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Events for November 2019 – March 2020

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Nov. 05, 2019 10am (MATH 350)	Geometry/Analysis Mitsuru Wilson (CU Boulder) X I will give an introductory talk on pseudo-differential operators and their symbols. I will also present my recent results for the case of associated vector bundles of a homogeneous space wherein I generalized Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions for a pseudo-differential operator is its symbol. I will draw analogies from $ℝ^{n}$ to the compact Lie group case and to vector bundles. Global symbolic calculus on homogeneous vector bundles.
Nov. 05, 2019 1pm (MATH 220)	Peter Mayr (CU Boulder) X It is easy to see that the direct product of two finitely generated groups or finitely generated lattices is finitely generated again. We investigate under which conditions generating sets for subdirect products are finite. Finiteness conditions for subdirect products 1
Nov. 05, 2019 2pm (MATH 350)	Lie Theory Mandi Schaeffer-Fry (MSU) X A theorem of Thompson studies the structure of finite groups with exactly one irreducible character degree not divisible by a given prime. I will discuss two recent variations of Thompson’s theorem. Namely, in joint work with E. Giannelli and C. Vallejo, we prove a variation for pairs of primes, and with E. Giannelli and N. Rizo, we prove a variation in the case that exactly two degrees are not divisible by a given prime. On Some Extensions of a Theorem of Thompson
Nov. 05, 2019 3pm (MATH 350)	Algebraic Geometry Abhiram Natarajan (Purdue) X Joint seminar with the Theoretical CS seminar, and the Topology seminar. One object that is pervasive throughout this talk is random polynomials, i.e. polynomials sampled from a specific distribution over the space of polynomials of fixed degree and dimension. We are going to be interested in bounds on the average Betti numbers of the zero sets of random polynomials in various situations. The first part of the talk is about the topological complexity of real algebraic sets restricted to sets that arise in o-minimal geometry. Given any sequence of smooth and compact hypersurfaces in R^{n-1}, we prove that (up to extracting subsequences) there exists a regular definable (in a particular o-minimal structure over R) hypersurface $Γ$ in $R P^{n}$ such that each hypersurface in the sequence appears as a component of the zero set of some polynomial on $Γ$ . On the other hand, we show that given $Γ$ , for "most" polynomials, a Bezout-type bound holds for the restriction of $Γ$ on the zero set of the polynomial. In the second part of the talk, we study the arrangement of hypersurfaces defined in projective space by the zero sets of s randomly and independently chosen (Kostlan) homogeneous polynomials. Our first theorem here uses a random spectral sequence which gives information about the top Betti number of the union of the zero sets of the polynomials. The next result here deals instead with the asymptotic structure of the union of the zero sets when we have s quadrics. It turns out that in this case, the problem of understanding the number of connected components of the union of zero sets is related to the connectivity of a certain random graph model, and can be studied in a precise way. Specifically, our second theorem gives an upper bound on the average number of connected components in a random arrangement of quadrics' zero sets. This is joint work with Saugata Basu and Antonio Lerario. Quantitative bounds on the topology of "random" sets in o-minimal geometry and real algebraic hypersurface arrangements
Nov. 05, 2019 4pm (MATH 350)	Topology Caroline Matson (CU Boulder) Heights of Formal Groups
Nov. 12, 2019 12:10pm (MATH …	Kempner Krzysztof Burdzy (University of Washington) X In lieu of an abstract I offer a problem. Consider three billiard balls of the same radius and mass, undergoing totally elastic reflections on a billiard table with no walls (the whole plane). All three balls can be given non-zero initial velocities. What is the maximum (supremum) possible number of collisions among the three balls? The supremum is taken over all initial positions and initial velocities. I will discuss this problem and its generalization to any finite family of balls in one, two and higher dimensions. Joint work with Jayadev Athreya and Mauricio Duarte. On the number of collisions of billiard balls
Nov. 12, 2019 1pm (MATH 220)	Peter Mayr (CU) Finiteness conditions for subdirect products 2
Nov. 12, 2019 4pm (MATH 350)	Topology Robin Deeley (CU Boulder) The Stratification of the Moduli Stack of Formal Groups
Nov. 13, 2019 5pm (Math 350)	MathClub Peter Mayr (CU Boulder) X Consider all computer programs of a fixed ``length'', say n. There are only finitely many of them.Some may run forever in an infinite loop. But among those that eventually halt, there must be some that run for the maximum number of steps. These programs are the busy beavers of length n, and the number of steps they run is denoted by BB(n). The function BB grows incredibly fast. So far its exact values are known only for n = 1,2,3,4. In fact it follows from the undecidability of the Halting Problem for Turing machines that there is no algorithm to compute BB(n) in general. Even worse, it was recently shown that BB (1919) cannot be determined using our usual foundations of Math (Zermelo-Fraenkel set theory). Clearly BB(1919) is some concrete finite number. However Math as we know it is not strong enough to prove which one. The busy beaver at the end of Math
Nov. 14, 2019 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. Wieler solenoids and the K-theory of the associated stable and stable Ruelle algebras
CANCELED Nov. 14, 2019	Probability Dan Stroock (MIT) X Suppose $X$ and $Y$ are independent, identically distributed real valued random variables and that $θ ∈ (0,1)$ . Then $θ X + 1-θ2 Y$ has the same distribution as $X$ and $Y$ if and only if these are centered Gaussian random variables. Equivalently, if $μ$ is a Borel probability measure on $ℝ$ , then $μ(ξ) = μ(θ ξ) μ(1-θ2 ξ)$ if and only if $μ$ is a centered Gaussian measure. In this lecture I will first prove this result in the case when $θ = 2-1/2$ and then give some applications of it. Some Applications of Gaussian Measures
Nov. 19, 2019 2pm (MATH 350)	Lie Theory Noah White (UCLA) X The RSK correspondence is a bijection between permutations (more generally nonnegative integer matrices) and certain pairs of tableaux. I will outline a geometric incarnation of this bijection coming from the representation theory of the general linear group. I will describe the link to the Calogero Moser system of particles on a line and to a conjecture of Bonnafe and Rouquier related to Kazhdan--Lusztig cells of finite reflection groups. This is joint work with Adrien Brochier and Iain Gordon. Particle collisions and RSK
Nov. 19, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) The Landweber Exact Functor Theorem
Nov. 21, 2019 2pm (MATH 350)	Functional Analysis Andrew Stocker (CU Boulder) X Dynamical systems that are expansive form a large class of examples, including all Smale spaces, shift spaces, and finitely presented systems. For Smale spaces it is well known that we can build C*-algebras from the stable, unstable, and homoclinic equivalence relations. However, we wish to extend these constructions to the class of finitely presented systems, for which we will need to use the more refined notion of local conjugacy to study the dynamics. We will be following Klaus Thompsen's work on relatively expansive dynamical systems. Expansive Dynamical Systems
Dec. 03, 2019 11am (MATH 350)	Number Theory Hanson Smith (CU) X A number field $K$ is monogenic if the ring of integers admits a power integral basis, i.e., if $OK=Z[α]$ for some $α ∈ OK$ . This talk will survey recent results on monogeneity with a focus on the speaker's recent results on radical extensions and torsion fields of abelian varieties. A Survey of Monogeneity
Dec. 03, 2019 1pm (MATH 220)	Peter Mayr (CU Boulder) X In the 1980s commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator. Since this commutator has particularly nice properties for algebras in varieties with modular congruence lattices, the structure theory for abelian, nilpotent, solvable,.. algebras is best developed in this setting. Unfortunately a variety of semigroups is congruence modular if and only if it consists of groups. We give an overview of properties of the binary and higher commutators for semigroups. In particular we investigate what the derived notions of nilpotence, supernilpotence, and solvability mean and how they relate to classical concepts in semigroup theory. Commutators for semigroups
Dec. 03, 2019 2pm (MATH 220)	Howie Jordan (CU Boulder) X The Axiom of Choice is one of the most pervasive principles of foundational mathematics, with much study focusing on the consequences and equivalences of it as well as the “disasters” present in its absence. In this talk, we will introduce an alternative principle, Determinacy, and see that the absence of Choice doesn’t have to be such a disaster after all. Determinacy is a game theoretic axiom that sits in an interesting position compared with Choice, directly contradicting it but sharing some of Choice’s implications and coming with its own novel consequences. We will explore some parts of the picture of set theory under Determinacy as compared with that under Choice, consider various forms of Determinacy, and comment on the consequences of our considerations for foundations of mathematics and logic. Determinacy and Choice
Dec. 03, 2019 2pm (MATH 350)	Lie Theory Sarah Salmon (CU) X This is an ongoing project with Dr. Eirini Chavli (University of Stuttgart), Dr. Maud DeVischer (City University of London), Dr. Alison Parker (University of Leeds), and Dr. Ulrica Wilson (Morehouse College). The walled Brauer algebra is a subalgebra of the Brauer algebra. As a diagram algebra, the walled Brauer algebra can be represented using diagrams of the Brauer algebra that satisfy certain conditions. We will discuss previous work has been done to describe the center of this algebra in the cases where the algebra is semisimple. It was conjectured that the center in the nonsemisimple case can be described the same way. We will describe current work on this conjecture. On the center of the walled Brauer algebra
Dec. 03, 2019 3pm (MATH 350)	Algebraic Geometry Patricio Gallardo (Washington University St. Louis) X It is known, thanks to the work of Deligne and Mostow from 1986, that some ball quotients can be interpreted as moduli spaces of points in the line. Such interpretation has been expanded to K3 surfaces thanks to the work of Kondo and Dolgachev. Within this context, we will describe ongoing work with Matt Kerr and Luca Schaffler, which allows us to give geometric interpretations to the toroidal compactifications of such ball quotients. Geometric interpretation of toroidal compactifications of Deligne-Mostow ball quotients
Dec. 03, 2019 4pm (MATH 350)	Topology Eva Belmont (Northwestern University) X Motivic homotopy theory over R is interesting in part because of its connection to ordinary stable homotopy theory and to C_2-equivariant homotopy theory. In this talk I will review some of these connections, and discuss work in progress with Dan Isaksen to compute R-motivic stable homotopy groups of spheres using an Adams spectral sequence. One of our main applications is to a variant of the Mahowald invariant which can be computed using knowledge of the R-motivic Adams spectral sequence. The R-motivic Adams spectral sequence Sponsored by the National Science Foundation
Dec. 04, 2019 4pm (MATH 350)	Grad Student Seminar Hanson Smith (CU Boulder) X In this talk we'll survey the development and study of number fields from a historical perspective. By focussing on questions and conjectures, some solved and some unsolved, we'll gain an appreciation for number fields and their utility in algebraic number theory. This talk will assume very little background besides a basic knowledge of rings and fields. Problems with Number Fields
Dec. 04, 2019 5pm (Math 350)	MathClub David Grant (CU Boulder) X In digital communications, all messages are sequences are zeroes and ones, but because of physics (for examples, radio waves from a cell tower bouncing off the Flatirons), sometimes a 0 gets interpreted as a 1, and vice versa. To combat errors, engineers build redundancy into transmitted messages. For example, sending 101 as 1111111111 000000000 1111111111 allow a few errors in reception, while still allowing a message to be faithfully decoded. Indeed if we received that transmission as 1101110111 0011000001 1111001101 we can use “majority decoding” to correctly determine that the original message was indeed 101. The problem is that with TOO much redundancy transmission becomes slow and wastes power. This is where the MATH comes in: how do you add enough redundancy to code words, so that the message can be recovered despite a few errors, but not SO much that the process is inefficient? These possible transmitted message are called {\it codewords} and sets of the them are called {\it codes}. We’ll discuss properties of codes like the Hamming distance, which measures the number of places in which two codewords differ, and a “weight enumerator” of the code that records how many codewords have a given distance from the 0 vector. The freaky thing is that if we define the “dual code” $C⊥$ to $C$ as all the code words that have a trivial dot product (mod 2) with all the codewords of $C$ , then the weight enumerator is $C⊥$ can easily be read off from the weight enumerator of $C$ , by an amazing theorem of Jessie MacWilliams. We will present her famous theorem and provide as many details of the proof as we can. CAN YOU HEAR ME NOW?
Dec. 10, 2019 2pm (MATH 350)	Lie Theory Emily King (CSU) Lie Algebras, Coxeter Groups, and Quantum Information Theory
Dec. 10, 2019 4pm (MATH 350)	Topology Agnes Beaudry (CU) Morava Stabilizer Groups and Lubin Tate Theory
Jan. 14, 2020 10am (MATH 350)	Geometry/Analysis Svetlana Roudenko (Florida International University) X We consider Zakharov-Kuznetsov (ZK) equation, a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation, which was introduced in physical context by V. Zakharov and E. Kuznetsov back in 1972. One of the questions they asked was existence of solitons in higher dimensions (the KdV is restricted as the one-dimensional model) and their stability. In this talk I will discuss behavior of solitary waves in the 3d ZK equation, proving that solutions in the energy space that are orbitally stable are also asymptotically stable, that is, as time goes to infinity, they converge to a rescaling and shift of the solitary wave Q(x-t,y,z) in a certain rightward moving window. This is a joint work with Luiz Gustavo Farah, Justin Holmer, and Kai Yang. Soliton stability in higher-dimensional generalization of KdV equation. Sponsored by the Meyer Fund
Jan. 14, 2020 2pm (MATH 220)	Eric Samperton (University of Illinois at Urbana-Champaign) X Algebraic invariants of a manifold can often be formulated as constraint satisfaction problems, where local constraints are induced by the topology via the combinatorics of a triangulation of the manifold. There are many interesting complexity theoretic questions concerning such invariants that are basically wide open. In the first part of this talk, I’ll explain why I believe 3-dimensional manifolds are the most interesting from this perspective. I’ll then quickly discuss what is known about some important decision problems in 3-manifold topology (homeomorphism, unknot recognition, sphere recognition, etc), before turning to the things I think about the most: invariants of 3-manifolds coming from topological quantum field theories. Computational complexity and invariants of 3-dimensional manifolds
Jan. 14, 2020 2pm (MATH 350)	Lie Theory Shawn Burkett (Kent State University) X Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius--Wielandt group is a triple $( G,H,L)$ where $H/L$ is almost a Frobenius complement for $G$ ; A Camina pair is a pair $( G,N)$ where $N$ is almost a Frobenius kernel for $G$ ; A Camina triple is a triple $( G,N,M)$ where $( G,N)$ and $( G,M)$ are almost Camina pairs. In this talk we discuss triples $( G,N,M)$ where $( G,N)$ and $( G,M)$ are almost Frobenius groups. A Frobenius group analog for Camina triples Sponsored by the Meyer Fund
Jan. 14, 2020 4pm (MATH 350)	Topology Jonathan Campbell (Duke University) X In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two n-dimensional polytopes, $P$ , $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble the pieces into $Q$ . The scissors congruence problem, aka Hilbert Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent iff they have the same area. In three dimensions, there is volume AND another invariant, the Dehn Invariant. In higher dimensions, very little is known. I'll give an introduction to this very classical problem, and explain how homotopy theory can be used to get purchase on it. Prerequisites: The discussion of Hilbert's Third Problem and Dehn's invariant will be widely accessible. No knowledge of algebraic K-theory will be assumed. This is all joint work with Inna Zakharevich. Hilbert's Generalized Third Problem and Homotopy Theory
Jan. 15, 2020 3pm (Math350)	Math Phys Markus J. Pflaum organizational meeting, overview
Jan. 19, 2020 9:30am (MATH 3…	Several talks (by people) X Coffee and bagels for participants start at 9:30. We plan to start at 10am. We plan to end at 4pm. Birkhoff Seminar Sponsored by the Meyer Fund
Jan. 21, 2020 10am (Math 350)	Noncomm Geometry Vladimir Baranovsky (University of California, Irvine) X For an algebraic or holomorphic symplectic variety M with a chosen deformation quantization of functions we consider a coherent sheaf F and try to find its deformation quantization as well. It turns out that existence of such quantization imposes nontrivial conditions on the Chern Character of F. These will be derived in the talk from the Algebraic Index Theorem. Chern character of quantizable sheaves
Jan. 21, 2020 11am (MATH 220)	Functional Analysis Elizabeth Gillaspy (University of Montana) X Higher-rank graphs (k-graphs) are a combinatorial model for C-algebras; indeed, much of the structure of k-graph C-algebras is encoded in the underlying combinatorial data of the k-graph. However, different k-graphs can give rise to isomorphic or Morita equivalent C-algebras. In this talk, we present several ways to modify the structure of a k-graph which preserve the Morita equivalence class of the associated C-algebra. Our constructions are inspired by the analogous work for graph C*-algebras of Bates and Pask, as well as by the textile system approach to describing k-graphs. This is joint work with C. Eckhardt, K. Fieldhouse, D. Gent, I. Gonzales, and D. Pask. Moves on higher-rank graphs preserving Morita equivalence Sponsored by the Meyer Fund
Jan. 21, 2020 1pm (MATH 220)	Dmitriy Zhuk (Moscow State University) Quantified Constraint Satisfaction Problems Sponsored by the Meyer Fund
Jan. 21, 2020 2pm (MATH 350)	Functional Analysis Karen Strung (Czech Academy of Sciences, Prague) X Given a Hilbert C-correspondence over C(X), one can construct the associated Cuntz-Pimsner algebra. In the case that the correspondence is full, nonperiodic, and minimal, the resulting C-algebra is simple and unital. An example of such a correspondence is obtained by taking the right Hilbert C(X)-module of continuous sections of a vector bundle over X and twisting the left multiplication by a minimal homeomorphism. As is the case of crossed products by minimal homeomorphisms, one can identify "orbit-breaking" C-subalgebras. In this talk I will discuss these Cuntz--Pimsner C-algebras and their orbit-breaking subalgebras, their relationship to one another, and when they can be classified by Elliott invariants. This is joint work with Adamo, Archey, Forough, Georgescu, Jeong and Viola. Simple constructions from Hilbert C(X)-correspondences Sponsored by the Meyer Fund
Jan. 21, 2020 2pm (MATH 220)	Alexandr Kazda (Charles University Prague) The category of valued relational structures Sponsored by the Meyer Fund
Jan. 21, 2020 4pm (MATH350)	Topology Hessel Posthuma (Amsterdam) X In this talk I will explain how to define so-called “higher genera” for a proper cocompact action of a Lie group satisfying some additional assumptions. These higher genera are numbers, whose definition is inspired by index theory for Dirac type operators in Noncommutative Geometry, and generalize the signature and A-hat genus for compact manifolds. The main theorem states that the higher signatures are equivariant homotopy invariants and that vanishing of the higher A-hat numbers is a necessary condition for the existence of an invariant spin structure together with a metric of positive scalar curvature. This talk is based on joint work with P. Piazza. Higher genera for proper action of Lie groups Sponsored by the Meyer Fund
Jan. 22, 2020 3pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) Recapitulation of Hilbert spaces and their uses in QM and QFT
Jan. 22, 2020 4pm (MATH 350)	Grad Student Seminar Andrew Campbell (CU Boulder) X In 1898 Ladislaus Bortkiewicz showed that the number of Prussian soldiers killed accidentally by horse kicks in a year follows a Poisson distribution. Since then the Poisson distribution and Poisson point processes have found numerous applications in engineering, physics, finance, biology, ecology, etc. We will ignore almost all of these and instead look at the Poisson process in random geometric graphs and random linear operators. On the number of Prussian soldiers killed each year by horse kicks
Jan. 28, 2020 12:10pm (MATH …	Kempner Kate Stange and Sebastian Bozlee (CU Boulder) X During Fall 2019, ICERM held a special semester on Illustrating Mathematics, including computer visualization, 3d printing, and virtual reality, among other media, with an eye toward advancing research, as well as art, outreach and teaching. We will give a brief overview of the program and then present a few highlight projects. Mathematical topics will include algebraic number fields, Fuchsian groups, Thurston geometries and elliptic curves. Illustrating Mathematics
Jan. 28, 2020 3pm (MATH 350)	Algebraic Geometry Srimathy Srinivasan (University of Colorado) X We will discuss the notion of good reduction of reductive groups, it’s implications and what is known so far. I will also give an outline involving main ideas on the proof of a conjecture of Chernousov, Rapinchuk, Rapinchuk for good reduction of unitary groups of quaternionic skew-hermitian forms. Good reduction of unitary groups of quaternionic skew-hermitian forms
Jan. 28, 2020 4pm (MATH 350)	Topology Bob Gompf (University of Texas at Austin) X One of the most surprising discoveries in 4-manifold topology was the existence of smooth manifolds homeomorphic, but not diffeomorphic, to Euclidean 4-space. For fundamental reasons, this phenomenon can only occur in 4 dimensions. We will survey the subject, from its origin to recent developments regarding symmetries of such manifolds. This talk should be accessible to a broad audience. Exotic Smooth Structures on R^4
Jan. 29, 2020 3pm (Math350)	Math Phys Robin Deeley (CU Boulder) An Introduction to C*-Algebras
Jan. 29, 2020 4pm (MATH 350)	Grad Student Seminar Juan Moreno (CU Boulder) X The idea behind obstruction theory is that, given a fiber bundle, the existence of a section is determined by a particular cohomology class of the base space. This class is an obstruction in the sense that if it is zero in the cohomology ring, then a section exists. In this talk, we will consider the particular problem of finding linearly independent sections of vector bundles. This problem turns out to be equivalent to finding a single section of an associated bundle (Stiefel bundle). We will find that the corresponding obstruction to the existence of this section generally lies in a 'twisted' version of cohomology. Obstruction theory and local coefficients
Jan. 29, 2020 5pm (Math 350)	MathClub Elizabeth (Boo) Grulke (CU Boulder) X How students think of mathematics is related to their approach to learning and falls into one of two categories, fragmented or cohesive conceptions of math. This mixed-method study investigated using online discussions to encourage development of cohesive math conceptions in a face-to-face math course at a large metropolitan university. Students participated in either concept-connecting (n=71) or context-focused (n=79) online discussions. While students’ conceptions of mathematics in both groups became more cohesive, their surface approaches to learning also significantly increased. However, deep approaches to learning significantly increased only for the concept-connecting group. Women were found to enjoy and find the discussions more useful than men and women in concept-connecting discussions demonstrated higher exam scores and overall course grades. A Mixed-Method Study of Online Discussion in Mathematics
Feb. 04, 2020 11am (MATH 350)	Number Theory VaNTAGe X From Rachel Pries: In spring 2020, I am starting a virtual seminar on open conjectures in number theory and arithmetic geometry. The seminar will provide open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG. The purpose of the seminar is to provide a viable way for researchers to learn about cutting-edge research in NT&AG without the expense and environmental impact of travel. Another aim of the seminar is to advance understanding on the most exciting open problems in this field. Please spread the news! More information about this seminar is on this webpage: https://sites.google.com/view/vantageseminar If you would like to join the e-mail list for this seminar, please go to this google group and apply for membership: https://groups.google.com/forum/#!forum/vantageseminar This seminar will be run through blue jeans technology and can support 100 connections. Please consider reserving a room with a projector or a video conferencing room if there are several people at your institution who are interested in watching. The talks will typically occur on the first and third Tuesdays at 1 pm ET = 12 CT = 11 MT = 10 am PT. The first topic of the seminar is CLASS GROUPS OF NUMBER FIELDS. The first talks of the seminar will focus on this survey paper. On a conjecture for ?-torsion in class groups of number fields: from the perspective of moments - https://arxiv.org/abs/1902.02008 Jan 21: Lillian Pierce. On some questions in number theory, from the perspective of moments Feb 4: Melanie Matchett-Wood. Conjectures for number field counting Feb18: Caroline Turnage-Butterbaugh. Moments of zeta and the vertical distribution of its zeros March 3: David Zureick-Brown. Moduli spaces and arithmetic statistics If you have any suggestions or advice about this seminar, please send me an email. Best wishes, Rachel VaNTAGe Seminar
Feb. 04, 2020 3pm (MATH 350)	Algebraic Geometry Sebastian Bozlee (University of Colorado) X Let C be a nodal curve, and let E be a union of semistable subcurves of C. We consider the problem of contracting the connected components of E to singularities in a way that preserves the genus of C and makes sense in families. In order to do this, we introduce the notion of mesa curve, a nodal curve with a logarithmic structure and a nice subcurve. Resulting singularities include the elliptic Gorenstein singularities. Contractions of log curves
Feb. 04, 2020 4pm (MATH 350)	Topology Connor McCranie (CU Boulder) Catastrophe Theory
Feb. 05, 2020 3pm (Math350)	Math Phys Markus J Pflaum (CU Boulder) Algebraic Quantum Mechanics
Feb. 05, 2020 4pm (MATH 350)	Grad Student Seminar Cherry Ng (CU Boulder) X Everything is more exciting with a group action on it! In this talk we'll give some examples of ordinary cohomology computations and see how each component changes when we add in a group action. No prior knowledge of homology will be required, and many pictures will be drawn. What is equivariant cohomology?
Feb. 11, 2020 11am (MATH 350)	Number Theory Nikolay Moshchevitin X It is well known that a real number is badly approximable if and only if the partial quotients in its continued fraction expansion are bounded. Motivated by a recent wonderful paper by Ngoc Ai Van Nguyen, Anthony Poels and Damien Roy (where the authors give a simple alternative solution of Schmidt-Summerer's problem) we found an unusual generalization of this criterion for badly approximable d-dimensional vectors. On badly approximable numbers
Feb. 11, 2020 12:10pm (MATH …	Kempner Maria Gillespie (CSU) X Given four lines in three-dimensional space, how many lines pass through all four of them? Enumerative questions like these that arise in geometry can be surprisingly difficult to solve using elementary methods. In this talk, we describe the modern method of using moduli spaces, cohomology rings, and combinatorics to answer such problems. With this framework in mind, we will then describe recent joint work with Renzo Cavalieri and Leonid Monin, in which the combinatorics of "parking functions" was used to calculate the degree of a certain embedding of the moduli space of genus-zero stable curves into a product of projective spaces. How to park your car on a moduli space
Feb. 11, 2020 1pm (MATH 220)	Peter Mayr (CU Boulder) X We give an explicit example of a Promise Constraint Satisfaction Problem PCSP for Boolean structures that can be reduced to a tractable CSP for a structure of size 3 but not for any smaller structure. This was found in an REU in Summer 2019 with Deng, El Sai, Manders, Nakkirt and Sparks. Reducing small PCSP to large CSP
Feb. 11, 2020 3pm (Math 350)	Algebraic Geometry Andreas Gross (CSU) X Tropical curves are piecewise linear objects arising as degenerations of al gebraic curves. The close connection between algebraic curves and their tro pical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic mod uli spaces of curves are reflected in their tropical counterparts. In my ta lk, I will report on joint work in progress with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations betwe en them. I will explain how some of the expected identities cannot be recov ered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place. Moduli spaces of tropical curves and tropical psi classes
Feb. 11, 2020 4pm	Topology Andre Davis (CU Boulder) The Nerve Theorem
Feb. 12, 2020 3pm (Math350)	Math Phys Alex Nita (CU Boulder) Pseudodifferential Operators
Feb. 12, 2020 4pm (MATH 350)	Grad Student Seminar Connor Mccranie (CU Boulder) X Of all mathematical definitions, perhaps one of the most pervasive is that of the continuous function, where a small change in input results in a small change in output. But there is an abundance of natural phenomena that aren't like this: water suddenly freezes with a continuous change in temperature, a star suddenly goes supernova, a camel's back breaks with N+1 straws. Catastrophic behavior is the antithesis of continuity, where a small perturbation of input induces a fundamentally different output. In this talk, we will examine how catastrophes are connected to critical points of functions, and we will state a classification theorem for catastrophes of low dimension. There will be lots of desmos examples and 3D pictures. Catastrophe Theory: Are continuous functions overrated?
Feb. 12, 2020 5pm (Math 350)	MathClub Nat Thiem (CU Boulder) X Matrices are everywhere, but they can also be cumbersome to work with. A standard strategy therefore is to replace complicated matrices with simpler matrices while preserving all the desired properties of the original. This talk explores the basic ground rules for such a simplification, and presents some families of answers. These can vary widely from simple generalizations of row-reduced echelon form to rich combinatorial structures. In some key cases we even find that there is provably no solution. Rewiring matrices using combinatorial tools
Feb. 18, 2020 11am (MATH 350)	Number Theory VaNTAGe X In spring 2020, I am starting a virtual seminar on open conjectures in number theory and arithmetic geometry. The seminar will provide open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG. The purpose of the seminar is to provide a viable way for researchers to learn about cutting-edge research in NT&AG without the expense and environmental impact of travel. Another aim of the seminar is to advance understanding on the most exciting open problems in this field. Please spread the news! More information about this seminar is on this webpage: https://sites.google.com/view/vantageseminar If you would like to join the e-mail list for this seminar, please go to this google group and apply for membership: https://groups.google.com/forum/#!forum/vantageseminar This seminar will be run through blue jeans technology and can support 100 connections. Please consider reserving a room with a projector or a video conferencing room if there are several people at your institution who are interested in watching. The talks will typically occur on the first and third Tuesdays at 1 pm ET = 12 CT = 11 MT = 10 am PT. The first topic of the seminar is CLASS GROUPS OF NUMBER FIELDS. The first talks of the seminar will focus on this survey paper. On a conjecture for ?-torsion in class groups of number fields: from the perspective of moments - https://arxiv.org/abs/1902.02008 Jan 21: Lillian Pierce. On some questions in number theory, from the perspective of moments Feb 4: Melanie Matchett-Wood. Conjectures for number field counting Feb18: Caroline Turnage-Butterbaugh. Moments of zeta and the vertical distribution of its zeros March 3: David Zureick-Brown. Moduli spaces and arithmetic statistics If you have any suggestions or advice about this seminar, please send me an email. Best wishes, Rachel VaNTAGe Seminar
Feb. 18, 2020 2pm (MATH 220 …	Algebraic Geometry Colleen Robles (Duke University) X Abstract: The cohomology (with complex coefficients) of a compact kahler manifold M admits an action of the algebra sl(2,C), and this action plays an essential role in the analysis of the cohomology. In the case that M is a hyperkahler manifold Verbitsky and Looijenga—Lunts showed there is a family of such sl(2,C)’s generating an algebra isomorphic to so(4,b_2-2), and this algebra similarly can tell us quite a bit about the cohomology of the hyperkahler. I will describe some results of this nature for both the Hodge numbers and Nagai’s conjecture on the nilpotent logarithm of monodromy arising from a degeneration. This is joint work with Mark Green, Radu Laza and Yoonjoo Kim. A refinement of the Lefschetz decomposition for hyperkahler manifolds Sponsored by the Meyer Fund
Feb. 18, 2020 4pm (MATH 350)	Topology Yining Zhang (CU Boulder) X In this talk, I will give a brief introduction to the subject of string topology discovered by Chas and Sullivan, and explain the connections to Hochschild homology and cohomology. If time permits, I will try to interpret string topology operations from the point of view of "derived noncommutative geometry" and briefly mention some joint works with Yuri Berest and Ajay Ramadoss. An introduction to string topology
Feb. 19, 2020 3pm (Math350)	Math Phys Andrew Hamilton (CU Boulder, Astrophysical & Planetary Sciences) X As first noticed in the 1970s, the charges of fermions follow patterns that allow the Standard Model U(1)xSU(2)xSU(3) to be embedded in Grand Unified groups SU(5) and SU(2)xSU(2)xSU(4), which in turn embed in SO(10), or equivalently in its covering group Spin(10). Spin(10) seems to know about Lorentz transformations: notably, the Dirac pseudoscalar miraculously equals the Spin(10) pseudoscalar. But Spin(10) lacks a time dimension. To accommodate time, 2 dimensions must be adjoined (the complex structure of spinors means that they live naturally in even dimensions). The main result is: The Dirac (Poincare) and SM algebras are commuting subalgebras of the spin algebra associated with the group Spin(11,1) of transformations of spinors in 11+1 spacetime dimensions. Each of the 3 spatial dimensions of 3+1 dimensional spacetime proves to be a 5-brane. The Standard Model and Beyond
Feb. 19, 2020 4pm (MATH 350)	Grad Student Seminar Sebastian Bozlee (CU Boulder) X Suppose you fill in a matrix with the Fibonacci numbers, left-to-right, top-to-bottom. What will its rank be? The answer is surprisingly simple. We will then explore how much this answer extends when the matrix is filled in with other sorts of sequences defined by linear recurrences. On the ranks of recurrence matrices
Feb. 25, 2020 1pm (Math 220)	Mark Pullins (CU) Generically 2-transitive actions of linear algebraic groups with solvable point stabilizers
Feb. 25, 2020 2pm (MATH 220)	Michael Wheeler (CU Boulder) X We will explore the structure of SOP_2 types (defined by Malliaris and Shelah) and types whose realizations fill cuts in regular ultrapowers of infinite linear orders. We will see that both sorts of types and their distributions can be characterized by certain classes of graphs. This will extend to a new characterization of good ultrafilters. Part 1 will motivate the study of these types in terms of Keisler's order on countable complete first-order theories. Graph-like Types and Good Ultrafilters, Part 1
Feb. 25, 2020 4pm (MATH 350)	Topology Florian Naef (MIT) X Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points. String topology and the configuration space of two points
Feb. 25, 2020 4pm (Math 350)	Noncomm Geometry Florian Naef (MIT) X Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points. String topology and the configuration space of two points
Feb. 26, 2020 3pm (Math350)	Math Phys Andrew Hamilton (CU Boulder, Astrophysical & Planetary Sciences) X Spinors -- spin 1/2 particles -- come already quantized. They come in two species, row spinors and column spinors, which already have the properties of fermion creation and annihilation operators: you can multiply a row spinor by a column spinor or vice versa, but you cannot multiply a row spinor by a row spinor or a column spinor by a column spinor. The anticommutation property of spinors can be traced to the fact that the spinor metric in 4 dimensions is antisymmetric. This seminar explores building quantum field theory on a foundation of the known properties of spinors, rather than on postulated rules. Spinors have properties beyond those invoked by qft; for example, the algebra of outer products of spinors yields (is isomorphic to) the multivector algebra of strings and branes of all dimensions. Might spinors point to a way forward for qft at unification scales? Founding Quantum Field Theory on Spinors
Feb. 26, 2020 5pm (Math 350)	MathClub Invited: Internal CU Math REU (Diversity Committee from CU Boulder) X The Math Club and Diversity Committee are having an event to talk to undergraduates about math research and the various opportunities available to the undergraduate students through CU. Our goal is to widen the pool of applicants to the REU and Experimental Math Lab programs and to create an inclusive and equitable opportunity for our students. Math Research Demystified
Mar. 03, 2020 11am (MATH 350)	Number Theory VaNTAGe X In spring 2020, I am starting a virtual seminar on open conjectures in number theory and arithmetic geometry. The seminar will provide open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG. The purpose of the seminar is to provide a viable way for researchers to learn about cutting-edge research in NT&AG without the expense and environmental impact of travel. Another aim of the seminar is to advance understanding on the most exciting open problems in this field. Please spread the news! More information about this seminar is on this webpage: https://sites.google.com/view/vantageseminar If you would like to join the e-mail list for this seminar, please go to this google group and apply for membership: https://groups.google.com/forum/#!forum/vantageseminar This seminar will be run through blue jeans technology and can support 100 connections. Please consider reserving a room with a projector or a video conferencing room if there are several people at your institution who are interested in watching. The talks will typically occur on the first and third Tuesdays at 1 pm ET = 12 CT = 11 MT = 10 am PT. The first topic of the seminar is CLASS GROUPS OF NUMBER FIELDS. The first talks of the seminar will focus on this survey paper. On a conjecture for ?-torsion in class groups of number fields: from the perspective of moments - https://arxiv.org/abs/1902.02008 Jan 21: Lillian Pierce. On some questions in number theory, from the perspective of moments Feb 4: Melanie Matchett-Wood. Conjectures for number field counting Feb18: Caroline Turnage-Butterbaugh. Moments of zeta and the vertical distribution of its zeros March 3: David Zureick-Brown. Moduli spaces and arithmetic statistics If you have any suggestions or advice about this seminar, please send me an email. Best wishes, Rachel VaNTAGe Seminar
Mar. 03, 2020 1pm (Math 220)	Mark Pullins (CU) Generically 2-transitive actions of linear algebraic groups with solvable point stabilizers
Mar. 03, 2020 2pm (MATH 220)	Michael Wheeler (CU Boulder) Graph-like Types and Good Ultrafilters, Part 2
Mar. 03, 2020 4pm (MATH 350)	Topology Sam Zhang (CU Boulder) The Morse-Smale complex
Mar. 04, 2020 3pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) Wightman Axioms of QFT
Mar. 04, 2020 4pm (MATH 350)	Grad Student Seminar Chase Meadors (CU Boulder) X Commutative algebra and its related field of algebraic geometry are extremely vast fields full of rich theory and daunting notation. However, if we restrict our attention to the simplest varieties (pun intended) of objects, many concepts and problems become completely combinatorial in nature. In particular, we will explore the so-called Stanley-Reisner duality between monomial ideals in a polynomial ring and simplicial complexes, discuss algebraic characterizations of topological properties and vice versa, and explore connections to graph theory. Combinatorial Commutative Algebra
Mar. 10, 2020 1pm (Math 220)	Mark Pullins (CU) Generically 2-transitive actions of linear algebraic groups with solvable point stabilizers
Mar. 10, 2020 2pm (MATH 350)	Lie Theory Gabe Feinberg (Washington College) X A fully commutative element of a Coxeter group is one for which any reduced word can be obtained from another by applying only commutation relations. For example, in a type A Coxeter group, these are exactly the 321-avoiding permutations, known to be counted by the Catalan numbers. For other types, these fully commutative elements have been studied and enumerated by John Stembridge, and have been found to be related to Temperley-Lieb algebras and Khovanov-Lauda-Rouquier algebras. In this talk, we’ll extend the theory to more general complex reflection groups. We’ll deal with some difficulties in generalizing the notion of full commutativity and see some interesting combinatorial structures, including the related Catalan triangle. Fully commutative elements of complex reflection groups Sponsored by the Meyer Fund
Mar. 10, 2020 2pm (MATH 220)	Michael Wheeler (CU Boulder) Graph-like Types and Good Ultrafilters, Part 3
Mar. 10, 2020 3pm (Math 350)	Algebraic Geometry Leo Herr (Utah) X Costello's Pushforward Formula relates virtual fundamental classes on different spaces. It's useful in enumerative geometry, where the goal is to perform computations on these classes within Cohomology or Chow Groups. More specifically, it's often used to compare different ways of performing curve counts or compare curve counts between related spaces. Tragically, this formula has a missing hypothesis. In joint work with Jonathan Wise, we provide counterexamples, prove a correct version of Costello's result, and check that papers using this formula are correct. We obtain a sort of "generalization" by weakening the hypotheses while correcting them, and then another which pertains to log geometry. Attendees can expect an exposition of virtual fundamental classes, clarification of the state of Costello's Formula, and report on ongoing generalizations. Costello's Pushforward Formula
Mar. 10, 2020 4pm (MATH 350)	Topology Claudia Scheimbauer (TU Munich) X The Cobordism Hypothesis provides a beautiful interplay between extended topological field theories and dualizability conditions, allowing for a conceptual explanation of certain finiteness conditions appearing in representation theory, and, vice versa, a geometric understanding of algebraic objects. In turn, $E_{n}$ -algebras, which are algebras for the little disks operad, (e.g.~associative algebras and quantum groups) provide important examples. I will explain why every $E_{n}$ -algebra leads to a categorified topological field theory using dualizability arguments. Furthermore, I will explore extensions and further directions. The main result is joint work with Owen Gwilliam. E_n-algebras, extended topological field theories, and dualizability
Mar. 11, 2020 3pm (Math350)	Math Phys Robert Mayer (CU Boulder & University of Arizona) Euclidean Field Theory a la Osterwalder-Schrader and Wick rotation
CANCELED Mar. 11, 2020	MathClub Social Hour Foster a Math community that is respectful, accessible, and inclusive.
Mar. 12, 2020 2pm (MATH 350)	Functional Analysis Kathy Merrill (Colorado College) X Single wavelet sets, and thus single wavelets, are shown to exist for the actions of all crystallographic groups on $R2$ under all integer dilations. Examples of such sets satisfying the additional requirement that they are finite unions of convex sets are presented for each of the groups under dilation by two. Crystallographic wavelet sets
CANCELED Mar. 17, 2020	Geometry/Analysis Monique Chyba (University of Hawaii) X The copepod model is a simplification of the 3-link Purcell swimmer and is relevant to analyze more complex micro-swimmers. The mathematical model is validated by observations performed by a team from Hawaii, showing the agreement between the predicted and observed motions. Sub-Riemannian geometry will be introduced, assuming that displacements are minimizing the expanded mechanical energy of the micro-swimmer. The objective is to maximize the efficiency of a stroke (the ratio between the displacement produced by a stroke and its length). Using the Maximum Principle in the framework of Sub-Riemannian geometry, this leads to analyze family of periodic controls producing strokes to determine the most efficient one. Finally a robotic copepod is presented whose aim is to validate the computations and very preliminary results are given. Sub-Riemannian geometry, Hamiltonian dynamics, Micro-swimmers, Copepod nauplii and Copepod robot Sponsored by the Meyer Fund
Mar. 17, 2020 11am (MATH 350)	Number Theory Cliff Blakestad (Pohang University of Science and Technology) TBA
CANCELED Mar. 17, 2020	Kempner Uri Shapira (Israel Institute of Technology) X Given an integral vector, there are several geometric and arithmetic objects one can attach to it. For example, its direction (as a point on the unit sphere), the lattice obtained by projecting the integers to the othonormal hyperplane to the vector, and the vector of residues modulo a prime p to name a few. In this talk I will discuss results pertaining to the statistical properties of these objects as we let the integral vector vary in natural ways. Geometry of integral vectors
CANCELED Mar. 17, 2020	Nik Ruskuc (St Andrews, UK) The number of countable subdirect powers of a finite semigroup Sponsored by the Meyer Fund
Mar. 17, 2020 2pm (Zoom)	Michael Wheeler (CU Boulder) X https://cuboulder.zoom.us/j/359857929 Graph-like Types and Good Ultrafilters, Part 4
CANCELED Mar. 17, 2020	Algebraic Geometry Klaus Hulek (Institute for Algebraic Geometry, Leibniz University Hannover) X Bogomolov, Petrov and Tschinkel defined monodromy strata in the moduli space of elliptic K3 surfaces (BPT strata) and they also proved that these strata are rational. Here we compare (some of) these BPT strata to moduli spaces of lattice polarized K3 surfaces. More precisely, we classify all moduli spaces of lattice polarized K3 surfaces which dominate finite to one one of the BPT strata. This is closely related to Shimada's classification of connected components of the moduli of elliptic K3 surfaces. This is joint work with Michael L\“onne. Elliptic K3 surfaces - monodromy strata versus lattice polarizations Sponsored by the Meyer Fund
Mar. 18, 2020 3pm (notes wil…	Math Phys Daniel Spiegel (CU Boulder Physics) Algebraic Quantum Field Theory à la Haag-Kastler
Mar. 18, 2020 3pm (Zoom)	Thesis Defenses Trevor Jack X Join at https://cuboulder.zoom.us/j/511481743 We investigate the computational complexity of checking whether a finite semigroup given by generators has certain properties. We are concerned specifically with transformation semigroups, matrix semigroups, and partial bijection semigroups. Some key results include: (1) checking that a transformation semigroup is a group is in AC^0, (2) checking if a transformation semigroup satisfies a fixed equation is in NL, (3) checking whether an element is regular in a transformation semigroup is PSPACE-complete, (4) checking that a matrix is nilpotent is in P, and (5) checking membership in a partial bijection semigroup is PSPACE-complete. Computational Complexity of Semigroup Properties
CANCELED Mar. 18, 2020	Grad Student Seminar Levi Lorenzo (CU Boulder) X Given a polynomial f, the orbit of zero under f creates a dynamical sequence (0,f(0),f^2(0), ...). Given a sequence (a_n), a natural question to ask is: when does n divide a_n? , For reasons we will discuss, this question is especially nice to ask for such "dynamically generated" sequences, but even in such sequences, is not nicely answered in general. How does the so-called "divisibility set" depend on the degree of the polynomial? How does it depend on the constant term? What happens if we allow non-integer rational coefficients and examine the numerators of such a sequence? What if we work over a different number field? Index Divisibility in Dynamically Generated Sequences
CANCELED Mar. 19, 2020	Kempner Klaus Hulek (Institute for Algebraic Geometry at Leibniz University of Hannover) X Cubic hypersurfaces are of special interest in geometry. The fact that every smooth cubic surface contains precisely 27 lines is one of the most classical results of algebraic geometry. Cubic hypersurfaces of dimension 3 became famous for anther reason: they were the first example known of a variety X which admits a finite dominant map from a projective space onto X (X is unirational), but is itself not birational to projective space (X is not rational). In dimension 4 we see yet another phenomenon: the lines on a cubic hypersurface of dimension 4 form an irreducible holomorphic symplectic (hyperkähler) manifold. In this talk I will discuss some new results on cubic hypersurfaces. Cubic hypersurfaces
Mar. 31, 2020 1pm (Zoom)	Thesis Defenses Hanson Smith (University of Colorado Boulder) X This thesis is essentially a compendium of the author's completed work as a graduate student. It includes a variety of different investigations centering around monogeneity and division fields. In the defense, we will introduce these concepts with some historical backing. The speaker hopes this portion will be friendly to a relatively general audience. After moving through some classical results, we will discuss the author's earlier work (joint with Katherine Stange and T. Alden Gassert) on the monogeneity of partial torsion fields. This segues into the author's work on the monogeneity of quartic fields generated by trinomials. We will move to more recent work on the radical extensions. Finally, we will discuss results on the non-monogeneity of division fields of certain abelian varieties, including explicit non-monogenic families in the case of dimension 1. Join Zoom Meeting https://cuboulder.zoom.us/j/365441180 Meeting ID: 365 441 180 Everyone is invited. After the public presentation, the general audience will be asked to leave during private questioning, but a re-invite for a celebration/congratulations will be emailed out afterward, so please watch for that. Monogeneity and Torsion

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