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Events for March – July 2020

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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)	Grad Algebra/Logic 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)	Grad Algebra/Logic 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, -algebras, which are algebras for the little disks operad, (e.g.~associative algebras and quantum groups) provide important examples. I will explain why every -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	Math Club 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 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	PDE/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	Grad Algebra/Logic Nik Ruskuc (St Andrews, UK) The number of countable subdirect powers of a finite semigroup Sponsored by the Meyer Fund
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. 17, 2020 2pm (Zoom)	Michael Wheeler (CU Boulder) X https://cuboulder.zoom.us/j/359857929 Graph-like Types and Good Ultrafilters, Part 4
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
CANCELED Apr. 01, 2020	Math Phys Andrea Carosso (CU Boulder Physics) Lattice Theory of Quantum Matter and Fields
CANCELED Apr. 01, 2020	Grad Student Seminar Howie Jordan (CU Boulder) X We’ve all heard the hype: all you have to do is forget about “excluded middle” and then BOOM, your logic is constructive (in a certain sense). But like… why would you do that? And in what sense is it more constructive? If you aren’t aware, the Law (ha!) of Excluded Middle is the assertion that, for any particular sentence, either that sentence or its negation holds in your logic. In this talk we’ll look at what kind of logic you get when you drop excluded middle and discuss various motivations and interpretations of this, from the philosophical hooey of Brouwer to the structural interpretations arising from Gentzen’s methods. We will also look at a sample of some of the differences in practice and theory that arise when you don’t allow yourself the luxury of excluded middle in your proofs. Noo don’t exclude the middle you’re so constructive aha
CANCELED Apr. 01, 2020	Math Club Judy Packer (CU Boulder) TBA
Apr. 02, 2020 9:30am (Zoom)	Thesis Defenses Zhenhua Wang X Join the Zoom meeting at https://cuboulder.zoom.us/j/4348160464 The first topic is based on work in [17], we study the so-called “coin-turning” walks. Let S be the random walk obtained from “coin turning” with some sequence {pn}n?2, as introduced in [16]. We investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n?1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. In the second topic, we study equilibrium of stopping problem for a multi-dimensional continuous strong Markov processes X. We use the framework for definition of equilibrium as in [20], especially, the discount function is of decreasing impatience, which means it is log sub-additive and may not be exponential. We provide properties of nearly Borel equilibria, and build up the equivalence be- tween an equilibrium and its finely closure. Under certain regular conditions on X, we give results of optimality of an equilibrium and prove that there exists an optimal equilibrium among a quite general family M of equilibria with some proper regularity condition for M. Moreover, the optimal one can be achieved by taking the intersection of all equilibria. Random Walks and Stochastic Control
Apr. 02, 2020 10am (Zoom)	Thesis Defenses Mark Pullins (CU) X Join the Zoom meeting at https://cuboulder.zoom.us/j/511678834 Let G be a connected, nonabelian, reductive, linear algebraic group over an algebraically closed field of characteristic 0. Let T be a maximal torus of G, and let Q be a closed, connected, solvable subgroup of G whose unipotent radical is stable under the conjugation action of T . We determine necessary and sufficient conditions for the algebraic action of G on G/Q to be at least generically 2-transitive. For semisimple groups G, we determine when this action is precisely generically 2-transitive. Through examples, we show that there exist closed, connected, solvable Q such that gtd(G, G/Q) = 2, thereby showing how the conditions of our theorem can be met. We also show, through example, that not every closed, connected, solvable Q such that gtd(G, G/Q) = 2 must have its unipotent radical stable under conjugation by T. Generically 2-Transitive Algebraic Group Actions with Solvable Point Stabilizers
CANCELED Apr. 02, 2020	Grad Algebra/Logic Alexander Hulpke (Colorado State University) X Quotient algorithms have been a principal tool for the computational investigation of finitely presented groups as well as for constructing groups. I will describe a method for a nonsolvable quotient algorithm, that extends a known finite quotient with a module. This is joint work with Heiko Dietrich (Monash U., Melbourne, Australia) Towards a Nonsolvable Quotient Algorithm
Apr. 02, 2020 3pm	Thesis Defenses Athena Sparks X Promise Constraint Satisfaction Problems (PCSP) are a generalization of Constraint Satisfaction Problems (CSP) that provide a common framework for many classical decision problems like graph colorability, satisfiability of Boolean formulas, solvability of linear equations, etc. Informally, a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. In the formulation for a fixed template of relational structures A and B, the input of PCSP(A,B) is a structure I of the same type as that of A and B. The problem is to determine if there exists a homomorphism from I into A or if there does not exist a homomorphism from I into B. The “promise” is that exactly one of these two alternatives occurs for the input I. As with CSP, the computational complexity of a PCSP is determined by the higher arity symmetries or relation preserving functions, called polymorphisms, from A to B. All polymorphisms from A to B form a clonoid, a set of finitary functions from A to a target algebra B that is closed under the usual variable operations like identifying, permuting and introducing arguments as well as closed under operations from B. We adapt the established algebraic approach for analyzing CSP to investigate the complexity of PCSP and take steps towards classifying clonoids on finite sets in general. Join Zoom Meeting: https://cuboulder.zoom.us/j/700856074 Clonoids and Promise Constraint Satisfaction Problems
Apr. 02, 2020 3pm (MATH 350)	Topology Katharine Adamyk (CU Boulder) X The mod 2 Steenrod algebra, A, appears in stable homotopy theory as the algebra of stable operations on cohomology with coefficients in Z/2. In this talk, we present results that hold in the stable category of modules over A(1), a small subalgebra of A. In this category, the invariants Q_0- and Q_1-Margolis homology together detect a large amount of information about a given A(1)-module. We show that all bounded below A(1)-modules of finite type whose Q_1-Margolis homology is trivial are stably equivalent to direct sums of suspensions of a distinguished family of A(1)-modules. The classification theorem is then used to simplify computations of localized Ext groups and to provide necessary conditions for lifting A(1)-modules to A-modules. We give the first differential for a spectral sequence converging to an h_0-localized Ext group associated to a given bounded below A(1)-module of finite type. The classification theorem allows the computation of all differentials if the desired decomposition of a module with trivial Q_1-Margolis homology can be given explicitly. This is not always possible, but we give some examples where the higher differentials can be computed. The differentials in this spectral sequence detect obstructions to lifting an A(1)-module to an A-module. Join Zoom Meeting https://cuboulder.zoom.us/j/949702894 Defense : A classification of Q_0-local A(1)-modules
Apr. 06, 2020 2pm	Thesis Defenses Sebastian Bozlee X We use log geometric data to construct contractions of families of curves of arbitrary genus. We use log blowups and this contraction construction to resolve the rational map between the moduli space of stable curves and some alternate semistable compactifications of the moduli space of curves. In at least two cases, the target space appears to be new. Join Zoom Meeting https://cuboulder.zoom.us/j/401893441?pwd=Y3dFUVl1czF5c3p5SjM3dG1MSDBQUT09 Meeting ID: 401 893 441 Password: mesa An Application of Logarithmic Geometry to Moduli of Curves in Genus Greater Than One
Apr. 07, 2020 10am	Thesis Defenses Carly Matson (University of Colorado Boulder) X One of the main difficulties in studying the Galois groups of field extensions is that, in general, the set of roots of a polynomial need not have any additional algebraic structure. In 1965, Lubin and Tate showed that a certain type of power series f(x) over a p-adic field can be recognized as the "multiplication-by-pi" endomorphism of a formal group law with complex multiplication, or a larger-than-usual endomorphism ring. They used this to put a module structure on the roots of f(x) and to show that the splitting field of f(x) is Galois and abelian, culminating in a new proof of the main theorem of local class field theory. In this talk we will offer a new and more general construction of formal group laws with complex multiplication in higher dimensions and will examine how this differs from the one-dimensional case. Finally, we will once again see that the torsion points of such a formal group generate abelian extensions over p-adic fields. This defense will be held via zoom, and all are welcome. For zoom link information, please email David Grant. Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations. Higher dimensional formal group laws with complex multiplication
Apr. 07, 2020 11am (MATH 350)	Number Theory TBA Number Theory thesis defenses today
Apr. 07, 2020 1pm (Zoom)	Thesis Defenses Daniel Martin (University of Colorado Boulder) X We will take a geometric perspective on Euclideaneity in imaginary quadratic fields. The result is a pseudo-Euclidean algorithm with applications to continued fractions, the class group, and the special linear group in two dimensions. This defense will be held via zoom, and all are welcome: Due to "zoombombing" concerns, please email Katherine Stange or Daniel Martin for link and password. Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations. After the defense, an announcement will be made and congratulations can be sent via email. The geometry of imaginary quadratic fields
CANCELED Apr. 07, 2020	Lie Theory Bryan Gillespie (CSU) TBA Sponsored by the Meyer Fund
Apr. 09, 2020 11am	Noncomm Geometry Yining Zhang (CU Boulder) X Given a closed oriented manifold X, Chas and Sullivan constructed a Lie bracket on the (reduced) S^1-equivariant homology of the free loop space of X, called the string bracket. Furthermore, if the manifold is simply connected, there is a Hodge type decomposition on the S^1-equivariant homology induced by the n-fold coverings of the circle. It is natural to ask whether the string bracket preserve this decomposition. We provide a positive answer under the additional assumption that X is rationally elliptic. Our argument is based on algebraic models of string topology and analogous operations on Hochschild and cyclic homologies. This is a joint work with Yuri Berest and Ajay Ramadoss. Join Zoom Meeting https://cuboulder.zoom.us/j/251778522 Hodge decomposition of string topology
CANCELED Apr. 15, 2020	Math Phys Daniel Spiegel (CU Boulder Physics) Rigged Hilbert Spaces
CANCELED Apr. 15, 2020	Math Club Marty Walter (CU Boulder) TBA
Apr. 21, 2020 11am (MATH 350)	Number Theory Hershy Kisilevsky (Concordia (Montreal)) TBA
CANCELED Apr. 21, 2020	Kempner Rui Fernandes (University of Illinois) TBA
Apr. 21, 2020 2pm (MATH 350)	Lie Theory Anne Schilling (UC Davis) X We introduce a new crystal on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call -crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the -crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization in the 0-Hecke monoid to pairs of (transposes of) semistandard Young tableaux and prove several properties about this new insertion, in particular its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators. This is joint work with Jennifer Morse, Jianping Pan and Wencin Poh (https://arxiv.org/abs/1911.08732). [Zoom link: https://cuboulder.zoom.us/j/957933828] Crystal for stable Grothendieck polynomials Sponsored by the Meyer Fund
CANCELED Apr. 22, 2020	Math Phys Juan Moreno (CU Boulder) Foliations
CANCELED Apr. 22, 2020	Grad Student Seminar Rachel Chaiser (CU Boulder) TBA
Apr. 25, 2020 9am (MATH 350)	Number Theory Kirsten Eisenträger, Robert Lemke Oliver, Jackson Morrow (Penn State, Tuf…) Front Range Number Theory Day
CANCELED Apr. 28, 2020	Lie Theory Flor Orosz Hunziker (Harvard) TBA Sponsored by the Meyer Fund
CANCELED Apr. 29, 2020	Math Phys Mike Hermele (CU Boulder Physics) Mathematical Theory of Fractons
Jul. 08, 2020 3pm (Zoom room…	Thesis Defenses Taylor Klotz (TK) X In this thesis, we provide new insights into the theory of cascade feedback linearization of control systems. In particular, we present a new explicit class of cascade feedback linearizable control systems, as well as a new obstruction to the existence of a cascade feedback linearization for a given invariant control system. These theorems are presented in Chapter 4, where truncated versions of operators from the calculus of variations are introduced and explored to prove these new results. This connection reveals new geometry behind cascade feedback linearization and establishes a foundation for future exciting work on the subject with important consequences for dynamic feedback linearization. Geometry of Cascade Feedback Linearizable Control Systems

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