# T.A.G. Seminar

## Central Michigan Topology, Analysis and Geometry Seminar

### 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 108.

Date Speaker Title
Sep 6 Daniel Corey (Univ of Wisconsin--Madison) The tropical Ceresa class
Oct 4 Sonmez Sahutoglu (Univ of Toledo) On regularity of the Berezin Transform.
Oct 11 Hitesh Gakhar (MSU) Persistent homology and quasiperiodicity in time series data
Oct 18 Mihai Fulger (U of Connecticut) Ample divisors on self-products of curves.
Oct 25 Aaron Pixton (MIT) Kappa rings and boundary vanishing
Nov 1 Timothy Clos (BGSU) Universality of Automorphisms on the Ball of Bounded Holomorphic Functions on the Polydisk
Nov 8 Funda Gultepe (U of Toledo) Space filling curves, Cannon-Thurston maps and boundaries of curve complexes
Nov 15 Andreas Gross (Colorado State) Tropical Jacobians, Theta Divisors, and the Poincaré Formula
Dec 6 Scott Corry (Lawrence University) Genus Bounds for Harmonic Group Actions on Finite Graphs
Jan 24 Alex Waldron (MSU) Gauge theory and geometric flows
Jan 31 Mark Pengitore (OSU) Unipotent translation-like actions on lattices in semisimple Lie groups
Feb 21 Karol Koziol (U of Mich) An introduction to the Langlands program
Feb 28 Jordan Watts (CMU) Classifying Spaces of Diffeological Groups
THE SEMINAR IS CANCELED FOR THE REST OF THE SEMINAR. STAY SAFE AND HEALTHY, AND SEE YOU IN FALL !!

### Abstracts

Speaker: Daniel Corey
Title: The Tropical Ceresa Class
Abstract: Let $X$ be an algebraic curve of genus at least 2. The Ceresa cycle is the homologically trivial algebraic cycle $X - [-1]_*X$ in $Jac(X)$. It is trivial when $X$ is hyperelliptic, and Ceresa proved that it is not algebraically equivalent to 0 for a generic curve of genus at least 3 defined over the complex numbers. We define the the tropical Ceresa class m(C) for a tropical curve $C$. This agrees with the l-adic Galois cohomology class associated to the Ceresa cycle of a curve defined over $C((t))$ whose tropicalization is $C$. We give a concrete way of computing this class and determining when it vanishes. In particular, we show that m(C) is torsion when C is unweighted. This is work in progress with Jordan Ellenberg and Wanlin Li.

Speaker: Hitesh Gakhar
Title: Persistent homology and quasiperiodicity in time series data
Abstract: Classically, sliding window embeddings were used in the study of dynamical systems to reconstruct the topology of underlying attractors from generic observation functions. In 2015, Perea and Harer developed a technique for recurrence detection in time series data using sliding window embeddings of L^2 periodic functions and persistent homology---which is an algebraic and computational tool used to quantify multiscale features of shapes. We define a quasiperiodic function as a superposition of periodic functions with non-commensurate harmonics. It turns out that sliding window embeddings of quasiperiodic functions are dense in high dimensional tori. The study of persistence of Rips filtration on these embeddings motivated our work on persistent Künneth theorems---that is, results relating persistent homology of two filtered spaces to persistent homology of their products (we define two such products). In this talk, I will present some theoretical results and demonstrate that in certain cases, a Künneth theorem helps recover quasiperiodicity better.

Speaker: Sonmez Sahutoglu
Title: On regularity of the Berezin Transform.
Abstract: For the open unit disk $\mathbb{D}$ in the complex plane, it is well known that if $\varphi \in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetilde{\varphi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of more general domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. This is joint work with Zeljko Cuckovic.

Speaker: Mihai Fulger
Title: Ample divisors on self-products of curves.
Abstract: Ample divisors are the heart of projective algebraic geometry. They control embeddings of a variety $X$ in projective spaces. A limit of ample divisors is called nef. These generate a closed convex cone in $H^2(X,\mathbb R)$ called the nef cone. It is interesting to compute the nef cone of $X$ in as many cases as possible. For curves, it is 1-dimensional, generated by the class of any point on the curve. Already for surfaces it becomes a mysterious object. One simple example of a surface is $X=C\times C$, where $C$ is a curve. The nef cone of $X$ is in general unknown, even for classes that are symmetric under the natural involution. The conjectural description of the symmetric classes is implied by the classical Nagata conjecture. We construct new (non-symmetric) nef classes on $X$. This is in joint work with Takumi Murayama.

Speaker: Aaron Pixton
Title: Kappa rings and boundary vanishing
Abstract: The moduli space of compact type curves of genus g parametrizes stable algebraic curves with compact Jacobians. The full cohomology of this moduli space is not very well understood, but in 2009 Pandharipande determined the structure of the kappa ring, a small subring of its rational cohomology ring. After reviewing this work, I'll explain how to use it to construct a nonzero class that restricts to zero on every boundary divisor of the moduli space. I'll conclude by giving a couple conjectures about a similar class on the moduli space of stable curves.

Speaker: Timothy Clos
Title: Universality of Automorphisms on the Ball of Bounded Holomorphic Functions on the Polydisk
Abstract: Given a sequence of automorphisms of the polydisk, we show that the associated composition semigroup homomorphisms on the ball of bounded holomorphic functions on the polydisk admit a universal inner function if a certain condition on the automorphisms are satisfied.

Speaker: Funda Gultepe
Title: Space filling curves, Cannon-Thurston maps and boundaries of curve complexes
Abstract: Given a hyperbolic 3- manifold which fibers over the circle with hyperbolic surface fiber, the inclusion map between the fiber and the manifold can be extended continuously to a map, resulting in a space-filling Peano curve. Such continuous extension of a map, in particular extension to a map between corresponding boundaries is called a 'Cannon-Thurston map' . In this talk we will discuss existence of Cannon-Thurston maps in different settings. In particular, we will explain how to construct a Cannon-Thurston map for the boundary of 'surviving' curve complex of a surface with punctures. Joint work with Christopher Leininger.

Speaker: Andreas Gross
Title: Tropical Jacobians, Theta Divisors, and the Poincaré Formula
Abstract: Poincaré’s formula relates the homology classes of the loci of effective divisors on the Jacobian of a curve with powers of the homology class of the theta divisor. In the tropical world, there are analogous versions of Jacobians, theta divisors, and homology, so it is possible to state an analogous formula using these tropical objects. In my talk, I will present joint work with Farbod Shokrieh in which we prove that this tropical Poincaré formula does indeed hold.

Speaker: Scott Corry
Title: Genus Bounds for Harmonic Group Actions on Finite Graphs
Abstract: In this talk I will present a discrete version of a classical story: sharp linear genus bounds for the sizes of automorphism groups of Riemann surfaces. After introducing and motivating the notion of a harmonic group action, I will explain the genus bounds in the case of graphs, the analogy with Riemann surfaces, and finish with an explicit connection between automorphism groups of graphs and Riemann surfaces that goes beyond pure analogy.

Speaker: Alex Waldron
Title: Gauge theory and geometric flows
Abstract: I will give a brief introduction to two major areas of research in differential geometry: gauge theory and geometric flows. I'll then introduce a geometric flow (Yang-Mills flow) arising from a variational problem with origins in physics, which has been studied by geometric analysts since the early 1980s. I'll conclude by discussing my own work on the behavior of Yang-Mills flow in the critical dimension ($n = 4$).

Speaker: Mark Pengitore
Title: Unipotent translation-like actions on lattices in semisimple Lie groups
Abstract: The Gersten conjecture says that a group being hyperbolic is equivalent to having no Baumslag-Solitar subgroups. This is known to be false due to work of Brady. While there are some weaker versions still open, we are interested in a geometric reformulation of the Gersten conjecture using translation-like actions. To be more specific, the geometric Gersten conjecture asks whether hyperbolicity is equivalent to having no translation-like action by any Baumslag-solitar group. In this talk, we show that cocompact lattices in real semisimple Lie groups admit translation-like actions by cocompact lattices in the unipotent part of the Iwasawa decomposition of the original Lie group. In particular, we demonstrate that $\mathbb{Z}^n$ acts translation-like on the fundamental group of any closed hyperbolic $n+1$ manifold which provides counterexamples to the geometric Gersten conjecture. This is joint work with Ben McReynolds.

Speaker: Karol Koziol
Title: An introduction to the Langlands program
Abstract: One of the crowning achievements of 20th century mathematics is Class Field Theory, which has its origins in Gauss' Law of Quadratic Reciprocity, and which (among other things) gives a description of all abelian field extensions of the rational numbers. This turns out to be the tip of a rather large iceberg known as the Langlands Conjectures, a vast program aimed at bridging the fields of Number Theory, Representation Theory, and Algebraic Geometry. I'll give an introduction to some aspects of these conjectures, and try to indicate other related incarnations.

Speaker: Jordan Watts
Title: Classifying Spaces of Diffeological Groups
Abstract: Fix an irrational number $A$, and consider the action of the group of pairs of integers on the real line defined as follows: the pair $(m,n)$ sends a point $x$ to $x + m + nA$. The orbits of this action are dense, and so the quotient topology on the orbit space is trivial. Any reasonable notion of smooth function on the orbit space is constant. However, the orbit space is a group: the orbits of the action are cosets of a normal subgroup. Can we give the space any type of useful "smooth" group structure? The answer is "yes": its natural diffeological group structure. It turns out this is not just some pathological example. Known in the literature as the irrational torus, as well as the infra-circle, this diffeological group is diffeomorphic to the quotient of the torus by the irrational Kronecker flow, it has a Lie algebra equal to the real line, and given two irrational numbers A and B, the resulting irrational tori are diffeomorphic if and only if there is a fractional linear transformation with integer coefficients relating $A$ and $B$, and so it is of interest in many fields of mathematics. Moreover, it shows up in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles, the main topic of this talk. We will perform Milnor's construction in the realm of diffeology to obtain a diffeological classifying space for a diffeological group G, such as the irrational torus. After mentioning a few hoped-for properties, we then construct a connection 1-form on the G-bundle $EG \to BG$, which will naturally pull back to a connection 1-form on sufficiently nice principal G-bundles. We then look at what this can tell us about irrational torus bundles. (Joint work with Jean-Pierre Magnot)

Speaker: Debraj Chakrabarti
Title: Reproducing Kernel Hilbert Spacess
Abstract: In this expository talk, we discuss the definition and basic properties of Reproducing Kernel Hilbert Spaces, and give some examples of this interesting and important notion.

### Fall 2018 and Spring 2019

 Date Speaker Title Sep 7 Jordan Watts (CMU) An Introduction to Smootheology Sep 14 Anirban Dawn (CMU) An abstract Fejér theorem and some applications Sep 20 2:30 to 3:30 pm Note special day and time Place: Pearce 307 Nikita Nikolaev (Université de Genève) Abelianisation of Logarithmic $\mathfrak{sl}(2)$-Connections Sep 21 Rasul Shafikov (UWO) On rationally convex embeddings and immersions of real manifolds in complex spaces Sep 28 Yoav Len (Univ of Georgia) Lifting Tropical Intersections Oct 5 David Jensen (Univ of Kentucky) Linear Systems on General Curves of Fixed Gonality Oct 12 Jordan Watts (CMU) Differential Forms on Orbit Spaces Oct 19 Nicola Tarasca (Rutgers University) $K$-classes of Brill-Noether varieties and a determinantal formula Oct 26 James Heffers (UMich--Ann Arbor) Geometric Properties of Upper Level Sets of Lelong Numbers of Currents on $\mathbb{P}^2$ Nov 9 Alexander Izzo (BGSU) Polynomial hulls and analytic structure Nov 16 Matthew Satriano (Univ of Waterloo) Interpolating Between the Batyrev--Manin and Malle Conjectures Nov 30 Martino Fassina(UIUC) Type conditions for real hypersurfaces in $\mathbb{C}^n$ Dec 7 Nathan Priddis (Brigham Young University) BHK Mirror symmetry and beyond Jan 25 2pm, Pearce 138 Note special time and place Karen Smith (Univ of Michigan) AMS Grad Student Chapter event Resolution of Singularities Feb 8 Harry Richman (Univ of Michigan) Distribution of tropical Weierstrass points Feb 15 Trieu Le (Univ of Toledo) Algebraic properties of $m$-isometries Feb 22 Neha Prabhu (Queen's Univ, Canada) Fluctuations in the distribution of Hecke eigenvalues Mar 1 Mohit Bansil (MSU) Spectral Theory of Kohn Laplacians on Spheres Mar 15 Zhenghui Huo (Univ of Toledo) Weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle. Mar 29 Luke Edholm (Univ of Michigan) Derivatives of Integral operators and Mapping Properties. Apr 12 Seth Wolbert (Univ of Manitoba, Canada) Short exact sequences of Lie group(oid)s, weak actions, and presentations of actions on stacks Apr 19 4:15 pm Note special time Yuan Zhang (Purdue-Fort Wayne) A rigidity phenomenon for CR submanifolds Apr 26 Dmitry Zakharov (CMU) The failure of the local-to-global principle for cubic equations

### Abstracts

Speaker: Jordan Watts
Title: Introduction to Smootheology
Abstract: Smooth manifolds form a successful theory in geometry that uses topology to generalize real analysis from Euclidean space to much more general shapes. For example, think of how the 2-torus locally looks like a disk in the plane at every point. This is exactly what a manifold is: a space that locally looks like a disk/ball in Euclidean space. However, in practice, mathematics is full of singular spaces, where singularities spoil the local structure, but we still need to do analysis at this locations. Think of the apex of the double cone $z^2 = x^2+y^2$ in 3-space. In this talk, I will introduce you to a few attempts to generalize smooth manifolds to bigger categories in order to include singular and function spaces, and mention some of their pros and cons.

Speaker: Anirban Dawn
Title: An Abstract Fejér theorem and some applications
Abstract: A famous theorem of Fejér (1899) states that if $f$ is a continuous function on the unit circle $\mathbb{T}$, then the Cesàro means of the partial sums of the Fourier series of $f$ converge to the function $f$ uniformly. Du Bois-Reymond produced an example of a continuous function on $\mathbb{T}$ where the partial sums do not converge to the function, even pointwise. In this talk we will give a generalization of Fejér's result. Let $\mathbb{T}^n$ be the $n$-dimensional unit torus, $X$ be a quasi-complete locally convex topological vector space and $\sigma$ be a continuous representation of $\mathbb{T}^n$ on $X$. For $x \in X$, we define the Fourier series of $x$ with respect to the representation $\sigma$ and we show that the Cesàro means of the partial sums of the Fourier series converge to $x$ in the topology of $X$. After that we give some applications of this theorem to Fourier series and spaces of holomorphic functions.

Speaker: Nikita Nikolaev
Title: Abelianisation of Logarithmic $\mathfrak{sl}(2)$-Connections
Abstract: I will describe an approach to studying meromorphic connections on vector bundles called abelianisation. This technique has its origins in the works of Fock-Goncharov (2006) and Gaiotto-Moore-Neitzke (2013), as well as the WKB analysis. Its essence is to put rank-$n$ connections on a complex curve $X$ in correspondence with much simpler objects: connections on line bundles over an $n$-fold cover $\Sigma\to X$. The point of view is similar in spirit to abelianisation of Higgs bundles, aka the spectral correspondence: Higgs bundles on $X$ are put in correspondence with rank-one Higgs line bundles on a spectral cover $\Sigma\to X$. However, unlike Higgs bundles, abelianisation of connections requires the introduction of a new object, which we call the Voros cocycle. The Voros cocycle is a cohomological way to encode objects such as ideal triangulations that appeared in Fock-Goncharov, spectral networks that appeared in Gaiotto-Moore-Neitzke, as well as the connection matrices appearing in the WKB analysis. By focusing our attention on the simplest case of logarithmic singularities with generic residues, I will describe an equivalence of categories, which I call the abelianisation functor, between $\mathfrak{sl}(2)$-connections on $X$ satisfying a certain transversality condition and rank-one connections on an appropriate 2-fold spectral cover $\Sigma \to X$. This presentation is based on the work completed in my thesis (2018) and recent extensions thereof.

Speaker: Rasul Shafikov
Title: On rationally convex embeddings and immersions of real manifolds in complex spaces.
Abstract: A classical result of Duval-Sibony characterizes rationally convex totally real embeddings of real manifolds into $\mathbb C^n$ as those that are isotropic with respect to some Kahler form. In this talk I will describe some generalizations of this result for topological embeddings and immersions, and will discuss some applications.

Speaker: Yoav Len
Title: Lifting Tropical Intersections
Abstract: My talk is concerned with combinatorial aspects of intersection theory. When tropicalizing algebraic varieties, each of their intersection points maps to a tropical intersection point. Characterizing this locus is a fundamental problem in tropical ge- ometry. In my talk, I will appeal to non-Archimedean and poly- hedral geometry to characterize the locus in various cases. The solution leads to a combinatorial tool for counting multi-tangent hyperplanes of algebraic varieties, detecting dual defects, and for computing Newton polygons of dual varieties. I will not assume any knowledge in tropical geometry.

Speaker: David Jensen
Title: Linear Systems on General Curves of Fixed Gonality
Abstract: The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Speaker: Jordan Watts
Title: Differential Forms on Orbit Spaces
Abstract: Differential forms are mathematical constructs used in a wide variety of contexts: they are what you integrate in calculus, they provide extra structure to spaces such as curvature or a symplectic structure, and these in turn are used to describe physical systems. Moreover, differential forms are important for obtaining topological invariants, such as de Rham cohomology. In this talk, we will discuss differential forms in the context of symmetry: a famous result of Koszul states that the singular cohomology of an orbit space coming from a compact Lie group action is the same as the cohomology of the basic differential forms (forms that respect the symmetry). In fact, if the action is free (trivial stabilisers), then the de Rham complex of forms on the orbit space is isomorphic to the complex of basic differential forms. Using the theory of diffeology, this relationship between differential forms extends to the non-free case; that is, we allow the orbit space to have singularities. Consequently, we have a triple of isomorphic cohomology theories: that from basic differential forms, singular cohomology on the orbit space, and diffeological de Rham cohomology on the orbit space.

Speaker: Nicola Tarasca
Title: $K$-classes of Brill-Noether varieties and a determinantal formula
Abstract: Brill-Noether varieties for pointed curves parametrize linear series on curves with prescribed vanishing at marked points. I will present a formula for the Euler characteristic of the structure sheaf of Brill-Noether varieties for curves with at most two marked points. The formula recovers the classical Castelnuovo number in the zero-dimensional case, and previous work of Eisenbud-Harris, Pirola, Chan-López-Pflueger-Teixidor in the one-dimensional case. The result follows from a new determinantal formula for the K-theory class of certain degeneracy loci of maps of flag bundles. This is joint work with Dave Anderson and Linda Chen.

Speaker: James Heffers
Title: Geometric Properties of Upper Level Sets of Lelong Numbers of Currents on $\mathbb{P}^2$
Abstract: Lelong numbers are a useful tool for complex analysts wanting to look at the mass a current $T$ has at a given point. In this talk we look at the geometric properties of sets of points where a current $T$ has large" Lelong numbers, and see that the points where our current has large Lelong number can be contained in a small subspace of $\mathbb{P}^2$. The talk will start with introductory definitions and some simple examples, then look at the following result. Let $T$ be a positive closed current of bidegree $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $\alpha > 2/5$ and $\beta = (2-2\alpha)/3$ it has been shown that if $T$ has four points with Lelong number at least $\alpha$, the upper level set $E_{\beta}^+ (T)$ of points of $T$ with Lelong number strictly larger than $\beta$ is contained within a conic with the exception of at most one point. We will investigate some examples showing the sharpness of the assumptions of this results as well as look into some interesting components of its proof.

Speaker: Alexander Izzo
Title: Polynomial Hulls and Analytic structure
Abstract: It was once hoped that whenever a compact set in complex Euclidean space has a nontrivial polynomial hull, the hull must contain an analytic disc. This hope was shattered by a counterexample given by Stolzenberg in 1963. Over the 55 years since then, several additional constructions of hulls without analytic discs have been found. Nevertheless, the issue of analytic structure in polynomial hulls is still not well understood. I will present new results regarding (the absence of) analytic structure in polynomial hulls.

Speaker: Matthew Satriano
Title: Interpolating Between the Batyrev--Manin and Malle Conjectures
Abstract: The Batyrev-Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.

Speaker: Martino Fassina
Title: Type conditions for real hypersurfaces in $\mathbb{C}^n$
Abstract: To every point of a real hypersurface in complex space, one can attach an invariant which measures the maximum order of contact of the hypersurface with complex $q$-dimensional varieties. This number is known as the $q$-type and was first introduced by D'Angelo. Since real hypersurfaces are the boundaries of complex domains, their geometry has deep interactions with the function theory of $\mathbb{C}^n$. In particular, work of Kohn and Catlin has related the $q$-type to the local boundary regularity of the Cauchy-Riemann equations. In this talk I will compare different ways of measuring the $q$-type of a real hypersurface that appear in the literature. I will then show how this work can be exploited to obtain quantitative information on the regularity of some complex PDEs.

Speaker: Nathan Priddis
Title: BHK Mirror symmetry and beyond
Abstract: I will discuss a phenomenon from physics called Mirror symmetry, and a particularly nice version of mirror symmetry called BHK mirror symmetry after Berglund--Huebsch and Krawitz. Then I will discuss two variations on Berglund--Huebsch mirror symmetry for Borcea--Voisin models, and for nonabelian Landau--Ginzburg models.

Speaker: Karen Smith
Title: Resolution of Singularities
Abstract: Algebraic varieties are geometric objects defined by polynomials – you have known many examples since high school, where you learned that a circle can be defined by a polynomial equation such as $x^2+y^2=1$. Polynomials can define incredibly complicated shapes, such as a mechanical arm in medical software or Woody’s arm in Toy Story, but yet they can be easily manipulated by hand or by computer. For this practical reason, algebraic geometry – the study of algebraic varieties and the equations that define them – is a central research area within modern mathematics. It is also one of the oldest and most beautiful. In this talk, I hope to share my love of the subject, which stems from the way the geometry and algebra interact, including some open problems and my favorite tools for attacking them.

Speaker: Harry Richman
Title: Distribution of tropical Weierstrass points
Abstract: The set of (higher) Weierstrass points on a metric graph of genus g > 1 is an analogue of the set of N-torsion points on a circle. As N grows, the torsion points "distribute evenly" over a circle. This makes it natural to ask how Weierstrass points distribute on a graph, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on metric graphs (i.e. tropical curves) in analogy with Riemann surfaces (i.e. complex algebraic curves), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.

Speaker: Trieu Le
Title: Algebraic properties of $m$-isometries
Abstract: $m$-Isometries are Hilbert space bounded linear operators satisfying an operator equation that generalizes the notion of isometries. These operators were introduced by Agler in the early nineties. A few years ago, Bermudez, Martinon, and Noda proved that the sum of an isometry and a commuting nilpotent operator of order $s$ is a $(2s-1)$-isometry. This interesting result shows that any Jordan block whose entries on the main diagonal have modulus one is an $m$-isometry for an appropriate value of $m$. Their proof, which makes use of combinatorial arguments, is quite lengthy and complicated. In this talk, I will discuss my approach which not only provides an elegant proof of the above result but also allows us to obtain more general results.

Speaker: Neha Prabhu
Title: Fluctuations in the distribution of Hecke eigenvalues
Abstract: A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor et al) predicts that the normalised p-th Fourier coefficients of a typical Hecke eigenform follow the semicircle distribution as the primes p grow. On averaging over families of Hecke eigenforms, we study the distribution of the deviations about the average value, and obtain a central limit theorem. This is joint work with Kaneenika Sinha.

Speaker: Mohit Bansil
Title: Spectral Theory of Kohn Laplacians on Spheres
Abstract: A CR-manifold is a submanifold in $\mathbb{C}^M$ with extra structure stipulating that the dimension of the complex part of its tangent space is pointwise invariant under some complex structure map. The Kohn Laplacian $\Box_b$ is a second order differential operator intrinsically defined on any CR-manifold whose spectrum reveals important geometric information. In the case of the Rossi sphere $(\mathbb{S}^3, \mathcal{L}^t)$ showing that 0 is in the essential spectrum of $\Box_b$ is enough to conclude that $(\mathbb{S}^3, \mathcal{L}^t)$ is not embeddable into $\mathbb{C}^M$ for any $M > 0$. One approach is to study the Kohn Laplacian on the subspaces of spherical harmonics in $L^2(\mathbb{S}^{2N-1})$. When restricted to finite dimensional subspaces of spherical harmonics, $\Box_b$ can be expressed as a matrix and one can either explicitly compute or at least obtain bounds on the eigenvalues. Furthermore, when the CR structure is induced from the ambient manifold $\mathbb{C}^M$ (unperturbed operator), the spectrum of $\Box_b$ is explicitly computed on any sphere $\mathbb{S}^{2N-1}\subset\mathbb{C}^M$ by Folland. In this project we expand on the previous work and study asymptotics of the spectrum on the Rossi sphere, and we provide upper and lower bounds on the maximum eigenvalues on the invariant subspaces. In the unperturbed case we examine the eigenvalue counting function and obtain asymptotics analogous to Weyl’s law.

Speaker: Zhenghui Huo
Title: Weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle.
Abstract: The boundedness of the Bergman projection on the weighted $L^p$ space of the unit ball was first studied by Bekollé and Bonami in the 80s. Using modern dyadic harmonic analysis techniques, sharp weighted $L^p$ estimates were obtained for the Bergman projection on the upper half plane by Pott and Reguera in 2012, and on the unit ball in $\mathbb C^n$ by Rahm, Tchoundja, and Wick in 2016. In this talk, I will introduce the dyadic operator technique used for these results and give a sharp weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle. This work is joint with Brett Wick.

Speaker: Luke Edholm
Title: Derivatives of Integral operators and Mapping Properties.
Abstract: The Bergman projection on the ball in $\mathbb{C}^n$ is known to extend to a bounded operator on the $L^p$-Sobolev spaces for the full range $p\in(1, \infty)$ Similar results hold for many classes of smoothly bounded, pseudoconvex domains. In 2016, Edholm and McNeal showed a limited $L^p$ mapping range for the Bergman projection on a class of domains called generalized Hartogs triangles, but derivatives were not considered in this work. In this talk, we show that the study of Sobolev mapping properties of the Bergman projection on these domains (and any other Reinhardt domain) requires analysis of new kernels which are not conjugate symmetric. This leads to many surprising consequences. We give a complete answer to the $L^p$-Sobolev mapping properties of the Bergman projection on the generalized Hartogs triangles.

Speaker: Seth Wolbert
TitleShort exact sequences of Lie group(oid)s, weak actions, and presentations of actions on stacks
Abstract: It is a well known result from algebra that, given a short exact sequence of Lie groups $1\rightarrow K \rightarrow G \rightarrow H \rightarrow 1$, we can express $G$ as a semidirect product associated to an action of $H$ on $K$ exactly when there exists a splitting (i.e., a smooth homomorphism section) $\sigma:H\to G$. Of course, such a section may fail to exist. However, in the case where a smooth section $\sigma$ which is not a homomorphism exists, $H$ admits what might be called a weak action on $K$: an "action" which fails the axioms of a group action in a controlled way. More generally, one may regard $G$ as (a presentation of) an action of $H$ on $K$ regarded as stacks. In my talk, I will discuss these previously known results as well as my work to extend these results to yield an equivalence between Lie groupoid fibrations (the analogous object to a short exact sequence in the world of Lie groupoids) and actions of a Lie groupoid on a stack.

Speaker: Yuan Zhang
Title: A rigidity phenomenon for CR submanifolds
Abstract: In Projective Geometry, it is a classical result that submanifolds of $\mathbf P^N$ with vanishing second fundamental forms or with degenerate Gauss maps are necessarily totally geodesic. In this talk, we discuss some similar phenomenon of CR submanifolds in CR geometry.

Speaker: Dmitry Zakharov
Title: The failure of the local-to-global principle for cubic equations
Abstract: Consider the differential equation $f'(z)=g(z)$ for an unknown function $f$ on an open set $U$ in the complex plane. This equation can be solved as a formal power series at any point of $U$. However, these local solutions can be patched together into a global solution on all of U only if U is simply connected, in other words, only if $H^1(U)=0.$ We can ask a very similar question in number theory. Suppose that a polynomial equation $f(x,y,z)=0$ with integer coefficients has a solution in every completion of the rational numbers, in other words, in the real numbers and in the p-adic numbers for all primes $p$. Does it follow that the equation has a rational solution? A theorem of Hasse and Minkowski states that this is true if f is a quadratic polynomial, so we say that quadratic equations satisfy the local-global principle. In my lecture, I will describe an example, due to Selmer, of a cubic polynomial f for which the local-global principle fails.

## Fall 2017 and Spring 2018

### 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.