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Events for May – September 2014

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May. 01, 2014 3pm (Webber 20…	Algebraic Geometry Nicola Tarasca (Utah) X Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, Chen-Coskun, Cavalieri-Marcus-Wise. Double total ramifications for curves of genus 2
May. 08, 2014 3pm (Webber 20…	Algebraic Geometry Ekin Ozman (UT Austin) The p ranks of Prym varieties
Aug. 27, 2014 4:05pm (MATH 3…	Grad Student Seminar Paul Lessard (CU-Boulder) X Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of� topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These� particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions? In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category. In lieu of pizza, it is suggested that one attend the department picnic. From Invariants, to functors, to homotopy, to model categories, and back to homotopy: part 1
Sep. 02, 2014 12:10pm (MATH …	Kempner Nandor Sieben (Northern Arizona University) X In the first part of the talk, we will recall the basic theory of impartial games. We�ll learn how to win at familiar games like NIM and precision bowling. Then, we�ll apply this knowledge to study games in which two players alternately pick elements of a given group until the picked elements generate the group. How to beat your neighbor at precision bowling and group generation
Sep. 02, 2014 3pm (Math 350)	Algebraic Geometry Agnes Beaudry (University of Chicago) X In the first talk, I will introduce some of the concepts which play a central role in homotopy theory. I will give an introduction to the concepts of stable homotopy groups, spectra, generalized cohomology theories and localization. I will also explain how formal group laws enter the subject. For the majority of the talk, I will only assume that the audience is comfortable with the concepts of topological spaces and their cohomology. The second talk will be focused on chromatic homotopy theory. The philosophy of chromatic homotopy theory is that the stable homotopy groups of the sphere $S$ can be re-assembled from the homotopy groups of a family of spectra $L_{n} S$ . Roughly, $L_{n} S$ is the $n$ -th chromatic layer of $S$ . Each layer corresponds to the height of a formal group law. The homotopy groups of the chromatic layers are related to the cohomology of the group of automorphisms of the corresponding formal group laws. In this talk, I will give an overview of this computational program. �I will also talk about the role played by elliptic curves and their deformation theory at chromatic level $2$ . I will assume that the audience has heard of spectral sequences. Chromatic homotopy theory
Sep. 03, 2014 4:05pm (MATH 3…	Grad Student Seminar Paul Lessard (CU-Boulder) X Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions? In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category. From Invariants, to functors, to homotopy, to model categories, and back to homotopy: part 2
Sep. 09, 2014 11am (MATH 350)	Number Theory Alden Gassert (University of Colorado Boulder) X An iterated tower is a sequence of number fields that is generated by the iteration of a rational map defined over the base field. These dynamically generated number fields have received a growing attention over the past decade for a variety of reasons; in this talk I will discuss how these towers could lead to the first construction of infinite, yet finitely and tamely ramified, extensions. One way to ensure a successful construction is to find a tower in which the discriminants of the number fields does not grow too fast. I will demonstrate a method for how one might seek to study the growth of these number fields using the Chebyshev polynomials as an example. The iterated extensions arising from the Chebyshev polynomials are interesting in their own right. Arithmetic properties of iterated towers
Sep. 09, 2014 12:10pm (MATH …	Kempner Mati Rubin (Ben Gurion University) X A locally moving (LM) group for a regular topological space $X$ , is a group $G$ of homeomorphisms of $X$ which has the following property: For every nonempty open set $U$ , there is $g ∈ G$ such that: (1) $g$ is not the identity function, and (2) for every $x ∈ X ∖ U$ , $g(x) = x$ . A group $H$ is a locally moving (LM) group, if it is isomorphic to a group $G$ as above. Many groups of automorphisms of topological spaces, linear orderings, trees, Boolean algebras, measure algebras and some other types of structures are LM. Examples: (1) The group of all auto-homeomorphisms of a Euclidean space is LM. (2) The group of all $C∞$ homeomorphisms of a �� Euclidean space is LM. (3) The group of all automorphisms of the binary tree is LM. Using a certain theorem about LM groups, one can prove theorems of the following type: If $X$ and $Y$ are topological spaces such that $H(X)$ (= the group of all auto-homeomorphisms of $X$ ) is isomorphic to $H(Y)$ ,�� then�� $X$ is homeomorphic to $Y$ . Similar theorems for linear orderings, trees, Boolean algebras, measure algebras and some other types of structures are also true. A recent theorem states that: The first order theory of every locally moving group is undecidable. (This solves in much more generality a question of Mark Sapir about the R. Thompson groups). I shall state and explain that "certain theorem" mentioned above. Then I'll speak about consequences of that theorem including the undecidability theorem. If time allows, I shall mention some open problems. Locally moving groups, reconstruction and undecidability
Sep. 09, 2014 2pm (MATH 350)	Lie Theory Erica Shannon (CU) X Given a parabolic subgroup of a Coxeter group, we may hope to describe a set of nice coset representatives for this subgroup. Deodhar's Lemma shows us how to directly compute a set of coset representatives with the property that every prefix of a coset representative is also on the list. Time permitting, we'll discuss an application of Deodhar's Lemma to the 56-dimensional minuscule module for the simple Lie algebra of type E7. No previous knowledge of Coxeter groups will be assumed. Deodhar's Lemma and an Application to Representation Theory
Sep. 09, 2014 3pm (Math 350)	Algebraic Geometry Elizabeth Gillaspy (University of Colorado) X The goal of this talk is to explain my thesis research and why we care. In the first part of the talk, I will define the objects in the title and explain how the K-theory of groupoid C-algebras can tell us about other mathematical objects, from dynamical systems to string theory. My hope is that this will be a gentle introduction to the topic(s) at hand; no prior familiarity with groupoids or with C-algebras will be assumed. In the second part of the talk, I'll discuss the particular question I've investigated about the K-theory of twisted groupoid C-algebras, why I chose it, and the progress that I've made so far. Time permitting, I will sketch some of the proofs, so this part of the talk will be more technical than the first part. K-theory for twisted groupoid C-algebras
Sep. 09, 2014 11pm (Math 220)	Noncomm Geometry Alexander Gorokhovsky Introduction to cyclic (co)homology.
Sep. 10, 2014 4:05pm (MATH 3…	Grad Student Seminar John Willis (CU-Boulder) X Schemes are objects central to the study of algebraic geometry, but what are they? A scheme is a locally ringed space that admits an open cover by affine schemes, sure, but what is a locally ringed space? What is an affine scheme? I will try to answer these questions by motivating the definition of a scheme with basic real and complex analysis. The basic yoga is that schemes are weird, because commutative rings are weird; any non-abelian category is weird. But Riemann Surfaces are not weird at all, and much intuition for how schemes behave, and what sorts of things should be true about them, can be obtained through the lens of complex geometry where one has holomorphic structure to work with - contractible neighborhoods, differentiability, and other such familiar and friendly notions. Once we have drawn some pictures and seen, at least heuristically, why we should think of commutative rings as the natural dwelling place for functions, we will define abstractly what a scheme is and give some examples. What Is a Scheme?
Sep. 11, 2014 3pm (Webber 20…	Algebraic Geometry Chris Ormerod (Cal Tech) X There are many integrable ordinary differential equations that may be derived as isomonodromic deformations. Symplectic geometry gives us insight into how such system possesses a Hamiltonian description. Analogously, there are many integrable ordinary difference equations that are derived as discrete isomonodromic deformations and connection preserving deformations. By considering a geometric setting for differential and difference Lax pairs, we present some preliminary work that suggests the evolution of these systems possesses an analogous description. The geometry of discrete isomonodromy
Sep. 16, 2014 2pm (Math 220)	Functional Analysis Judith Packer (University of Colorado, Boulder) X We continue our discussion introducing groupoids. All are welcome. Introduction to Groupoids part 2 Sponsored by the Meyer Fund
Sep. 16, 2014 2pm (MATH 350)	Lie Theory Megan Ly (CU) X The Huneke-Wiegand conjecture, a long-standing open problem in commutative algebra has prompted much recent study.� In the category of numerical monoid algebras, Garcia-Sanchez and Leamer provide a reduction theory that equates the Huneeke-Wiegand conjecture to finding certain irreducible arithmetic sequences within numerical monoids.� This talk focuses on Leamer monoids, whose elements correspond to arithmetic sequences of a given step size within a fixed numerical monoid, and its factorization theory.� In this talk we will provide a description for the first element $x0$ in the Leamer monoid for numerical monoids generated by two integers using Apery sets.� We apply our results to prove the Huneke-Wiegand Conjecture for all numerical monoid algebras of two generators and provide direction for potentially proving the conjecture for numerical monoid algebras with more generators. The Factorization Theory of Leamer Monoids
Sep. 16, 2014 11pm (Math 220)	Noncomm Geometry Alexander Gorokhovsky Introduction to cyclic (co)homology (contd.)
Sep. 18, 2014 3pm (Weber 201…	Algebraic Geometry Sofia Tirabassi (University of Utah) X We give a complete classification of varieties of maximal Albanese dimension . chi equal to 1 and irregularity 2dimX-1. This is a joint work with Z. Jiang and M. Lahoz. Varieties with Chi equal to 1
Sep. 23, 2014 11am (MATH 350)	Number Theory William Chen (Macquarie University) X The birth of geometric discrepancy theory is usually attributed to the fundamental work of Klaus Roth in 1954. The present scope of the subject is due in no small part to the many important results of Wolfgang Schmidt in the 1960s and 1970s. It is fair to say that every subsequent worker in the subject have been motivated and encouraged, directly or indirectly, by these two pioneers and their work. Lower bound results in discrepancy theory exhibit the limitations to just point distributions, whereas upper bound results lead to point distributions that are close to best possible under such limitations. In this expository talk, we shall discuss some of the main results obtained over the last 60 years, and introduce to the audience a number of lower and upper bound techniques. We shall also briefly discuss some open questions. A quick tour of discrepancy theory
Sep. 23, 2014 12:10pm (MATH …	Kempner Gerard Misiolek (Notre Dame) X I will describe some recent results on local ill-posedness (in the sense of Hadamard) of the Cauchy problem for the incompressible Euler equations of hydrodynamics. Ill-posedness of the incompressible Euler equations in borderline spaces
Sep. 23, 2014 2pm (Math 220)	Functional Analysis Judith Packer (Department of Mathematics, CU Boulder) X We continue our introductory lectures on groupoids. In this lecture, we will introduce left Haar systems, and hopefully will discuss C*-algebras associated to locally compact groupoids having a left Haar system. Introduction to Groupoids, Part 3 Sponsored by the Meyer Fund
Sep. 23, 2014 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X In this series of two talks, the goal is to study combinatorial realizations of representations induced from unipotent subgroups. The first talk will build a combinatorial framework for the representations of the finite general linear groups via symmetric functions. The goal is to have a setting where representations have a pleasing combinatorial existence even without knowing the algebraic underpinnings. The second talk will then find the induced characters of unipotent groups in this combinatorial setting. This is joint work with S. Andrews. The combinatorial representation theory of the finite general linear groups
Sep. 23, 2014 11pm (Math 220)	Noncomm Geometry Alexander Gorokhovsky Introduction to cyclic (co)homology (contd.)
Sep. 24, 2014 4:05pm (MATH 3…	Grad Student Seminar Shawn Burkett (CU-Boulder) X Algebraic group theory is, essentially, the intersection of algebraic geometry and group theory. Algebraic geometry, in a classical sense, is the study of algebraic sets (the vanishing loci of systems of polynomials) in affine or projective space. An algebraic group is an abstract group which is also an algebraic set, where the group operations are given by polynomial functions. Among the most widely studied are the affine algebraic groups, those which may be embedded in GL(V), the automorphism group of some (finite dimensional) vector space V. After defining affine algebraic groups, we will discuss some of the important subgroups which, in many ways, control the structure of the groups. Time permitting, we will discuss finite groups of Lie type, which are rational points of a particular type of linear algebraic group. An Introduction to Algebraic Groups
Sep. 25, 2014 3pm (Weber 201…	Algebraic Geometry Mark Shoemaker (University of Utah) X Mirror symmetry is the name given to a broad collection of mathematical conjectures arising from a type of duality in physics. Roughly it states that certain information about a smooth variety M should by expressible in terms of different properties of a ``mirror manifold�� W. Rephrasing questions about M in terms of W has led to striking solutions to many previously intractable problems in geometry. In this talk I will give a friendly introduction to the mathematics of mirror symmetry, and outline some ways in which given a manifold M one can construct its mirror W. Not all of these constructions agree however, and sometimes one is left with different spaces which should be mirror to the same M. This ``multiple mirror�� phenomenon has interesting mathematical implications. In particular I will show that in certain settings, the mirror manifolds which arise are in fact birational to one another. An introduction to mirror symmetry and the question of multiple mirrors
Sep. 25, 2014 4pm (Weber 201…	Algebraic Geometry Olivia Dumitrescu (Hanover Universitaat) X In the first part of the talk I will discuss the Eynard-Orantin theory to the family of spectral curves of Hitchin fibrations over a smooth base curve C of genus at least 2. In particular, we study rank 2 holomorphic Higgs field and we identify the spectral curve of the Eynard-Orantin as the spectral curve of Hitchin fibration. In the second part we generalize this construction to meromorphic Higgs fields with possibly singular spectral curve in the compactified cotangent bundle. We identify the spectral curve of the Eynard-Orantin recursion with a divisor in a ruled surface over the curve C. We further apply recursion to quantize spectral curve. We present as simple examples of our theory meromorphic Higgs bundles over the rational curve, and we obtain well-known classical equations as Airy function and Catalan recursion. This talk is based on my joint work with Motohico Mulase, U.C. Davis. Quantization of singular spectral curves for meromorphic Higgs bundles of rank 2
Sep. 26, 2014 4:15pm (MATH 3…	Probability Carl Mueller (U. Rochester) Hitting questions and multiple points for stochastic PDE (SPDE) in the critical case
Sep. 30, 2014 2pm (Math 220)	Functional Analysis Judith Packer (University of Colorado) X We continue our discussion of r-discrete groupoids with left Haar systems and the C*-algebras that arise from them. Introduction to Groupoids, part 4 Sponsored by the Meyer Fund
Sep. 30, 2014 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X In this series of two talks, the goal is to study combinatorial realizations of representations induced from unipotent subgroups.� The first talk will build a combinatorial framework for the representations of the finite general linear groups via symmetric functions.� The goal is to have a setting where representations have a pleasing combinatorial existence even without knowing the algebraic underpinnings.� The second talk will then find the induced characters of unipotent groups in this combinatorial setting.� This is joint work with S. Andrews. Supercharacters and the generalized Gelfand--Graev representations
Sep. 30, 2014 3pm (Math 350)	Algebraic Geometry Matthew Grimes (University of Colorado) X Recent work on the log minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program.� Our main result is the construction of a universal moduli space of slope semi-stable sheaves which compactifies the moduli space of vector bundles over the moduli space of pseudo-stable curves. The first half of the talk will emphasize accessibility and focus on background material: geometric invariant theory, moduli problems, and the log-minimal model program.� In the second segment, we will discuss the construction of the universal moduli space of vector bundles. Universal moduli spaces of vector bundles and the log minimal model program on the moduli of curves
Sep. 30, 2014 11pm (Math 220)	Noncomm Geometry Alexander Gorokhovsky Introduction to cyclic (co)homology (contd.)

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