly activity, service, and leadership development. The focus is on how to cultivate and use a portfolio document as a tool for planning, implementing, and reflecting on professional growth. We will start with an overview of the nature of “professional development” for those seeking faculty roles. Then, in small groups, we will look at some examples of teaching portfolios of college faculty. The third and final activity includes consideration of your own professional destination(s), how the documentation of your journey in a professional portfolio might look, and some initial planning about how to get there (wherever “there” is). Meeting ID: 813 9009 4107 Passcode: 265706 On Being the Architect of Your Own Professional Future
Dec. 07, 2021 1pm (Zoom)	Martin Goldstern (Technical University Vienna, Austria) X Georg Cantor's "Continuum Hypothesis" (CH) postulates that every infinite set S of reals is either countable or equinumerous with the set of all reals. Using the axiom of choice this means that the "continuum" (the cardinality of the set of reals) is equal to aleph1, the smallest uncountable cardinal. David Hilbert's first problem asked if CH is true; we know now that neither CH nor non-CH can be proved from the usual axioms of set theory (ZFC). Paul Cohen's method of forcing allows us to build universes (structures satisfying ZFC) where the continuum is arbitrarily large. There are many relatives of the continuum, such as the answers to these questions: How many nulls sets (Lebesgue measure zero) do we need to cover the real line? How many points do we need to get a non-null set? How many sequences (or convergent series) do we need to eventually dominate all sequences (convergent series)? etc. All these cardinals are located in the closed interval between aleph1 and the continuum. In my talk I will present some of these cardinals and hint at the methods used to construct universes where these cardinals have prescribed values, or satisfy strict inequalities. Cardinals below the continuum
Jan. 13, 2022 3pm (Zoom)	Probability Andreas Kyprianou (University of Bath) X Consider a general class of branching Markov processes with non-local offspring distribution, or a non-local general superprocess. Let k be a non-negative integer. We will show that, when the mean semigroup possesses a Perron-Frobenius type asymptotic decomposition, remarkably precise asymptotic growth results for its k-th moments and the k-th moments of its occupation measure can be established as t -> oo. The method relies on a combinatorial decomposition of the moments, correctly identifying and treating the leading order terms and using ergodic properties that follow as a result of the Perron-Frobenius assumption. This is joint work with Emma Horton and Isaac Gonzalez. Asymptotic Moments of Spatial branching processes
CANCELED Jan. 18, 2022	Topology Araminta Gwynne (MIT) TBA
Jan. 20, 2022 1pm (Zoom)	Probability Stanislav Volkov X The discrete version of the famous Bak-Sneppen model (https://en.wikipedia.org/wiki/Bak-Sneppen_model) is a Markov chain on the space of {0,1} sequences of length n with periodic boundary conditions, which runs as follows. Fix some 0 Barbay and Kenyon (2001) claimed that the fraction of zeros in the stationary distribution becomes negligible when n goes to infinity whenever p>0.54. This result is indeed correct, however, its proof is not. I shall present the rigorous proof of the Barbay and Kenyon's result, as well as some better bounds for the critical p. Rigorous bounds for the discrete Bak-Sneppen model
Jan. 25, 2022 1pm (Zoom)	Charlotte Aten (University of Rochester) X In my recent work with Semin Yoo we produced a generalization of a construction of Herman and Pakianathan which assigns to each finite noncommutative group a closed surface in a functorial manner. We give a pair of functors whose domain is a subcategory of a variety of n-ary quasigroups. The first of these functors assigns to each such quasigroup a smooth, flat Riemannian manifold while the second assigns to each quasigroup a topological manifold which is a subspace of the metric completion of the aforementioned Riemannian manifold. I will give examples of these constructions, draw some pictures, and argue that all homeomorphism classes of smooth orientable manifolds arise from this construction. I will then discuss a connection with the Evans Conjecture on partial Latin squares, give its implication for orientable surfaces, and state a related problem applicable to our construction for compact n-manifolds. Orientable smooth manifolds are essentially quasigroups
Jan. 26, 2022 5pm (MATH 350)	Grad Student Seminar Ian Miller (CU Boulder) X Sobolev spaces -- I think they are pretty neat -- I'll tell you 'bout 'em. In developing a theory of partial differential equations, we must choose a setting for solutions. For example, we could look for smooth, or analytic solutions, or solutions in $C^{k}$ -- the space of functions which are $k$ times continuously differentiable. Introducing a notion of weak solution allows us to work in spaces with more structure. Typical settings for these weak solutions are the Sobolev spaces $H^{k}$ . We will discuss the definitions of weak solutions and Sobolev spaces, as well as applications of the Riesz representation theorem and Banach contraction theorem to PDE through Sobolev spaces. Sobolov Spaces and the Question "What is a Solution to a Partial Differential Equation?"
Jan. 27, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and other mathematicians. Rather than give all of the lectures myself, I would like to enlist the help of the graduate student participants. In addition to the actual lectures (on Thursdays), I will also host a weekly (schedule TBD) informal working/problem session focused on finer details and working with explicit examples. The intended audience/participants are second year graduate students, but all who are interested are welcome. We will hold an organizational meeting on Thursday, Jan 27, with the intention of determining background and interest and to schedule a time for the working sessions. If you are unable to attend in person, please email michael.woodbury@colorado.edu so that I can share a Zoom link with you. To all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper or papers) that you think your current or future grad students would benefit to learn, please share your ideas with me. I have many ideas myself, but I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Ulam Lectures
Jan. 27, 2022 11:15am (Math …	Noncomm Geometry Vladimir Baranovsky (UCI) X we plan to start with a brief overview of some results on characteristic classes of modules over deformation quantization (in the case of a symplectic algebraic variety). Then we apply these results and the Categorical Riemann-Roch formula to representation theory; bounding the number of simple finite dimensional modules over algebras which arise from studying resolutions of singular symplectic varieties. Deformation quantization and traces of modules
Feb. 01, 2022 1pm (Zoom)	Ross Willard (University of Waterloo) X Many years ago, Kiss proved that the commutator relation [alpha,beta]=0 can be characterized in congruence modular varieties by a simple condition involving a certain kind of 4-ary term, which is now called a Kiss term. Seven years ago, Kearnes, Szendrei and I claimed to extend this characterization to varieties having a difference term, and we used this at a key step in proving our finite basis theorem for finite algebras in varieties having a difference term and having a finite residual bound. It was recently brought to our attention that the published proof of our extension of Kiss’s result has a significant gap, bringing into question the validity of our finite basis theorem. In this talk I will sketch a new (correct) proof of this extension. This is joint work with Keith Kearnes and Agnes Szendrei. Characterizing [alpha,beta]=0 using Kiss terms
Feb. 01, 2022 3pm (MATH 350)	Topology Juan Moreno (CU Boulder) X This meeting is to kick-off a learning seminar on modular tensor categories (MTCs). The meeting will consist of a short overview of the topic after which participants will have the opportunity to sign up for talks. MTCs, and more generally fusion categories, were designed to model the physics of anyon systems which offer a potential framework for quantum computation. Anyons are particles which arise as excitations in (2+1)D topological phases of matter. Such phases (or at least some subclass of them) are believed to have a low-energy description of a (2+1)D topological quantum field theory (TQFT). The goal for this seminar is to study MTCs and their relationship with TQFTs. While their motivation is primarily in the realm of physics, these objects are interesting as stand-alone mathematical objects: (2+1)D TQFTs give rise to 3-manifold and knot/link invariants, and many important examples of MTCs come from representation theory and the theory of quantum groups. Organizational Meeting
Feb. 02, 2022 4pm (Zoom)	Diversity Diversity Committee Math Research Demystified
Feb. 03, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and other mathematicians. Rather than give all of the lectures myself, I would like to enlist the help of the graduate student participants. In addition to the actual lectures (on Thursdays), I will also host a weekly (schedule TBD) informal working/problem session focused on finer details and working with explicit examples. The intended audience/participants are second year graduate students, but all who are interested are welcome. I especially welcome students who may already have or want to learn some background in Lie algebras as this will be the topic for next week. This week, Thursday, Feb 3, I will give an introductory lecture. If you are unable to attend in person, please email michael.woodbury@colorado.edu so that I can share a Zoom link with you. To all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper(s) or book chapter(s)) that you think your current or future grad students would benefit to learn, please share your ideas with me. I have many ideas myself, but I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Introduction to Chevalley Groups
Feb. 08, 2022 1pm (Zoom)	Dragan Masulovic (University of Novi Sad) X Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In this talk we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a monad. We show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements. Dual Ramsey properties for classes of algebras
Feb. 08, 2022 3pm (MATH 350)	Topology Daniel Lyness (CU Boulder) X This is the first talk in the semester learning seminar on modular tensor categories and applications to topological phases of matter. In this talk, we will introduce the concept of topological field theories and give a few fundamental examples. An introduction to TQFTs
Feb. 08, 2022 5pm (MATH 350)	Grad Student Seminar Howie Jordan (CU Boulder) X Information. What is it? Where does it come from? How can I give you information about getting information? By talking at you while pizza powers your focus. (A proof of this will be constructed over the course of the talk). Information theory starts with the idea that we gain information by observing the outcome of something we were unsure about. If you learn which horse wins the race, you've gained information about the race. If you learn a politician lied, you didn't really gain any information about the politician. In this talk, I'll introduce the basic concepts and tools of classical information theory. In particular, we'll consider the notion of entropy and its origins in analyzing communication. Okay Ladies, Now Let's Get Information
Feb. 09, 2022 3:15pm (Math350)	Math Phys Howie Jordan (CU Boulder) X Information can be captured as a measure of how far a probability space is away from being a "sure thing", or equivalently, how much data is required to uniquely specify an event. A fair coin flip contains more information than an unbalanced coin. As the coin becomes more and more biased, a flip tends towards having no informational content. Classically, this leads to a rich theory including an analysis of the efficiency of encodings, compressions, and transmissions of information. It is exactly "A Mathematical Theory of Communication". Information becomes quantum when it starts being contextual and linear. Essentially, we cannot assign all events in the quantum event lattice consistent probabilities and information cannot be duplicated or destroyed. The theme for this semester's mathematical physics seminars is Quantum Information, and this first talk serves as an introduction to the foundations. We will compare the basic ingredients of quantum information theory to the classical counterpart, in particular their event structures and their notions of entropy. Over the course of the semester, we will cover such topics as decoherence in communication channels, distinguishing quantum states, information geometry, and the topological model of quantum computing. We aim to enhance our understanding by utilizing the duality between the boolean logic of subsets and the partition logic of quotients used respectively in probability and information theory, and to express these through a unifying sheaf theoretic formalism. Information, Classical and Quantum: An Introduction
Feb. 09, 2022 4pm (Math 350)	MathClub organized by Agnes Beaudry and Jonathan Quartin (Diversity Committee) X What's undergraduate math research anyways? Who is it for? It might not be what you think it's like and it might be for you, even if you haven't realized it yet! So come to the Math Club on Wednesday Feb 2 from 4 - 5 pm to learn more about undergraduate math research and the opportunities for math research experiences available to CU undergrads. And if you're wondering, does math research happen online? The answer is absolutely! Everyone is welcome! Demystifying Math Research
Feb. 10, 2022 9:05am (Math 3…	Ulam Breeann Wilson (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and many other mathematicians. As promised, I (Mike Woodbury) will not be giving all of the lectures. Our first "outside" speaker is our own Breeann Wilson who will teach us this week about the structure and classification of semisimple and simple Lie algebras over the complex field. In conjunction with and in preparation for Breeann's talk, we will also hold a problem/example session Tuesday at 4:05pm in Math 350. The goal of the Tuesday session is to see some concrete examples of Lie algebras and their root systems. Reminder to all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper or papers) that you think your current or future grad students would benefit to learn, please share your ideas with me (michael.woodbury@colorado.edu). I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Structure and classification of (semi)simple Lie algebras
Feb. 15, 2022 11am (TBD)	Diversity Stephanie Roberts (CISC) Undocually workshop
Feb. 15, 2022 1pm (Zoom)	Bill De Witt (University of St Andrews, UK) X The number of subdirect powers of an finite algebraic structure is a question that has appeared at various points in recent history. The situation is known in full for groups, and to some extent in semigroups. We answer the question for unary algebras, and look at how the situation is more complicated for infinite algebras. We then discuss what these results can tell us about the general question, and how it ties into other topics such as boolean separation. The number of countable subdirect powers of finite unary algebras
Feb. 15, 2022 3pm (MATH 350)	Topology Howie Jordan (CU Boulder) X Fusion categories are complex linear categories with tensor products, duals, and satisfying simplicity conditions. These categories have a finite class of simple objects out of which all other objects are built, and these along with their duals form the representations of the pure states of particles and anti-particles in TQFTs. By adding a braiding onto the monoidal structure and twist maps compatible with the braiding we arrive at a ribbon category. A final nondegeneracy condition gives us the notion of a Modular Tensor Category (MTC). In particular, MTCs have a nondegenerate inner product structure allowing for probabilities to be assigned to states inline with traditional quantum theory. In this talk, we unpack the definitions of fusion and ribbon categories and build up to the notion of Modular Tensor Category as a nondegenerate ribbon fusion category. We will consider these axiomatically as well as skeletal versions in terms of 6j symbols and diagrammatic calculi. Fusion categories and MTCs
Feb. 16, 2022 3:20pm (MATH220)	Math Phys David Jonas (CU Boulder Chemistry Department) X Zoom Meeting ID: 940 0255 330 Password: 790356 Molecular quantum mechanics is usually understood in the context of a fast electron/slow nucleus approximation in which the fast electrons instantaneously adjust to slow motions of the nuclei. Recent experiments have shown that elementary steps in photosynthesis rely on non-adiabatic quantum mechanics. The non-adiabatic eigenfunctions that underlie this extraordinarily efficient process have remarkable nodal structures. These nodal structures were revealed by exactly factorizing the eigenfunctions into a vibrational marginal probability amplitude and an electronic conditional probability amplitude. Instead of the one-dimensional nodal curves familiar from adiabatic quantum mechanics, the two-dimensional non-adiabatic eigenfunctions from a model of photosynthetic energy transfer have zero-dimensional nodal points. Nodal points also occur for two-dimensional eigenfunctions from models that exhibit Berry’s geometric phase. We have made progress towards understanding the local geometry of these nodes by regarding the electronic probability amplitude as a vector field over an oriented surface of vibrational coordinates. For closed curves on the vibrational coordinate surface, proofs of the Poincaré-Hopf theorem establish that the electronic index around the curve equals the sum of the electronic indices of the nodes enclosed by the curve. The electronic indices of nodes are also related to the local radial exponent of the vibrational marginal probability amplitude and to Berry’s geometric phase. These results suggest that differential geometry and topology might help us to understand how photosynthesis generated the fossil fuels we burn and the oxygen we breathe. The Geometry of Nodes in Nonadiabatic Quantum Mechanics
Feb. 16, 2022 4pm (Math 350)	MathClub Agnès Beaudry (CU-Boulder) X This talk will be a friendly introduction to "categories", a concept that's everywhere in mathematics but that's often not introduced until the graduate curricula. The fact that we don't talk about it before then is just cultural, as the definitions are very basic and don't require that much math to formulate. I will give a lot of different examples, pooling from all kinds of classes you may or may not have taken. But worry not, even if you've taken very few math classes, there should be at least some examples that make sense. My hope is that, after this talk, if you hear the word "category" whispered by one of your peers, or casually dropped as a side remark by a professor, you'll feel like you know what that means. A Category? What's that?
Feb. 17, 2022 9:05am (Math 3…	Ulam Breeann Wilson (CU Boulder) X In preparation to begin studying the notes of Steinberg on Chevalley groups, Breeann will discuss the representation theory of simple Lie algebras. By the end of the talk, we hope to understand how every irreducible representation of a given simple Lie algebra is determined by a highest weight vector and that these are in correspondence to the (non-negative) integral span of certain fundamental weights. Everyone is invited to a problem session on Tuesday, February 15, in which we will continue looking at some of the specific examples of Lie algebras seen up to this point. We will also classify and construct the irreducible representations of sl(2,C). Representations of simple Lie algebras
Feb. 17, 2022 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X This talk will be the first in a series discussing KK-theory. We will follow Kasparov's paper "Equivariant KK-theory and the Novikov conjecture". In addition to organizing some of the future talks, I will discuss analytic K-homology as a warm up to Kasparov's bivariant theory. Analytic K-homology
Feb. 22, 2022 1pm (Zoom)	Jordan DuBeau (CU Boulder) X For an infinite algebra J in a countable algebraic language, we say J is Jónsson if it has no proper subalgebra of the same cardinality as J. This talk explores Jónsson algebras in a particular variety: the variety of Jónsson-Tarski algebras. When a Jónsson algebra of size $ℵ1$ was constructed in this variety, it showed that minimal varieties can contain uncountable Jónsson algebras. We will describe that construction and two further results, demonstrating exactly which cardinalities are possible for Jónsson Jónsson-Tarski algebras, and how many pairwise nonisomorphic Jónsson Jónsson-Tarski algebras exist. We discuss implications for other varieties and Jónsson algebras in general. Jónsson Jónsson-Tarski algebras
Feb. 22, 2022 3pm (MATH 350)	Topology Daniel Spiegel (CU Boulder) X The goal for this talk is to cover invariants of MTCs and provide detailed examples such as the quantum double D(G) of a finite group G. Invariants and examples of MTCs
Feb. 23, 2022 5pm (MATH 350)	Grad Student Seminar Andrew Campbell (CU Boulder) X In this talk we'll consider what can be said about the eigenvalues of A+B, given the eigenvalues of A and the eigenvalues of B. We'll begin by exploring the challenges of this problem and why the answer should look very different depending on whether or not A and B are Hermitian. We will then dive into the world of representation theory and combinatorics for a remarkable complete description of the set of all possible eigenvalues for Hermitian A and B. Finally, now that we know the set of all possible eigenvalues we'll see how free probability can give a sense of what they typically look like even for non-Hermitian A and B. What are the eigenvalues of A+B?
Feb. 24, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X Due to the delayed start last week, we were unable to hold the scheduled Ulam lecture. Breeann will thus speak during the problem session time slot (Tuesday 4:05-4:55pm) on representations of sl(2,C) and sl(3,C). On Thursday, Mike will discuss the representations of g_2. Note that beginning next week, we will start on the notes of Steinberg on Chevalley groups. These can be purchased during the month of February through the AMS for a discounted price of just under $20. The root decomposition and representation theory of the exceptional Lie algebra g_2
Feb. 24, 2022 2:30pm (MATH 3…	Functional Analysis Levi Lorenzo (CU Boulder) X As we continue our discussion of analytic K-homology, we encounter Fredholm modules in a motivating example. As seen, operator algebras allow us to recover topological spaces in the noncommutative setting, and K-homology is a valuable invariant for these noncommutative spaces. This week, we look at spaces with additional structure and attempt to recover that structure as algebraic data. Namely, if we have a manifold with a (smooth, Riemannian, Spin, Spin^c) structure, we port that information into the noncommutative world using Spectral Triples. We see how Spectral Triples open the doors to the calculus of noncommutative spaces and how a Spectral Triple determines the topological data of a space in the form of a Fredholm module and a class in K-homology. Noncommutative Spin Geometry and K-homology

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