AWM at JMM 2017

Special Session Abstracts

Contents

Friday, January 6, 2017, 8:00 am – 10:50 am (International C, International Level, Marriott Atlanta Marquis)

AMS-AWM Special Session on Symplectic Geometry, Moment Maps and Morse Theory, I

8:00 a.m. Contraction of a Hamiltonian K-space

Christopher Allen Manon*, George Mason University (1125-51-840)

I will describe how to construct the contraction X0 of a Hamiltonian K-space X. In terms of symplectic and algebraic geometry, the contraction X0 is very similar to X yet it comes equipped with a Hamiltonian K × T action for T ⊂ K a maximal torus. I’ll also discuss how contraction emerges algebraically from the horospherical contraction operation of Popov, and its relationship to recent work of Harada and Kaveh on Newton-Okounkov bodies. This is joint work with Joachim Hilgert and Johan Martens.

8:30 a.m. On geometric quantization of b-symplectic manifolds

Victor Guillemin, Massachusetts Institute of Technology;
Eva Miranda, Universitat Politecnica de Catalunya;
Jonathan Weitsman*, Northeastern University (1125-58-1001)

We study a notion of pre-quantization for b-symplectic manifolds. We use it to construct a formal geometric quantization of b-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional T-modules.

9:00 a.m. Vanishing theorems in the cohomology ring of the moduli space of parabolic bundles

Elisheva Adina Gamse*, University of Toronto (1125-51-2305)

Let Σ be a compact connected oriented 2-manfiold of genus g, and let p be a point on Σ. We define a space Sg(t) consisting of certain irreducible representations of the fundamental group of &Sigma\p , modulo conjugation by SU(N). This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if Σ has a Kähler structure then Sg(t) is the moduli space of parabolic vector bundles of rank N over Σ. For N=2, Weitsman considered a tautological line bundle on Sg(t), and proved that the (2g)th power of its first Chern class vanishes, as conjectured by Newstead. In this talk I will present his proof and then outline my extension of his work to SU(N) and to SO(2n + 1).

9:30 a.m. Positivity in Schubert Calculus, with Symplectic Analogs

R Goldin*, George Mason University (1125-51-2040)

Schubert calculus is the study of (complex) intersections of a specific set of varieties in the flag manifold; the intersection numbers and their generalizations to equivariant cohomology and K-theory are manifestly positive, defined in an appropriate sense. I will present results toward finding positive formulas for (some of) the structure constants in the corresponding rings, with generalizations to the symplectic category. Part of this work is joint with Allen Knutson, and part with Susan Tolman.

10:00 a.m. Invariants of pairs in SL(4,C) and SU(3,1)

Sean D Lawton*, George Mason University;
Krishnendu Gongopadhyay, Indian Institute of Science Education & Research, Mohali (1125-14-1016)

We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3,1).

10:30 a.m. Discussion

Friday, January 6, 2017, 1:00 pm – 5:50 pm (International C, International Level, Marriott Atlanta Marquis)

AMS-AWM Special Session on Symplectic Geometry, Moment Maps and Morse Theory, II

1:00 p.m. Beyond toric blow-ups

Sonja Hohloch, University of Antwerp;
Silvia Sabatini, University of Cologne;
Daniele Sepe, Federal Fluminense University;
Margaret Symington*, Mercer University (1125-53-2941)

Blowing up and down is an important tool in the study of symplectic manifolds. In dimension four, equivariant blow-ups of symplectic four-manifolds equipped with a T2-action or an S1-action are well understood. In this talk I will describe a blow-up that respects an S1 ×R-action but no T2-action. This blow-up can be “big” in the sense that given certain toric manifolds of dimension four, the exceptional sphere introduced may have greater area than can be achieved by a toric blow-up. I will discuss both topological and symplectic aspects of this blow-up, and explain how it can be implemented on certain completely integrable Hamiltonian systems with two degrees of freedom.

1:30 p.m. GKM graphs for odd dimensional manifolds with torus actions

Chen He*, Northeastern University (1125-53-844)

Let torus T act on a manifold M , if the equivariant cohomology H∗T(M) is a free module of H∗T (pt), then according to the Chang-Skjelbred Lemma, H∗T (M) can be determined by the 1-skeleton M1 consisting of fixed points and 1-dimensional orbits. Goresky, Kottwitz and MacPherson considered the case where M is an algebraic manifold and M1 is 2-dimensional, and introduced a graphic description of equivariant cohomology. In this paper, we follow those ideas to consider the case where M is an odd-dimensional (possibly non-orientable) manifold and M1 is 3-dimensional, and give a similar graphic description of equivariant cohomology.

2:00 p.m. Convexity property of Hamiltonian transversely symplectic manifolds

Yi Lin*, Georgia Southern University (1125-53-762)

In this talk, we introduce the notion of a Hamiltonian action on a transversely symplectic foliation. This provides a framework to study the Hamiltonian actions on many interesting singular ( possibly non-Hausdorff) symplectic spaces, such as symplectic orbifolds, symplectic quasi-folds ( by E. Prato), and the leaf spaces of characteristic Reeb foliations in both contact and co-symplectic geometries. We explain that under reasonable conditions, the components of a mo- ment map introduced by us are still Morse-Bott functions with even indexes. This in particular leads to a foliated version of the Atiyah-Guillemin-Sternberg-Kirwan convexity theorem. This talk is based on a recent joint work with R. Sjamaar.

2:30 p.m. Norm-square localization for Hamiltonian LG-spaces

Yiannis Loizides*, University of Toronto;
Eckhard Meinrenken, University of Toronto (1125-53-1577)

Let G be a compact, connected, simply connected Lie group, and let LG denote the loop group. There is a one-one correspondence between proper Hamiltonian LG-spaces and compact quasi-Hamiltonian G-spaces. One of the authors ([M]) has proposed a definition for the quantization of a quasi-Hamiltonian G-space as an element in the twisted K- homology of G (the latter is related to the ring of positive energy representations of LG via the Freed-Hopkins-Teleman theorem). We prove a ‘norm-square localization’ formula for the quantization of a quasi-Hamiltonian G-space, with terms indexed by the components of the critical set of the norm-square of the moment map of the corresponding Hamiltonian LG-space. An important application is to give a new proof of the quantization-commutes-with-reduction theorem for quasi-Hamiltonian spaces.

3:00 p.m. Dirac geometry of folded symplectic and b-symplectic structures

Geoffrey Scott*, University of Toronto (1125-53-2984)

The simplest examples of singularities that appear in presymplectic geometry and Poisson geometry are described by folded symplectic forms and b-symplectic forms, respectively. In this talk, I will show how certain techniques and celebrated results from folded and b-symplectic geometry generalize to the context of Dirac geometry, which is a common generalization of both presymplectic and Poisson geometry.

3:30 p.m. C0-characterization of symplectic embeddings via Lagrangian embeddings

Stefan Müller*, Georgia Southern University (1125-53-101)

We prove that an embedding of a (small) ball into a symplectic manifold is symplectic if and only if it preserves the shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back of a fixed primitive of the symplectic form by a Lagrangian embedding of a fixed manifold and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods. The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). In particular, we derive a new proof of the well-known C0-rigidity of symplectic embeddings (and diffeomorphisms). An advantage of our techniques is that they avoid the cumbersome distinction between symplectic and anti-symplectic, and also work well in the contact setting (which will be discussed only if time permits). We moreover demonstrate that the shape is often a natural language in symplectic topology. The talk is based on the preprint arXiv:1607.03135.

4:00 p.m. Product Structures for Generating Family Cohomology of Legendrian Submanifolds

Ziva Myer*, Bryn Mawr College (1125-53-1053)

In contact geometry, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one invariant, Generating Family Cohomology, by constructing a product structure on the cohomology groups. The construction uses moduli spaces of Morse flow trees – spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian. This product structure lays the foundation for an A-infinity algebra structure for the Legendrian that shows, in particular, that this product gives Generating Family Cohomology a ring structure.

4:30 p.m. On Fillability of Higher Dimensional Contact Manifolds Supporting Iterated Planar Open Books

Bahar Acu*, University of Southern California and University of California, Los Angeles (1125-53-176)

The Weinstein conjecture asserts that certain vector fields carry closed orbits. It was proven for all closed 3-dimensional manifolds by Taubes, but it is still open in higher dimensions. In this talk, we show that a (2n + 1)-dimensional contact manifold supporting an iterated planar open book decomposition, which will be defined in the talk, satisfies the Weinstein conjecture.

5:00 p.m. Symplectic and Contact Imprimitivity

François Ziegler*, Georgia Southern University (1125-53-3063)

A famous theorem of Mackey characterizes those unitary G-modules V that are induced from a closed subgroup H ⊂ G by the presence of a system of imprimitivity based on G/H: that is, a G-invariant, commutative C∗-subalgebra of End(V ) whose spectrum is, as a G-space, homogeneous and isomorphic to G/H. In this work, we similarly characterize those hamiltonian G-spaces X that are induced from H (in the sense of Kazhdan-Kostant-Sternberg, 1978) by the presence of a (symplectic) system of imprimitivity based on G/H: that is, a G-invariant, Poisson commutative subalgebra f of C∞(X), consisting of functions whose hamiltonian flow is complete, and such that the image of the moment map X → f∗ is homogeneous and isomorphic to G/H. Likewise, we characterize induced Kostant-Souriau bundles over hamiltonian G-spaces by the presence of a (contact) system of imprimitivity. This result is a key ingredient in the Mackey ‘normal subgroup analysis’ of hamiltonian and Kostant-Souriau G-spaces.

5:30 p.m. Discussion

Saturday, January 7, 2017, 8:00 a.m. – 12:00 p.m.

AMS-AWM Special Session on Special Session on Number Theory, I

8:00 am Ring-LWE for the number theorist

Yara Elias, Max Planck Institute for Mathematics;
Ekin Ozman, Boğaziçi University;
Kristin E. Lauter, Microsoft Research;
Katherine E Stange*, University of Colorado Boulder (1125-11-747)

The talk will give an overview for number theorists of the Ring-Learning-With-Errors problem, a number theoretical hard problem proposed for post-quantum cryptography. I will review joint work on this problem that came about as part of Women in Numbers 3.

8:30 am Curves with many automorphisms

Irene Bouw, Ulm University;
Wei Ho, University of Michigan;
Beth Malmskog, Villanova University;
Renate Scheidler, University of Calgary;
Padmavathi Srinivasan, Georgia Institute of Technology;
Christelle Vincent*, University of Vermont (1125-14-159)

For p an odd prime, we study a certain class of Artin-Schreier curves. The automorphism group of these curves contains a large extra special group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta function of the curves after extending the base field to contain the field of definition of the automorphisms. We find that over fields of square cardinality, these curves are either maximal or minimal, and we classify which curves fall into which category.

9:00 am The inverse Galois problem for symplectic groups

Valentijn Karemaker*, University of Pennsylvania
Sara Arias-de-Reyna, University of Seville;
Cécile Armana, Université de Franche-Comté;
Marusia Rebolledo, Université Blaise Pascal Clermont-Ferrand 2;
Lara Thomas, Université de Franche-Comté;
Núria Vila, University of Barcelona (1125-11-641)

The inverse Galois problem asks whether any finite group occurs as a Galois group. Given any prime number ?, we will construct a three-dimensional abelian variety A/ℚ such that the Galois representation attached to its ?-torsion realises the symplectic group GSp(6, ??) as a Galois group. This solves the inverse Galois problem for an infinite family of groups. The abelian variety will be the Jacobian variety of a curve whose behaviour at two distinct primes p and q satisfies certain congruence conditions.

9:30 am Explicit constructions of Ramanujan Bigraphs

Cristina M Ballantine*, College of the Holy Cross;
Brooke Feigon, The City College of New York;
Radhika Ganapathy, University of British Columbia;
Janne Kool, Max Plank Institute for Mathematics;
Kathrin Maurischat, University of Heidelberg;
Amy Wooding, McGill University (1125-20-746)

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of SU(3,ℚp). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

10:00 am Classification of Elliptic Fibrations of a Singular K3 Surface

Marie José Bertin, Jussieu Institute of Mathematics, Pierre and Marie Curie University, France;
Alice Garbagnai, University of Milan, Italy;
Ruthi Hortsch*, Bridge to Enter Advanced Mathematics;
Odile Lecacheux, Jussieu Institute of Mathematics, Pierre and Marie Curie University, France;
Makiko Mase, Tokyo Metropolitan University, Japan;
Cecília Salgado, Federal University of Rio de Janeiro, Brazil; Ursula Whitcher, University of Wisconsin-Eau Claire (1125-11-783)

We classify, up to automorphism, the elliptic fibrations on the singular K3 surface X associated with the Laurent polynomial
x + 1⁄x + y + 1⁄y + z + 1⁄z + x ⁄y + y⁄x +y⁄z + z⁄y + z⁄x + x⁄z
the transcendental lattice of which is isometric to ⟨6⟩ ⊕ ⟨2⟩.
In the paper, we give each elliptic fibration by Dynkin diagrams characterizing its reducible fibers, and the rank and
torsion of its Mordell-Weil group. We will review this and explain Nishiyama’s method, which was used to obtain this classification.

10:30 am Symmetries of Rational Functions Arising in Ecalle’s Study of Multiple Zeta Values

Adriana Salerno*, Bates College;
Damaris Schindler, Hausdorff Center for Mathematics;
Amanda Tucker, SUNY Geneseo (1125-11-1323)

In Ecalle’s theory of multiple zeta values he makes frequent use of certain properties that express symmetries of rational functions in several variables. We focus on the properties of push-invariance, circ-neutrality, and alternality. Ecalle states and uses several implications about the relations between these symmetries. In this talk we will introduce these concepts and prove two results: first, that push-invariance and circ-neutrality imply the first alternality relation, but not the more general alternality relations, and second, that alternality does, indeed, imply circ-neutrality.

11:00 am Orbital Integrals and Shalika Germs for ??n and ??2n

Sharon M Frechette*, College of the Holy Cross;
Lance Robson, Vancouver, British Columbia (1125-11-1139)

Shalika germs were introduced as a tool for studying orbital integrals, objects that play a large role in harmonic analysis on p-adic groups. The Shalika germ expansion expresses regular semisimple orbital integrals in terms of nilpotent ones, in a neighborhood of the origin. Exact values of Shalika germs elude computation, except for those of a few Lie algebras of small rank. We prove that Shalika germs on ??n and ??2n belong to a class of motivic functions defined by Cluckers and Loeser by means of a first-order language of logic (Denef-Pas language). The proof involves Nevins’ combinatorial matching between two parametrizations of nilpotent orbits: a parametrization involving partitions, and DeBacker’s parametrization arising from the Bruhat-Tits building. As a result, we establish bounds on the Shalika germs that are uniform in p. This is joint work with Julia Gordon and Lance Robson.

11:30 am , Transfer of Transfer

Thomas C. Hales, University of Pittsburgh;
Sharon Frechette, College of the Holy Cross;
Lance Robson, Vancouver, BC (1125-22-1605)

In the WIN project, we proved that Shalika germs for the special linear and for symplectic Lie algebras are ”motivic” (as explained in Sharon Frechette’s talk). In this talk, I will explain the broader context for this project, in particular, how Shalika germs appear in the proof of the Fundamental Lemma, and discuss subsequent work with T. C. Hales on transferring the Fundamental Lemma for smooth functions to positive characteristic. Our WIN project left some open questions in the general case; they can be circumvented for the the transfer of the Fundamental lemma, but an answer would greatly simplify the proof; I will survey these questions.

Saturday, January 7, 2017, 1:30 pm – 4:50 pm (A704, Atrium Level, Marriott Atlanta Marquis)

AMS-AWM Special Session on Special Session on Number Theory, II

1:30 pm Obstructions to the Hasse principle on Enriques surfaces

Francesca Balestrieri, University of Oxford;
Jennifer Berg*, Rice University;
Michelle Manes, University of Hawai’i;
Jennifer Park, University of Michigan (1125-11-712)

In 1970, Manin showed that the Brauer group can obstruct the existence of rational points, even when there exist points everywhere locally. Later, Skorobogatov defined a refinement of this Brauer-Manin obstruction, called the étale-Brauer obstruction. We show that this refined obstruction is necessary to understand failures of the Hasse principle on Enriques surfaces, thereby completing the case of Kodaira dimension 0 surfaces.

2:00 pm Galois action on Fermat curves

Rachel Davis, University of Wisconsin;
Rachel Pries, University of Colorado;
Vesna Stojanoska, UIUC;
Kirsten Wickelgren*, Georgia Institute of Technology (1125-11-1229)

Consider the Fermat curve xp + yp = 1 where p is an odd prime. Let K = Q(ζp) be the cyclotomic field. We extend work of Anderson about the action of the absolute Galois group GK on a relative homology group of the Fermat curve. Anderson proved that the action factors through Q = Gal(L/K) where L is the splitting field of 1 − (1 − xp)p. For p satisfying Vandiver’s conjecture, we find an explicit formula for the action of q ∈ Q on the relative homology. This is joint work by R. Davis, R. Pries, V. Stojanoska, and K. Wickelgren.

2:30 pm Fermat curves and Heisenberg extensions

Rachel Davis*, University of Wisconsin-Madison;
Rachel Pries, Colorado State University;
Vesna Stojanoska, University of Illinois at Urbana-Champaign;
Kirsten Wickelgren, Georgia Institute of Technology (1125-11-1253)

Using the explicit formula for the Galois action on Fermat curves from the first talk of this pair, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the outcome.

3:00 pm Kneser-Hecke operators for codes over finite chain rings

Amy Feaver, The King’s University;
Anna Haensch*, Duquesne University;
Jingbo Liu, University of Hong Kong;
Gabi Nebe, RWTH Aachen (1125-11-166)

There is a well known correspondence between lattices and codes. Via this classical construction, the weight enumerator for codes corresponds to the theta series for lattices, where one counts the number of codewords by composition, and the other counts the number of vectors in a lattice of a certain length. In this talk, we will explore how some of the attendant machinery of theta series are born out in this correspondence. In particular, we will consider the Kneser-Hecke operator, a code theoretic analogue of the classical Hecke operator.

3:30 pm Generalized Legendre Curves and Quternionic Multiplication

Alyson Deines, Center for Communications Research;
Jenny G. Fuselier, High Point University;
Ling Long, Louisiana State University;
Holly Swisher, Oregon State University;
Fang-Ting Tu*, Louisiana State University (1125-11-305)

We are going to construct abelian surfaces with quaternionic multiplication from certain generalized Legendre curves
yN =xi(1−x)j(1−λx)j, λ ∈ ℂ, N, i, j, k ∈ ℕ.
For a given generalized Legendre curve with parameter λ, we denote Jλnew the primitive part of its Jacobian variety. In this talk, we will give a criterion for End(Jλnew) containing quaternion algebra when N = 3, 4, 6.

4:00 pm p-adic q-expansion principle and families of automorphic forms on unitary groups of arbitrary signature

Ana Caraiani, Universität Bonn;
Ellen Eischen, University of Oregon;
Jessica Fintzen*, University of Michigan / University of Cambridge;
Elena Mantovan, Caltech;
Ila Varma, Columbia University (1125-11-1108)

We discuss a variant of the q-expansion principle (called the Serre-Tate expansion principle) for p-adic automorphic forms on unitary groups of arbitrary signature. We outline how this can be used to produce p-adic families of automorphic forms on unitary groups, which has applications to the construction of p-adic L-functions. This is done via an explicit description of the action of certain differential operators on the Serre-Tate expansion.

4:30 pm Bad reduction of genus 3 curves with Complex Multiplication

Irene Bouw, Ulm University;
Jenny Cooley, Department of Education, Warwick, England;
Elisa Lorenzo-Garcia, Leiden University;
Kristin Lauter, Microsoft Research;
Michelle Manes*, University of Hawaii at Manoa;
Rachel Newton, University of Reading;
Ekin Ozman, Boğaziçi University (1125-11-435)

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes ? of M such that the stable reduction of C at ? contains three irreducible components of genus 1.