Rice University Math Department Calendar

     January 2013          February 2013            March 2013     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
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                                               31

4:00 pm Monday, February 18, 2013
Stulken Topology Seminar: Transverse knots, infinite cyclic covers and Heegaard Floer homology
by Shea Vela-Vick (LSU) in HB 427

In recent years, Heegaard Floer theory has proven an invaluable tool for studying contact manifolds and the Legendrian and transverse knots they contain. I plan to discuss a method for defining a variant of Heegaard Floer theory for infinite cyclic covers of transverse knots in the standard contact 3-sphere. Thi s invariant takes the form of a Z[t,t^-1]-module and generalizes one defined in joint work with Baldwin and Vertesi for transverse knots braided about open book decompositions. In this talk, I will discuss how our invariant is constructed and present some basic properties and computations. This is joint work with Tye Lidman, Robert Lipshitz and Sucharit Sarkar.Host Department: Rice University-Mathematics
Submitted by shelly@rice.edu

4:00 pm Wednesday, February 20, 2013
Geometry-Analysis: H\"older continuity of an extended CMV matrix
by Darren Ong (Rice) in HB227

Abstract: The CMV matrix is an operator on $\ell^2(\mathbb Z_+)$ that serves as a unitary analogue of the self-adjoint Jacobi matrix. It is a central tool in the study of orthogonal polynomials on the unit circle. We may extend the CMV matrix to an operator on $\ell^2(\mathbb Z)$, and this extended matrix is useful for understanding quantum random walks. We prove that power law bounds on the formal eigenvectors of the one-sided CMV matrix implies that the two-sided extension has spectral measure that is H\"older continuous, using techniques developed in the self-adjoint case by Damanik, Killip and Lenz. Our result implies spreading in the corresponding quantum random walk Host Department: Rice University-Mathematics
Submitted by hardt@rice.edu

4:00 pm Monday, March 4, 2013
Topology Seminar: Relations among characteristic classes of manifold bundles
by Ilya Grigoriev (Stanford) in HB 427

The characteristic classes of surface bundles with fiber of genus g coincide with the elements of the cohomology of the classifying space of surface bundles, denoted BDiff \Sigma_g. The n-th cohomology of this space is known in the "stable range" (n <= (2g-2)/3) by theorems of Madsen-Weiss, Harer, and others. In this range, the map from a free algebra generated by the so-called "kappa-classes" to H^*(BDiff \Sigma_g) is an isomorphism. Recently, Soren Galatius and Oscar Randal-Williams have obtained similar results for the case where the surface \Sigma_g is replaced with a certain high-dimensional manifold, namely the connect sum of g copies of the product of spheres S^k x S^k. Outside the stable range, the kernel of the above-mentioned map for surfaces has been studied by Morita, Faber, Looijenga, Pandharipande and many others. In this talk, I will describe a vast family of elements in the kernel that also works in the the high-dimensional case (for odd k >= 1). The kernel is large enough to imply that the image of this map ("the tautological subring") is finitely-generated for all odd k, rationally, even though there are infinitely many kappa classes. It also implies upper bounds on the stable range of cohomology for fixed g and k. Host Department: Rice University-Mathematics
Submitted by andyp@rice.edu

4:00 pm Tuesday, March 5, 2013
Algebraic Geometry Seminar: Hodge theory and derived categories of cubic fourfolds
by Nicolas Addington (Duke University) in HB 227

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove that they coincide generically: Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and we show that Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.Host Department: Rice University-Mathematics
Submitted by btl1@rice.edu

4:00 pm Thursday, March 7, 2013
Colloquium: Fat, exhausted, integer homology spheres
by Jeff Brock (Brown) in HB 227

Perelman's groundbreaking proof of the geometrization conjecture for three-manifolds has foregrounded the problem of exploring tighter correspondences between geometric and algebraic invariants of three-manifolds. In this talk, we address the question of how homology interacts with hyperbolic geometry in 3-dimensions, providing examples of hyperbolic integer homology spheres that have large injectivity radius on most of their volume. (Indeed such examples can be produced that arise as (1,n)-Dehn filling on knots in the three-sphere). Such examples fit into a conjectural framework of Bergeron, Venkatesh and others and provide a counterweight to phenomena arising in asymptotic L^2 invariants of families of covers of hyperbolic manifolds. This is joint work with Nathan Dunfield.Host Department: Rice University-Mathematics
Submitted by jtanis@rice.edu

4:00 pm Monday, March 18, 2013
Topology Seminar: SL(n, Q) has no volume-preserving actions on (n − 1)-dimensional compact manifolds
by Dave Witte-Morris (Lethbridge) in HB 427

As part of the "Zimmer program," numerous authors have studied volume-preserving actions of the group SL(n,A) on compact manifolds, where A is either the ring Z of integers or the field R of real numbers. On the other hand, very little seems to be known about the intermediate case where A is the field Q of rational numbers. As a first step in this direction, we show that SL(n,Q) has no nontrivial, C-infinity, volume-preserving action on any compact manifold of dimension strictly less than n. The proof has two main ingredients: a theorem of Zimmer tells us that the action of any "S-arithmetic" subgroup must extend (a.e.) to a measurable action of its profinite completion, and the Congruence Subgroup Property provides a very nice description of this profinite completion. This is joint work with Robert J. Zimmer of the University of Chicago.Host Department: Rice University-Mathematics
Submitted by andyp@rice.edu

4:00 pm Tuesday, March 26, 2013
Algebraic Geometry Seminar: Etale Homotopy and Diophantine Equations
by Tomer Schlank (MIT) in HB 227

From the view point of algebraic geometry solutions to a Diophantine equation are just sections of a corresponding map of schemes X- > S. When schemes are usually considered as a certain type of "Spaces" . When considering sections of maps of spaces f:X->S in the realm of algebraic topology Bousfield and Kan developed an Obstruction-Classification Theory using the cohomology of the S with coefficients in the homotopy groups of the fiber of f. In this talk we will describe a way to transfer Bousfield - Kan theory to the realm of algebraic geometry. Thus yielding a theory of homotopical obstructions for solutions for Diophantine equations. This would be achieved by a generalizing the étale homotopy type defined by Artin and Mazur to a relative setting X → S . In the case of Diophantine equation over a number field i.e. when S is the spectrum of a number field, this theory can be used to obtain a unified view of classical arithmetic obstructions such as the Brauer-Manin obstruction and descent obstructions. If time permits I will present also applications to Galois theory. This is joint work with Yonatan Harpaz.Host Department: Rice University-Mathematics
Submitted by btl1@rice.edu

4:00 pm Tuesday, April 2, 2013
Algebraic Geometry Seminar: The McKay correspondence via tilting
by Morgan Brown (University of Michigan) in HB 227

Let $G$ be a subgroup of $SL_n(\mathbb{C})$. When $n=2$, $\mathbb{C}^n/G$ has a unique minimal resolution, and the classical McKay correspondence relates the representation theory of $G$ with the structure of this resolution. For $n=3$, Bridgeland, King, and Reid used categorical techniques to show that $\mathbb{C}^n/G$ has a distinguished crepant resolution Y=$G$-Hilb. Specifically, they showed that there is an equivalence between the derived categories $D^G(\mathbb{C}^n) and $D(Y)$. I will show how one can use this equivalence to describe the coherent sheaves on $Y$ in terms of $G$-equivariant objects on $X$ by a process called tilting. This is joint work with Ian Shipman. Host Department: Rice University-Mathematics
Submitted by btl1@rice.edu

4:00 pm Thursday, April 4, 2013
Colloquium: Symmetries of the Field of Algebraic Numbers
by Avner Ash (Boston College) in HB 227

One of the central concerns of Algebraic Number Theory is understanding the solution sets to systems of polynomial equations with integer coefficients (Diophantine equations). For example, the equation x^n+y^n=z^n has no solutions in positive integers if n > 2 (Fermat's Last Theorem, or FLT). We gain a lot by first studying single polynomials in one variable, and we do it by looking at the symmetries satisfied by their roots. We then combine this with a study of how polynomials interact with the prime numbers, and we make comparisons wtih data (lists of Hecke operator eigenvalues, one for each prime) coming from various geometric objects. This leads to Reciprocity Laws, of which the first was Quadratic Reciprocity. I will give an introduction to these subjects, explaining how Reciprocity Laws are brought to bear on the study of Diophantine equations. There has been a lot of progress in this area in the last few years, but the best example is still Wiles' proof of FLT.Host Department: Rice University-Mathematics
Submitted by jtanis@rice.edu

4:00 pm Friday, April 5, 2013
Special Lecture : Infinite Determinantal Measures
by Alexander Bufetov (Rice-CNRS-Steklov-IITP-HSE) in HB227

Abstract: Infinite determinantal measures introduced in the talk are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal process and a convergent, but not integrable, multiplicative functional. The main result of the talk gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes. The talk is based on the preprint arXiv:1207.6793.Host Department: Rice University-Mathematics
Submitted by hardt@rice.edu

4:00 pm Tuesday, April 9, 2013
Algebraic Geometry Seminar: An asymptotic Mukai model of M6
by Evgeny Mayanskiy (Penn State) in HB 227

http://arxiv.org/pdf/1303.6843.pdf.

Host Department: Rice University-Mathematics
Submitted by btl1@rice.edu

4:00 pm Thursday, April 18, 2013
Colloquium: Combinatorial richness of multiplicatively large sets
by Vitaly Bergelson (Ohio State) in HB 227

Many famous results of additive combinatorics deal with combinatorial richness of additively large sets in N = {1,2,...}. For example, the celebrated Szemeredi's theorem states that any subset of N which has positive upper density in N, contains arbitrarily long arithmetic progressions. One is naturally inclined to inquire whether there are interesting results pertaining to MULTIPLICATIVELY large sets in N. The goal of this talk is to introduce and juxtapose various notions of additive and multiplicative largeness and to discuss multiplicative analogs of Szemeredi's theorem and its extensions. As we will see, the methods of ergodic theory are well suited to tackle the problems which naturally arise in this context. In particular, we will show that multiplicatively large sets have a very rich combinatorial structure, both multiplicative and (somewhat surprisingly) additive. The talk is intended for a general audience.Host Department: Rice University-Mathematics
Submitted by jtanis@rice.edu

3:00 pm Friday, April 19, 2013
Geometry-Analysis Seminar: A New Compactness Theorem for Gromov-Hausdorff and Intrinsic Flat Convergence
by Christina Sormani (Grad.Center CUNY) in HB 227

Abstract. The Tetrahedral Compactness Theorem states that sequences of Riemannian manifolds, $M^m_j$, with a uniform upper bound on volume and on diameter that satisfy a uniform tetrahedral property have subsequences which converge in the Gromov-Hausdorff and Intrinsic Flat sense to countably H^m rectifiable metric spaces of the same dimension as the initial sequence. In general the Gromov-Hausdorff and Intrinsic Flat limits of Riemannian manifolds do not agree and Gromov-Hausdorff limits are not usually countably H^m rectifiable. In addition to the presentation of this new theorem, there will be a review of the notions of the Gromov-Hausdorff and Intrinsic Flat convergence and their relationshio in a variety of settings.Host Department: Rice University-Mathematics
Submitted by hardt@rice.edu