# CMU Analysis and Geometry Seminar

## Fall 2017 and Spring 2018

### Old Seminar pages

Spring 2014, Fall 2015, Spring 2015, Fall 2014 , Fall 2016 and Spring 2017,

### Organizers

If you would like to give a talk, please email one of us!

### Time and Place

Fridays, 3:00–4:00pm, Pearce Hall, Room 223.

### Schedule

 Date Speaker Title Sep 1 Debraj Chakrabarti (CMU) Bergman spaces on Reinhardt Domains Sep 8 Anirban Dawn (CMU) Partitions of Unity Sep 22 Dmitry Zakharov (CMU) Divisors on graphs Sep 29 Dmitry Zakharov (CMU) Tropical Brill-Noether theory Oct 6 Ilya Kachkovsky (Michigan State University) Band edges of 2D periodic Schrodinger operators Oct 13 Sid Graham (CMU) The Prime Number Theorem Oct 20 Martin Ulirsch (University of Michiagn, Ann Arbor) Tropical geometry of the Hodge bundle Oct 27 Mythily Ramaswamy (TIFR, India) Control of PDE models Nov 3 Tim Reynhout (CMU) Partition of Unity for Symplectic Volumes of Ribbon Graph Complexes. Nov 10 Sivaram Narayan Complex Symmetric Composition Operators on the Hardy Space Nov 15 Chaya Norton (Concordia University) Differentials with real periods and the geometry of M_g SPECIAL DATE AND TIME: 2 pm at PE 223 Nov 17 Anthony Vasaturo (University of Toledo) Carleson measures and Douglas' question on the Bergman space on the disk Dec 1 Luke Edholm (Univ of Michigan, Ann Arbor) The Leray Operator on Two Dimensional Model Domains Dec 8 Adam Coffman (IUPU Fort Wayne) An Example for Green's Theorem with Discontinuous Partial Derivatives Jan 19 Felix Janda (Univ of Michigan, Ann Arbor) Moduli of meromorphic functions on algebraic curves Feb 9 Nathan Grieve (Michigan State University) On complexity of rational points and arithmetic of linear series Feb 16 Anirban Dawn (CMU) A Theorem of Grothendieck Feb 23 Tanuj Gupta (CMU) Hörmander's theorem for the Cauchy-Riemann operator: the one-variable case Mar 30 Zeljko Cuckovic (University of Toledo) $L^p$ Regularity of Bergman Projections on Domains in $\mathbb{C}^n$ Apr 13 Matthew Woolf (UIC) Stable Cohomology of Moduli Spaces of Sheaves on Surfaces Apr 20 Steven Rayan (University of Saskatchewan) Asymptotic geometry of hyperpolygons Apr 27 Eric Bucher (MSU) Introducing cluster algebras and their applications

### Abstracts

Speaker:Debraj Chakrabarti
Title: Bergman spaces on Reinhardt domains
Abstract: Let $\Omega$ be a possibly non-smooth Reinhardt domain in $\mathbb{C}^n$ , and let $A^p(\Omega)$ be the Banach space of holomorphic functions on $\Omega$ whose $p$-th powers are integrable, $p\geq 1$. We study properties of $A^p(\Omega)$ as a linear space, for example, the question of convergence of Laurent series of functions in $A^p(\Omega)$ in the norm of $A^p(\Omega)$, and that of determining the dual of $A^p(\Omega)$. These questions have unsurprising answers when $\Omega$ is the unit disc in the plane. We show there are new phenomena in the general situation, some only partially understood. In particular we look at the special case of the Hartog's triangle, where some of the computations can be performed explicitly. This is joint work with Luke Edholm and Jeff McNeal.

Speaker:Anirban Dawn
Title: Partitions of Unity (Expository Talk)
Abstract: In this talk I will introduce a very interesting and well known topic of Differential Topology known as "Partitions of Unity". A partition of unity on a differential manifold $\mathcal{M}$ is a collection of $\mathcal{C}^{\infty}$ (smooth) functions $\{\phi_i : i \in \textit{I} \}$ on $\mathcal{M}$, where $\textit{I}$ is an arbitrary index set, not assumed to be countable, such that the collection of supports $\{\text{supp}(\phi_i) : i\in \textit{I} \}$ is locally finite. Moreover, for any point $p \in \mathcal{M}$ we have $\phi_i(p) \geq 0$ and $\sum_{i \in \textit{I}} \phi_i(p) = 1$. In my talk I will prove the existence of partitions of unity of $\mathcal{M}$ subordinate to an open cover $\{\mathcal{U}_{\alpha} : \alpha \in \Lambda\}$. The proof will need some ideas of topics from point set topology, for example refinement of a cover, local finiteness, paracompactness, which will also be discussed. After that I plan go over the construction of a $\mathcal{C}^{\infty}$ function with compact support on $\mathcal{M}$ which I need to prove the existence. At the end, I will try to talk about some applications of partitions of unity.

Speaker:Dmitry Zakharaov
Title: Divisors on graphs
Abstract: It has long been understood in combinatorics that there is a remarkable similarity between graphs and algebraic curves, also known as Riemann surfaces. In the last decade or so, it was recognized that this relationship is not a coincidence: graphs are one-dimensional algebraic varieties from the point of view of tropical mathematics. In the first talk, I will introduce divisor theory on graphs. A divisor on a graph is a linear combination of the vertices with integer coefficients. We define an equivalence relationship on divisors, using so-called chip-firing moves, which is similar to linear equivalence of divisors on curves. We can then define the standard objects of curve theory — meromorphic functions and their divisors, the complete linear system associated to a divisor, the genus of a graph, the canonical class — for graphs, and prove analogues of theorems such as the Riemann—Roch theorem and Clifford's theorem. This talk will be elementary and will not use any concept more advanced than that of an abelian group.

Speaker:Dmitry Zakharaov
Title: Tropical Brill-Noether theory
Abstract: In the second talk, I will explain the relationship between the theories of divisors on graphs and on algebraic curves. Given a family of smooth algebraic curves degenerating to a singular curve, the intersections between the irreducible components of the singular curve define a graph, called the dual graph of the family. Baker’s specialization lemma then establishes a relationship between the divisor theory on the degenerating family and on the dual graph. This relationship enables us to reduce algebro-geometric questions, such as the existence or non-existence of line bundles with prescribed properties, to combinatorial problems on graphs, which are often more tractable. Conversely, it is possible to establish new results in graph theory using known results in algebraic geometry.

Speaker:Ilya Kachkovskiy
Title: Band edges of 2D periodic Schrodinger operators.
Abstract: : The band-gap structure of the spectra of periodic Schrodinger operators can be described in terms of spectral band functions which manifest dispersion relations between the energy and the quasimomentum. It is widely believed that, by a small perturbation of the potential, one can make all the extrema of these functions non-degenerate, i. e. isolated and having non-vanishing Hessians. This is also equivalent to the notion of effective mass being well defined for generic potentials. We establish the “isolated” part for all sufficiently regular periodic potentials, without the need of a perturbation. The talk is based on the joint work with Nikolay Filonov.

Speaker:Sid Graham
Title: The Prime Number Theorem.
Abstract: For a positive real number $x$, let $\pi(x)$ denote the number of primes that are less than or equal to $x$. The Prime Number Theorem (PNT) states that $\pi(x)$ is asymptotically $x/\log x$. This was first proved independently by Hadamard and de la Vall\'ee Poussin in 1896. In 1899, de la Vall\'ee Poussin gave a version of PNT with an explicit error term. In this talk, I will sketch a proof of the Prime Number Theorem. It turns out that PNT is equivalent to showing that the Riemann-zeta function has no zeros with real part 1. The primary tools in the proof are the Hadamard Factorization Theorem and the Functional Equation of the Riemann-zeta function.

Speaker: Martin Ulirsch
Title: Tropical geometry of the Hodge bundle
Abstract: The Hodge bundle is a vector bundle over the moduli space of curves whose fiber over a smooth curve (of genus g) is the space of abelian differentials on this curve. We may define a tropical analogue of its projectivization as the moduli space of pairs (\Gamma, D) consisting of a stable tropical curve \Gamma and an effective divisor D in the canonical linear system on \Gamma. This tropical Hodge bundle turns out to have dimension 5g-5, while it is a classical fact that the projective Hodge bundle has dimension 4g-4. This means that not every pair (\Gamma, D) in the tropical Hodge bundle arises as the tropicalization of a suitable element in the algebraic Hodge bundle. In this talk I am going outline a comprehensive (and completely combinatorial) solution to the realizability problem, which asks us to determine the locus of points in the tropical Hodge bundle that arise as tropicalizations. Our approach is based on recent work of Bainbridge-Chen-Gendron-Grushevsky-Möller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne. This talk is based on joint work with Bo Lin, as well as on an ongoing project with Martin Möller and Annette Werner.

Speaker:Mythily Ramaswamy
Title: Control of PDE models (Colloquium Talk)
Abstract: Starting with a brief introduction to control of ODE systems, I will discuss similar issues for PDE systems. The focus will be on linear viscoelastic fluid flow models, a system of coupled partial differential equations for velocity and stress.

Speaker: Tim Reynhout
Title: Partition of Unity for Symplectic Volumes of Ribbon Graph Complexes.
Abstract: As an example of Anirban's talk on partitions of unity, we will briefly develop the idea of ribbon graph complexes and the tools necessary to create the partition of unity used for calculating their volume. Ribbon graphs are combinatorially defined objects which can be viewed as graphs on open or closed Riemann surfaces. This allows for results to be extended to the moduli space of Riemann Surfaces.

Speaker:Sivaram Narayan
Title: Complex Symmetric Composition Operators on the Hardy Space
Abstract: We say that a bounded operator $T$ on a complex Hilbert space $H$ is complex symmetric if there exists a conjugation (i.e., a conjugate linear, isometric involution) $J$ such that $T=JT^*J$. In this talk, we will first discuss a few general results about complex symmetric operators on a Hilbert space. We will then focus for most of the talk on the complex symmetry of composition operators $C_\varphi f=f\circ \varphi$ induced on the Hardy space $H^2$ by analytic self-maps $\varphi$ of the open unit disk $\mathbb{D}$. We show that there are complex symmetric composition operators on $H^2$ induced by $\varphi$ that are linear-fractional but not automorphisms. In doing so, we answer a recent question of Noor, and partially answer the original problem posed by Garcia and Hammond. This is a joint work with Sievewright and Thompson. We will briefly mention the work in progress (with Sievewright and Tjani) on the weighted Hardy spaces.

Speaker: Chaya Norton
Title: Differentials with real periods and the geometry of $M_g$
Abstract: A differential on a Riemann surface is called real normalized (RN) if the result of integrating the differential along any closed loop is real. Alternatively the imaginary part of the anti-derivative of the differential is a well-defined harmonic function on the Riemann surface, and in this form RN differentials have been studied by Maxwell. In the 80s Krichever introduced RN differentials in the development of the spectral theory of the non-stationary Schrodinger operator. The moduli space M_g is roughly the collection of genus g Riemann surfaces up to isomorphism. Relatively recently Grushevsky and Krichever have shown that RN differentials provide a useful perspective to study the geometry of M_g by noticing that for any fixed singular part of a differential at marked points, there exists a unique RN differential. We will introduce these objects and attempt to outline the perspective it provides on the geometry of M_g and vanishing tautological classes. In joint work with Grushevsky and Krichever we developed the degeneration theory for RN differentials.

Speaker:Anthony Vasaturo
Title: Carleson measures and Douglas' question on the Bergman space on the disk
Abstract: Motivated by Douglas' question about the invertibility of Toeplitz operators on the Hardy Space, we study a related question concerning the Berezin transform and averaging function of a Carleson measure for the weighted Bergman space of the disc. As a consequence, we obtain a necessary and sufficient condition for the invertibility of Toeplitz operators whose symbols are averaging functions of these Carleson measures.

Speaker: Luke Edholm
Title: The Leray Operator on Two Dimensional Model Domains
Abstract: One major difference between complex analysis in one and several variables is the lack of a true analogue to the one-variable Cauchy transform, $\mathbf{C}$. However, by looking at domains satisfying a convexity condition, we are able to construct the Leray transform, $\mathbb{L}$, which shares many of $\mathbf{C}$'s familiar properties. A significant amount of recent work has been done to study the mapping properties of $\mathbb{L}$ in various settings. This talk will focus on a family of model domains in $\mathbb{C}^2$, and discuss new techinques used in the analysis of the Leray operator. These models can be used to locally approximate a very general class of domains, and it is expected that the theorems in the model case will carry over to the general case. I will also discuss what these results mean in terms of dual CR structures on hypersurfaces in projective space. This is joint work with Dave Barrett.

Title: An Example for Green's Theorem with Discontinuous Partial Derivatives
Abstract: Green’s Theorem in multivariable calculus is usually stated with a hypothesis that the partial derivatives are continuous. I will present an example of a function where the partial derivatives exist but are discontinuous, to which a stronger version of Green’s Theorem applies.

Speaker:Felix Janda
Title: Moduli of meromorphic functions on algebraic curves
Abstract: I will discuss three different ways to compactify (the class of) the locus of meromorphic functions on algebraic curves with prescribed zeros and poles. One of them is called the double ramification cycle.

Speaker:Nathan Grieve
Title: On complexity of rational points and arithmetic of linear series Abstract: I will survey recent results which pertain to diophantine and arithmetic aspects of linear series on projective varieties. One theme is that complexity of rational points should be measured on rational curves. These results are consequences of Schmidt's Subspace Theorem. Further, I will explain how these theorems can be interpreted using ideas from toric geometry. For example, I will indicate connections to the theories of Chow forms and Okounkov bodies.

Speaker:Anirban Dawn
Title: A Theorem of Grothendieck.
Abstract: It is an important problem in analysis to find the duals of function spaces. One of these is the Fréchet space $\mathcal{O}(G)$, the space of all holomorphic functions on an open set $G \subset \mathbb{C}$. Following work by da Silva Dias and Köthe, it was shown by Grothendieck (1953) that $\mathcal{O}(G)^{*}$, the dual of $\mathcal{O}(G)$, is naturally isomorphic to $\mathcal{O}_{0}(\hat{\mathbb{C}} \setminus G)$, the space of holomorphic functions in a neighborhood of $\hat{\mathbb{C}} \setminus G$ which vanish at $\infty \in \hat{\mathbb{C}}$. We will prove the result for the special case when $G$ is the open unit disc and will describe the main ideas of the proof in the general case.

Speaker:Tanuj Gupta
Title: Hörmander's theorem for the Cauchy-Riemann operator: the one-variable case.
Abstract: Many problems in complex analysis can be reduced to solving the inhomogeneous Cauchy-Riemann equations, which is referred to as a $\overline{\partial}$-problem. In 1965, L. Hörmander proved a remarkable estimate for the $\overline{\partial}$-problem in $L^2$ norms. In this talk we discuss the statement and proof of this result in the case of domains in the complex plane.

Speaker:Zeljko Cuckovic
Title: $L^p$ Regularity of Bergman Projections on Domains in $\mathbb{C}^n$
Abstract: Bergman projections and Bergman kernels are among the central objects in complex analysis. In this talk we will discuss the $L^p$ regularity of weighted Bergman projections on various domains in $\mathbb{C}^n$. Then we will show an $L^p$ irregularity of weighted Bergman projections on complete Reinhardt domains with exponentially decaying weights (joint work with Yunus Zeytuncu). Finally we establish estimates of the $L^p$ norms of Bergman projections on strongly pseudoconvex domains.

Speaker:Matthew Woolf
Title: Stable Cohomology of Moduli Spaces of Sheaves on Surfaces
Abstract: On any algebraic surface, there is a moduli space of torsion-free stable sheaves with fixed rank and Chern classes. In general, these moduli spaces can be very badly behaved. However, if you fix the rank and determinant and let the second Chern class go to infinity, things change. In this case, the moduli spaces become nonempty and irreducible of the expected dimension, and the singular locus has arbitrarily large codimension. In this talk, I will discuss work with Izzet Coskun that suggests these spaces are in fact converging, at least homotopically. Specifically, we will show that for certain surfaces, the Betti numbers of these moduli spaces stabilize. Moreover, there is a simple formula for the stable Betti numbers which is independent of rank and first Chern class. If time permits, we will discuss how to think of this as a variation of the Atiyah-Jones conjecture.

Speaker:Steven Rayan
Title: Asymptotic geometry of hyperpolygons
Abstract: Nakajima quiver varieties lie at the interface of geometry and representation theory and provide an important class of examples of Calabi-Yau manifolds. I will discuss a particular instance, hyperpolygon space, which arises from a certain shape of quiver. The simplest of these is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we can speculate on how this classification might be extended by studying the geometry of hyperpolygons at "infinity". This talk represents previous work with Jonathan Fisher and ongoing work with Hartmut Weiss.

Speaker:Eric Bucher
Title: Introducing cluster algebras and their applications
Abstract: Cluster algebras were first invented by Fomin and Zelevinsky in 2003 to study total positivity of canonical bases. Since their inception, these mathematical objects have popped up in a large variety of seemingly unrelated areas including: Teichmuller theory, Calabi-Yao categories, integrable systems, coordinate rings of Grassmannians and the study of high energy particle physics. In this talk we will lay the basic groundwork for working with cluster algebras as well as discuss a few of their applications to the above areas. This talk is intended to be introductory so no background or definitions will be assumed. The intention is to have everyone walk away having learned about this new and fascinating mathematical object.