Postdoc Seminar

Postdoc Seminar @ Northwestern University

Organizers: Adi Glücksam and Noah Mark Riggenbach
Time and Place: Fortnightly on Thursdays at 10AM, Lunt 102.

What is it?

It's a place for you to practice future talks, discuss you research, or anything that you find interesting. and share it with fellow postdocs in an informal laid back atmosphere.
If you would like to join the mailing list, please email one of the organizers (see emails above).

May 30th- Martin Bishop

$A_r$ singularities and the integral Chow ring of $ \overline{\mathcal M}_{g,n}(1,3)$ Martin Bishop
Abstract: In this talk we discuss how the consideration of $A_r$ singularities, singularities which are of the form $k[x,y]/(y^2-x^{r+1})$, arose from the historical development of $\mathcal M_{g,n}$ and $\overline{\mathcal M}_{g,n}$, and how the inclusion of these singularities enables computations of the integral Chow ring of $\overline{\mathcal M}_{g,n}$ for new values of $(g,n)$.

Future Talks:

TBD

Past Talks: Spring 2024:

May 23rd

Wondering about the shapes of Wandering domains Adi Glücksam
Abstract: Complex dynamics explores the evolution of points under iteration of functions of complex variables. For a transcendental entire function, wandering domains are regions of stability that are not eventually periodic. In this talk I will explore the possible shapes of wandering domains and expose our idea of how to construct such shapes for functions of finite order. This talk is based on work in progress with L. Pardo-Sim ́on.

May 16th

Does Ricci flow preserve the Anosov property? Solly Coles
Abstract: In a 2004 paper, Manning asked whether the Anosov property for geodesic flows is preserved when the defining metric evolves under Ricci flow. A closely related question is whether a metric with no conjugate points can develop them under Ricci flow.
In joint work (in progress) with Keith Burns and Dong Chen, we construct a metric on a surface that acquires conjugate points when it evolves under Ricci flow. We use this to give a negative answer to Manning's original question. In this talk, I will describe the connection between conjugate points and the Anosov property, before outlining our construction.

May 9th

SYZ fibration duality and non-archimedean geometry Hang Yuan
Abstract: We will describe a precise formulation of SYZ conjecture based on a toy model of the fibration duality between the natural logarithm map in complex and non-archimedean geometry. We will also talk about how explicit examples can be produced.

April 25th

The X-Ray Transform on Euclidean and Hyperbolic Disks via Projective Equivalence Joey Zou
Abstract: The X-ray transform is an operator which takes in a function and a geodesic and returns the integral of the function along the geodesic. We discuss recent joint works (with Mishra-Monard and Eptaminitakis-Monard) studying sharp mapping properties of weighted X-ray transforms on the Euclidean disk and hyperbolic disk. The latter work uses a projective equivalence between the Euclidean and hyperbolic disks via the Beltrami-Klein model, where one can view geodesics in the hyperbolic disk as Euclidean geodesics up to reparametrization, allowing us to pass results about the X-ray transform on one space to the other.

April 18th

Zeros of Dirichlet L-functions Hua Lin
Abstract: We discuss why zeros of the Riemann zeta function and families of Dirichlet L-functions are important objects of study. Specifically, we talk about the one-level density of zeros of Dirichlet L-functions over function fields and its connection to random matrices and questions of non-vanishing.

April 11th

Pinned squares and prong-matchings Bradley Zykoski
Abstract: In this talk, I will first summarize key aspects of the BCGGM multi-scale compactification of strata of abelian differentials. I will then discuss how a significant technical detail of this construction (“prong-matching”) is exhibited at the level of $L^\infty$-Delaunay triangulations.

April 4th

Extending our Understanding of Ambient Obstruction Solitons Erin Griffin
Abstract: This talk will discuss the speaker's upcoming paper joint with Poddar, Sharma, and Wylie that defines and investigates what we call extended ambient obstruction solitons. Focusing on how this new notion enables us to further the study the traditional notion of ambient obstruction solitons as examined in the speaker's 2021 paper, we ultimately show that all closed ambient obstruction solitons are ambient obstruction flat. We conclude by looking at how this notion applies to more explicit examples of Bach solitons and extended Bach solitons in dimension $n=4$.

Past Talks: Winter 2024:

March 7th

The Helmholtz Solution Operator is Algebraically Bounded Most of the Time Joey Zou
Abstract: We discuss recent works by Lafontaine-Spence-Wunsch and Spence-Wunsch-Zou, which consider the solution operator to the Helmholtz equation (an eigenvalue problem for the Laplacian operator) with a one-parameter family of parameters (either a spectral parameter or varying the wave speed). The norm of the solution operator depends on the frequency parameter, and in certain cases, corresponding to trapping for Hamiltonian dynamics for a certain Hamiltonian, the norm can be very large (i.e. super-algebraic) in terms of the frequency. The works discussed show that, even in the trapping cases, this does not happen "most" of the time, i.e. the norm is algebraically bounded excluding a small measure set of parameters. The proof uses techniques from complex analysis, viewing the solution operator as a meromorphic function of the parameter, and using Jensen's formula and a modified version of Hadamard's three lines theorem (originally used in works of Tang and Zworski) to bound the number of poles of this meromorphic function, as well as its size away from poles.

February 29th

Growth of Torsion of Elliptic Curves over Imaginary Quadratic Fields with Class Number 1 Irmak Balcik
Abstract: In this talk, we will examine the effect that quadratic base change of an elliptic curve $E$ over a number field $K$ has on its torsion subgroup $E(K)_{\text{tor}}.$ If $K=\mathbb{Q},$ it has been extensively studied. As a step toward number fields greater than $\mathbb{Q}$ , we will start with elliptic curves defined over a fixed quadratic field. The proofs require the use of several techniques ranging from the Galois action to the study of quadratic points on certain modular curves.

February 22nd

Counting periodic orbits by their linking numbers Solly Coles
Abstract: Given an Anosov flow on a closed 3-manifold M, we fix a periodic orbit $K$ (which is a knot in M), and count the number of other periodic orbits with a prescribed length and homology class in the complement $M/K$. When M is a homology 3-sphere, this homology class corresponds to the linking number with K. We show that this counting problem can be rephrased in terms of symbolic dynamics and invariant measures. With this rephrasing, one can apply a result of Babillot-Ledrappier to obtain asymptotics for the count, as a function of orbit length.

February 8th

Diffeomorphisms of disks, special values of zeta functions, and algebraic K-theory Noah Mark Riggenbach
Abstract: As Chuck Weibel says in the preface of his book "The $K$-book:" "The ... interplay of algebra, geometry, and topology in K-theory provides a fascinating glimpse of the unity of mathematics." In this talk I will review the some of this interplay, specifically the connection of diffoemeorphisms of disks with the K-theory of the sphere spectrum and zeta values with $K$-theory of the integers. I will also talk about work, joint with Mejia and Ray, refining the known connection between these two objects and discuss a Corollary (pointed out to us by Mandell) using this to generalize results of Blumberg-Mandell-Yuan.

February 1st

Multi-fractal spectrum of planar harmonic measure Adi Glücksam
Abstract: In this talk, I will define various notions of the multi-fractal spectrum of harmonic measures and discuss finer features of the relationship between them and properties of the corresponding conformal maps. Furthermore, I will describe the role of multifractal formalism and dynamics in the universal counterparts. (it looks like my talk from October 19th, but it is different!)
This talk is based on a joint work with I. Binder.

Past Talks: Fall 2023:

November 9th

The Laplacian is self-adjoint on $L^2$, and other half-truths I learned Joey Zou
Abstract: Sometimes analysts and geometers will say something like "the Laplacian is self-adjoint on $L^2$". This is an abbreviated version of a much subtler statement involving what the domain and adjoint of an unbounded operator really means; such considerations usually involve boundary conditions. In this talk I will go over these considerations, discuss the idea of a "boundary triplet" which allows one to characterize all self-adjoint realizations of an operator, and apply this idea to current research regarding a degenerate version of the Laplacian.

October 19th

October 12th

Coordinate Systems for Orbit Closures of Translation Surfaces Bradley Zykoski
Abstract: Translation surfaces are locally Euclidean geometric structures that arise in multiple contexts, most notably in billiards dynamics and in complex algebraic geometry. Their Euclidean geometry determines a natural triangulation, the Delaunay triangulation, that gives rise to local systems of coordinates on the moduli space of all translation surfaces. This moduli space admits an action by $GL(2,\mathbb R)$, and orbit closures of this action are widely studied (see e.g. the Magic Wand Theorem and the Illumination Problem). I am interested in understanding the intersections of these orbit closures with the Delaunay coordinate charts. I will present a finiteness result for low-dimensional orbit closures, discuss the conjecture of how this result generalizes, and discuss how this result extends my dissertation work on the topology of these moduli spaces.

September 28th

The unreasonable good behavior of NK Noah Mark Riggenbach
Abstract: This is a modified version of the talk I will be giving at the midwest topology seminar (MTS) this weekend. The first half will be mostly background on what NK is and what we know about it, and will be a more detailed version of what I will go through for the first part of my talk at MTS. I will then use the back half to talk about some of the amazing properties of NK which is part of my joint work with Elmanto but will not be covered in much/any detail in my MTS talk.

Past Talks: Spring 2023:

May 11th

Waves on Lorentzian Manifolds: A Geometric Surfari Katrina Morgan
Abstract: We will explore the wave equation on various Lorentzian manifolds. This choice of geometry is particularly relevant for studying wave propagation in the context of relativity, where the geometry of spacetime is described by a Lorentzian metric. The goal of the talk is to explore how properties of the manifold impact wave dispersion and propagation. In particular we will discuss what is known about waves on globally hyperbolic spacetimes, which have well-behaved causal structures. Furthermore, we will see what happens in a particular case where global hyperbolicity fails and strange causal phenomenon are allowed. Wave dispersion on asymptotically flat manifolds will also be discussed.

May 4th

Hippopotomonstrosesquipedaliophobia (the fear of long words) Itay Glazer
Abstract: Every word $w(x_1,...,x_r)$ in a free group, such as the commutator word $w=xyx^{-1}y^{-1}$, induces a word map $w:G^r\rightarrow G$ on every group $G$. For $g\in G$, it is natural to ask whether the equation $w(x_1,...,x_r)=g$ has a solution in $G^r$, and to estimate the "size" of this solution set, in a suitable sense. When $G$ is finite, or more generally a compact group, this becomes a probabilistic problem of analyzing the distribution of $w(x_1,...,x_r)$, for Haar-random elements $x_1,...,x_r \in G$. When $G$ is an algebraic group, such as $SL_n(\mathbb C)$, one can study the geometry of the polynomial map $w:SL_n(\mathbb C)^r\rightarrow SL_n(\mathbb C)$, using algebraic methods. Such problems have been studied in the last few decades, in various settings such as finite simple groups, compact p-adic groups, compact Lie groups, and simple algebraic groups. Analogous problems have been studied for Lie algebra word maps as well. In this talk, I will mention some of these results, and explain the tight connections between the probabilistic and algebraic approaches. Based on joint works with Yotam Hendel, Raf Cluckers and Nir Avni.
Notes

April 20th

Parameterized s-cobordisms Noah Mark Riggenbach
Abstract: One of the major results of mathematics in the 20th century was the h-cobordism theorem, proven by Smale, and the s-cobordism theorem, proven by Barden, Mazur, and Stallings. These thereoms give rise to a complete invariant to distinguish between particularly nice cobordisms of a given manifold and was used in the solution of the Poincaré conjecture in high dimensions. If one has a family of cobordisms instead of a single cobordism, an extension of this invariant was constructed by Waldhausen, Jarhen, and Rognes, but the group that this invariant is valued in can be quite mysterious. In this talk I will discuss my work on computing these groups for negatively curved manifolds.

April 6th, 10AM, Lunt 102

Wave propagation on rotating cosmic string backgrounds Katrina Morgan
Abstract: A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.

Multi-fractal spectrum of planar harmonic measure Adi Glücksam
Abstract: In this talk, I will discuss various notions of the multi-fractal spectrum of harmonic measures and their relationships with properties of the corresponding conformal maps. I will discuss some finer features of this relationship. This talk is based on a joint work in progress with I. Binder (University of Toronto).

March 2nd, Lunt 101, 10AM

Unbounded fast escaping wandering domains Adi Glücksam
Abstract: Complex dynamics explores the evolution of points under iteration of functions of complex variables. In this talk I will introduce into the context of complex dynamics, a new approximation tool allowing us to construct new examples of entire functions and show new possible dynamical behaviours. In particular, we answer a question of Rippon and Stallard from 2012 about unbounded wandering domains with unbounded orbits, and provide a collection of examples supporting a conjecture of Baker. This talk is based on a joint work with V. Evdoridou, and L. Pardo-Sim ́on, and will be given in `colloquium style', i.e., all the definitions and basic concepts will be discussed during the talk.