GEOMETRY/TOPOLOGY SEMINAR (PAST TALKS)

Ever since I learned that a coffee cup accepts graph embeddings that a sphere cannot, I can’t stopped telling my students. Linking in the Gauss-Bonnet theorem, Morse theory, a little crochet, and some careful considerations of non-euclidean pizza, and you have a great conversation with a class. After several iterations, I have tested a few topics and can share what works for me and what draws yawns and crickets. I hope to describe ideas for other things I haven’t tried yet. Audience members are encouraged to share their experiences.

Dimension 4 is unlike the other dimensions. There are many simply-connected closed 4-manifolds that admit infinitely many distinct smooth structures, and surprisingly, there are no smooth 4-manifolds known to have only finitely many smooth structures. Also, classification problems for smooth, simply-connected 4-manifolds are far from fully understood. For example, the generalized Poincaré conjecture is true topologically in all dimensions, but unknown smoothly in dimension 4. Attempts to resolve this conjecture have led to constructions of 4-manifolds that are homeomorphic, but not diffeomorphic (such manifolds are called "exotic"), with the particular goal of constructing exotic 4-spheres. In this first talk, I will mostly focus on the simply-connected symplectic 4-manifolds, describe the topological invariants, and give examples.

Continuing from last time, we will get into more details about symplectic manifolds. Then, we will provide background about Lefschetz fibrations and mapping class groups, which have a very nice and useful interaction with symplectic manifolds. Finally, we will introduce the tools we use to construct exotic 4-manifolds, like the symplectic connected sum and Luttinger surgery.

In this part, we will talk about some results we obtained and what we have covered in the first two parts. First, we will present a new construction of symplectic 4-manifolds that are homeomorphic but not diffeomorphic to \((2h+2k-1)\mathbb{CP}^{2}\#(6h+2k+3)\overline{\mathbb{CP}}^{2}\) with \((h,k) \neq (0,1)\), via Lefschetz fibrations and Luttinger surgery on the product manifolds \(\Sigma_g \times T^2\). In the second half, we will construct families of Lefschetz fibrations over \(S^2\) using finite order cyclic group actions on \(\Sigma_g \times \Sigma_g\). These are joint works with Anar Akhmedov.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Complex hyperbolic space is a fairly natural complex analog of the more familiar real hyperbolic space. However, despite the similarities in construction, the geometric features of these two spaces can be quite different, and questions of the form "If [statement] is true in real hyperbolic space, is it also true in complex hyperbolic space?" might require vastly different techniques to answer. In this talk, I plan to give a gentle introduction to (complex) hyperbolic geometry.

As we saw last time, the boundary of complex hyperbolic 2-space is topologically \(S^3\), but due to the natural action of \(\operatorname{PU}(2,1)\), it geometrically inherits the structure of the (1-point compactification of the) Heisenberg group. In the early 2000's, Schwartz discovered that one could actually find a real hyperbolic 3-manifold in this strange \(S^3\), and so it seems natural to ask which other 3-manifolds can arise in the boundary of complex hyperbolic 2-space. In this talk, I will introduce Falbel's program for finding \(\operatorname{PU}(2,1)\) representations of 3-manifold groups and summarize some known results about 3-manifolds.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.

The mapping class group is an algebraic invariant of a topological space that detects the symmetries of that space. In the case of surfaces, this group has a deep connection with the fundamental group and the associated Teichmuller space. In this talk I'll give a introduction to the mapping class group along with some examples and intuition for some interesting results surrounding it. Time permitted, I'll also discuss some associated results for so-called "big" mapping class groups.

In the 1940's, Jakob Nielsen set about analyzing and classifying the elements in the mapping class group for closed orientable surfaces. In the 1970's, while working on his famous Geometrization Conjecture, Thurston managed to successfully complete what we now call the Nielsen-Thurston Classification of mapping classes. In this talk I'll motivate the classification on the torus, highlight some parallels with hyperbolic isometries, and present an overview of Thurston's proof for surfaces with genus g>1.

Although 4-manifolds are outside our imagination, there is a very nice way to encode the topological information in the so-called handlebody diagram. In this talk, we'll first start with basic descriptions and we'll see examples of handlebody diagrams for low-dimensional manifolds. Time-permitted, we will see how to obtain more suitable handlebody diagrams. Namely, we will modify it using two fundamental topological operations: handle cancelation/creation and handle sliding.

Last time we worked on handlebody diagrams of 1- and 2-dimensional manifolds. In the second part, we will describe how to obtain more suitable handlebody diagrams via two fundamental topological operations: handle cancelation/creation and handle sliding. In this talk we will see handlebody diagrams of 3- and 4-manifolds as well.

Braid group theory is an interesting and versatile subject with applications in many different fields of mathematics including algebra, topology, and quantum computation. In this talk, I will give an introduction to the braid groups and share my intuition for why and how these groups are used. In particular, I will discuss the representations of the braid groups and some of the motivating open questions that fuel my research. Many of the famous representations of the braid groups are parametrized by a variable \(q\) (these representations secretly come from quantum groups). I will share some of my results about choosing careful specializations of \(q\) with the aim of structural results about the image of the representation.

In this talk, I will discuss some of the building blocks of complex hyperbolic geometry. In doing so, I will point out some differences between real hyperbolic space and complex hyperbolic space including: the various analogous models in both settings, the difference in setup using symmetric bilinear forms and Hermitian forms, and the nature of distance, isometries, and geodesic subspaces. I will also discuss the connection of the boundary at infinity of the complex hyperbolic plane to the Heisenberg group. This introduction is aimed to provide the necessary background materials to discuss how to derive presentations for a special class of lattices in \(\mathbf{H}_{\mathbb{C}}^2\) known as the Picard modular groups.

In this talk, I will discuss a method for obtaining group presentations for a particular class of lattices in the complex hyperbolic plane known as the Picard modular groups. The method deals with finding an appropriate covering of \(\mathbf{H}_{\mathbb{C}}^2\) by translates of a collection of horoballs under the action of a discrete group, \(\Gamma\), and appealing to a theorem of Macbeath. We will focus on the application of this method to the case where our lattice has a single cusp, and as examples, we will derive presentations for the Picard modular groups when d=2 and d =11. The presentations for the Picard modular groups in the cases d=2,11 completes the list of presentations for Picard modular groups with entries from Euclidean domains. We will also discuss how the method changes when our lattice has more than one cusp.

First, we will give a brief introduction to symplectic manifolds and Lefschetz fibrations. Then we will talk about Mapping Class Groups and their connection with the Lefschetz Fibration. Finally, we will provide examples. This talk will be a preparation for the next talk, which I will talk about my recent work.

We construct families of Lefschetz fibrations over \(S^2\) using finite order cyclic group actions on the product manifolds \(\Sigma_g \times \Sigma_g\) for \(g>0\). We also obtain more families of Lefschetz fibrations by applying the rational blow-down operation to these Lefschetz fibrations. This is a joint work with Anar Akhmedov and Mohan Bhupal.

In this talk I will go through a combinatorial approach to understanding/measuring curvature. It is accessible to undergraduates and particularly interesting for those that want to understand a discrete version of the Gauss-Bonnet Theorem. There are a few related topics that I would like to cover in this short talk. As a theme, my focus will be how I engage undergraduates with topics that sit in the intersection of geometry, topology, and combinatorics.

At one time or another, most have probably found themselves bored in class with a pencil and ruler, doodling tilings of a plane with triangles. These triangular tilings can be realized in terms of group actions, and for a particularly nice family of triangles, we have the so-called "triangle groups." In this talk, I'll motivate triangle groups and we'll explore some of the interplay between the group theory and geometry (possibly with a sprinkling of arithmeticity, if time allows).

Last time we looked at groups generated by reflections in the sides of triangles and ultimately were able to reconcile the geometric intuition with the algebra and classify these triangles by arithmeticity. It's then natural to ask if we can play this same game with other (convex) polygons or other higher-dimensional polytopes. As one might expect, keeping track of the geometric features becomes considerably harder as the combinatorial complexity of the polygon increases. Instead, the slightly more natural approach (which generalizes to higher dimensions) is to study which polygons (or polytopes) can arise from a given algebraic construction.

In 1975, E. Vinberg produced an algorithm for finding the convex polytopes, and in her 2015 PhD thesis, A. Mark gives an algorithm that improves upon Vinberg's original ideas in the setting of totally real quadratic number fields and obtains some effective bounds. In this talk, I'd like to motivate the algebraic setting and outline Mark's algorithm. Provided time allows, I'd like to also discuss some recent joint work with Mark for extending her algorithm to the more general case of totally real (Galois) number fields.

Continued fractions arise when trying to find rational numbers that approximate some fixed real number \(x\). Using a process which emulates the Euclidean algorithm, one produces a (possibly-finite) sequence of integers which encodes a sequence of rational numbers converging to \(x\), and the terms in this integral sequence are called a the continued fraction digits of \(x\). Given that there are other mathematical objects that somehow generalize the integers and the real numbers (a lattice in a real Lie group, for example), it is natural to ask if there is also a reasonable notion of continued fractions within these objects. In 2013, Lukyanenko and Vandehey affirmatively answer this question in the context of the Heisenberg group, which has a particularly interesting geometry associated with it. In this talk I'll go over the authors' construction, survey some of their results as compared with classical results, and (time-permitted) discuss some open questions for further generalizations in this direction.

• Anton Lukyanenko and Joseph Vandehey. “Continued fractions on the Heisenberg group”. In: Acta Arith. 167.1 (2015), pp. 19–42. issn: 0065-1036.
  (arXiv preprint)

We present a basic theory toward a hypothetical undergraduate class on complex geometry and symplectic geometry. Conventional wisdom says that the visualization of spaces of more than three dimensions is not feasible. However, when one focuses their studies on complex vector spaces, (finite-dimensional vector spaces over the field of complex numbers, or \( \mathbb{C}^n \) ), there is a visualization. We introduce the notion of complex lines, complex angles, and the utility of the symplectic product via this presentation and demonstrate some applications

In pursuit of a proof that topologically stable maps form an open dense subset of \(C^{\infty}(M, N)\), Thom and Mather laid the foundation for the theory of stratified spaces. In this talk, I will go over some of the key ideas they developed.

Complex geometry is now inseparable from modern mathematical physics. In this talk I'll discuss why that is and also show some basic constructions and calculations you can perform for common string vacua.

The Hirzebruch signature theorem is a special case of the Atiyah-Singer index theorem, and shows that the signature of a compact, oriented, even dimensional manifold (the signature of the intersection form) is equal to the index of a first order operator of Dirac type, the signature operator (the square root of the Hodge - de Rham laplacian). Hirzebruch showed that the signature vanishes for every \(n\)-manifold arising as the boundary of some \( (n+1) \)-dimensional manifold. Even so, interesting phenomena can arise if one varies the underlying Riemannian structure, by utilizing the family version of the index theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula and Quillen’s determinant line bundle. Such is the case of a two-torus and varying the conformal class of flat Riemannian metrics. By reinterpreting as the variation of complex structure on an elliptic curve, we construct an elliptic fibration over a rational base, the so-called j-line. Then the RRGQ formula for the family of (complexified) fibrewise signature operators results in a generalized cohomology class that is known in physics as a measure of the local and global anomaly. The anomalies will be explicitly computed, and if time permits, it will be shown how the anomalies can be resolved by combining several anomalous operators.

Based off https://arxiv.org/abs/1907.07313.

This talk will be a very basic introduction to connections (also called gauge fields in physics). We start with the definition of a connection, curvature, and gauge equivalence of connections. We then briefly discuss how Maxwell's equations can be written in terms of connections, together with their direct generalization as Yang-Mills equations. We also introduce the Chern-Simons functional as another example of a gauge equation.

It is a classical result of Grossman that mapping class groups of finite type surfaces are residually finite. In recent years, residual finiteness growth functions of groups have attracted much interest; these are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity. Residual finiteness growth functions detect many subtle properties of groups, including linearity. In this talk, I will discuss some recent joint work with Thomas Koberda on residual finiteness growth for mapping class groups, adapted to nilpotent and solvable quotients of the underlying surface group.
(Joint with Thomas Koberda)

Persistence theory is a very recent phenomena in mathematics originally developed independently by Ferri, Robins, and Edelsbrunner at the turn of the 21st century. This talk will focus more so on the work of the last author and the applications of persistence to Topological Data Analysis (TDA). The main tools here are persistent homology, simplicial complexes, and spectral sequences. There are also many applications outside of TDA such as in Riemannian Geometry and geometric group theory which we will explore if time permits.

Certain minimal blocking sets in various finite projective planes will be discussed. A blocking set is a set of points \( S \) for which every line of the plane meets \( S \) in at least one point, but \( S \) contains no line of the plane. A blocking set \( S \) is minimal if no proper subset of \( S \) is a blocking set. Along with the classical (Desarguesian) projective plane of order \(q\), coordinatized by a field of order \(q\), these sets will be sought in certain non-classical projective planes arising from a nearfield of order \(q\) . Such planes contain both \(q^2 + q + 1\) points and lines, where \(q\) is an integer power of a prime \(p\). For each plane, aspects of the \(p\)-code will also be discussed, such as the dimension, spanning sets from a geometric viewpoint, and the presence of the characteristic vectors of the aforementioned sets as codewords. The \(p\)-code of a projective plane \( \pi \) is the subspace of the vector space of dimension \(q^2 + q + 1\) over \( \mathbb{Z}_p \) spanned by the characteristic vectors of the lines of \( \pi \). In other words, the \(p\)-code of \( \pi \) is the rowspace over \( \mathbb{Z}_p \) of an incidence matrix of \( \pi \), where rows and columns are indexed by lines and points of \( \pi \), respectively.

We will consider operators in the classical sense as a linear map between two Hilbert spaces over a field of complex numbers and state (without proof) the classical Atkinson's theorem and Atiyah-Janich theorem. We will discuss a way to generalize these theorems when the field of complex numbers is replaced by an algebra of operators.