Union College Mathematics Conference Abstracts

Union College 2016 Mathematics Conference Abstracts

A PDF with times and locations of talks can be found here.

Plenary Talks
Algebraic Topology Talks
Applied Topology Talks
Category Theory Talks
Differential Geometry and Geometric Analysis Talks
Number Theory Talks

Plenary Talks

Christina Sormani

Converging Spaces and their ADM Mass

We will begin by introducing the ADM mass of an asymptotically flat spacelike manifold, emphasizing the example of a single star or black hole. Schoen and Yau proved in their famous positive mass theorem that any such manifold with nonnegative scalar curvature has mass greater than or equal to 0 and if it is 0 then the the manifold is Euclidean space. Dan Lee and I have conjectured that if such a manifold has mass close to 0 then the manifold is close to 0 in the intrinsic flat sense. I will describe what it means for a sequence of manifolds to converge in this intrinsic flat sense and various partial solutions of this conjecture completed in joint work with Dan Lee, Lan-Hsuan Huang and Iva Stavrov. I will also discuss related work with Philippe LeFloch and work of Jeff Jauregui with Dan Lee.

Ravi Ramakrishna

What is a Galois Representation?

Abstract: This will talk will survey Galois representations and their central place in modern number theory with particular emphasis on R=T theorems. I'll conclude with a recent result with Khare on ramification structure in Hecke algebras.

Vin de Silva

Persistence, Categories and Reeb Cosheaves

Topological persistence is perhaps the most well-known aspect of applied algebraic topology. In this talk, I shall emphasize the category-theory perspective. This unifies several different constructions: persistence modules, merge trees, Reeb graphs. The case of Reeb graphs is particularly fruitful: these are ordinarily thought of as topological graphs equipped with a non-degenerate real-valued function, but one can also interpret them as cosheaves over the real line. This motivates the definition of a metric on the space of Reeb graphs and, along the way, a semigroup of topological smoothing operators.
I thank my collaborators: Peter Bubenik, Jonathan Scott, Elizabeth Munch, Amit Patel, Anastasios Stefanou, Song Yu, Dmitriy Smirnov.

Algebraic Topology Abstracts

David Allen

Applications of homotopy theory to toric topology

In this talk I will describe explicit calculations of the higher derived functor of the indecomposable functor, $L_i(QA)$ (or dually, $R^i(P(--))$)for an augmented simplicial algebra A over a commutative ring. In certain degrees there are concrete representations that facilitate explicit calculations. These computations allow one to translate statements concerning torus actions on Quastitoric manifolds into statements concerning the orbits, which can then be analyzed using unstable methods. There is an interplay between the combinatorics, simplicial methods and commutative algebra that produces isomorphisms of these derived functors. A few applications will be discussed, the first answers in the negative a question posed by Bendersky on the existence of certain "nice" torus actions. These methods also provide additional insight into questions regarding cohomological rigidity.

Kristine Bauer

Abelian functor calculus and a higher order chain rule

The calculus of functors, first established by Goodwillie for homotopy functors of spaces or spectra, provides a Taylor tower of polynomial approximations for functors in the same way that the sequence of Taylor polynomials provides approximations for functions. There are many analogies between functor calculus and undergraduate calculus, not the least of which is the existence of a Faa di Bruno style chain rule in both contexts (due to work of Arone-Ching for functors). In many cases, calculus for functions has provided useful clues regarding the available structure of functor calculus. In the 1990's, Johnson and MCarthy established a variation of Goodwill's homotopy calculus for functors of abelian categories. Earlier this year, a team of mathematicians (B., Johnson, Osborne, Riehl and Tebbe) set out to find a higher order chain rule for abelian functor calculus. Motivated by work of Huang-Marcantognini-Young, we established a higher order chain rule for the directional derivatives of a functor. In the process, we discovered underlying structure which explains many of the analogies between the calculus of functors and undergraduate calculus. In short, abelian functor calculus is an example of a cartesian differential category in the sense of Blute-Cockett-Seely. I will explain the chain rule and this categorical structure in this talk.

Alyson Bittner

Spaces with Complexity One

The inductive construction of a $CW$-complex builds the space out of spheres. This process can be generalized to build $A$-cellular spaces out of some fixed space $A$. Given such a construction, we can ask if it is the most efficient construction in the sense that it requires the least ordinal number of steps to build the space out of copies of $A$, called the $A$-complexity. With certain assumptions on $A$, every space has $A$-complexity less than or equal to one. We will discuss the properties and significance of such spaces $A$ with the use of algebraic theories.

Rick Blute

Affine structures in tangent categories

We introduce the notion of affine object in a tangent category as an object equipped with a flat, torsion-free connection. We show that the subcategory of affine objects in a tangent category is still a tangent category. Furthermore, we show that in addition to the monad structure on any tangent category, there is a canonical comonad structure as well, and these two structures are related by a mixed distributive law.

Robin Cockett

Secant Theories

A classical way to obtain a differential uses the fact that the secant of a function is unique: the differential is then obtained as the secant of length zero. This led Kock and Dubuc to study "Fermat theories" in which this property is taken as an axiom. Their approach enshrined the idea that a differential is somehow unique. However, all is not as it seems. This talk will discuss secant theories, a generalization of Fermat theories in which the secant is not necessarily unique. Secant theories always give rise to Cartesian differential categories -- by setting the "length" of the secant to zero. They (together with Fermat theories) provide a very useful gateway into many of the classical examples of Cartesian differential categories.

John Harper

Iterated Suspension Spaces and an Integral Analog of Quillen's Rational Homotopy Theorem

In his landmark 1969 Annals paper, Quillen showed that the rational homotopy type of a simply connected space could be detected at the level of its singular rational chains, and furthermore, that rational chains fit into a derived equivalence with cocommutative dg coalgebras over the rationals, after restricting to 1-connected objects. In 1977 Sullivan subsequently proved the analogous result in the case of rational cochains and commutative dg algebras over the rationals. Since then topologists have worked on attempting to establish analogous results for finite fields (Kriz, Goerss, Mandell), and more recently some partial results have been established in the integral chains case (Mandell, Karoubi). Nevertheless, establishing that integral chains fit into a derived equivalence has proved resistant to all attacks. In this talk I will outline how we recently resolved, in the affirmative, the integral chains problem. If time permits, I will also describe how we recently resolved the problem of establishing a recognition principle for iterated suspension spaces, dual to the celebrated iterated loop spaces work of May and Beck. This is joint work with J. Blomquist and M. Ching.

Ben Knudsen

Betti numbers of configuration spaces of surfaces

We give explicit formulas for the Betti numbers of the unordered configuration spaces of an arbitrary surface of finite type. The basis for the computation, which is joint work with G. Drummond-Cole, lies in the theories of factorization homology and higher enveloping algebras.

Keith O'Neill

Smoothness in Codifferential Categories

Differential categories arise as models for differential linear logic. As such, they are equipped with the structure necessary for a broad investigation of algebraic smoothness, including noncommutative notions of algebraic smoothness. Motivated by the Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of a smooth algebra to its differential structure, we consider characterisations of smoothness relative to varying monads. To facilitate this, Andre-Quillen homology is developed specifically for the context of a codifferential category.

Christina Osborne

The first step towards higher order chain rules for abelian calculus

One of the most fundamental tools in calculus is the chain rule for functions. Huang, Marcantognini, and Young developed the notion of taking higher order directional derivatives, $\Delta_nf(x_0,\ldots,x_n)$, which has a corresponding higher order iterated directional derivative chain rule: $\Delta_nf\circ g=\Delta_nf(g(x_0),\Delta_1g(x_0,x_1),\ldots,\Delta_ng(x_0,\ldots,x_n))$. When Johnson and McCarthy established abelian functor calculus, they constructed the chain rule for functors which is analogous to the directional derivative when n=1. In joint work with Bauer, Johnson, Riehl, and Tebbe, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the HMY chain rule. Our initial investigation of this result involved a concrete computation of the case n=2, which will be presented in this talk.

Angelica Osorno

2-Segal spaces and the Waldhausen construction

The notion of 2-Segal spaces was introduced by Dyckerhoff and Kapranov as a higher dimensional version of Rezk's Segal spaces. In this talk we will explore the motivation for this notion, give examples, and show that it is related to a certain class of double categories via a version of Waldhausen's construction.

Amelia Tebbe

Computing Polynomial Approximations of Atomic Functors

A functor from finite sets to chain complexes is called atomic if it is completely determined by its value on a particular set. In this talk, we present a new resolution for these atomic functors, which allows us to easily compute their Goodwillie polynomial approximations. By a rank filtration, any functor from finite sets to chain complexes is built from atomic functors. Computing the linear approximation of an atomic functor is a classic result involving partition complexes. Robinson constructed a bicomplex, which can be used to compute the linear approximation of any functor. We hope to use our new resolution to similarly construct bicomplexes that allow us to compute polynomial approximations for any functor from finite sets to chain complexes.

Marco Varisco

Topological cyclic homology of group algebras of finite groups

In recent joint work with Wolfgang Lük, Holger Reich, and John Rognes [arXiv:1607.03557], we proved a general induction theorem for the topological cyclic homology ($\mathit{TC\,}$) of group algebras of finite groups, in the spirit of Artin and Brauer induction in the representation theory of finite groups. The theorem states that, for any finite group $G$, for any ring (or connective ring spectrum) $\mathbb{A}$, and for any prime number $p$, $\mathit{TC\,}(\mathbb{A}[G];p)$ is determined by $\mathit{TC\,}(\mathbb{A}[C];p)$ as $C$ ranges over the cyclic subgroups of $G$. Technically, we showed that the assembly map for the family of cyclic subgroups induces isomorphisms on all homotopy groups. This result allows us to attack explicit computations for non-cyclic finite groups; more precisely, to reduce such computations to the cyclic subgroups. A lot is known about $\mathit{TC\,}$ of cyclic groups, but for non-abelian groups most of the previously known methods of computation do not apply. In this talk I will describe explicitly how this works in the smallest example: that of the symmetric group $\Sigma_3$.

Juan Villeta-Garcia

Stabilizing Spectral Functors of Exact Categories

Algebraic K-Theory is often thought of as "the" universal additive invariant of rings (or more generally, exact categories). Often, however, functors on exact categories don't satisfy additivity. We will describe a procedure (due to McCarthy) that constructs a functor's universal additive approximation, and apply it to different different local coefficient systems, recovering known invariants of rings (K-Theory, THH, etc.). We will talk about what happens when we push these constructions to the world of spectra, and tie in work of Lindenstrauss and McCarthy on the Taylor tower of Algebraic K-Theory.

Aliaksandra Yarosh

Twisted Morava K-theory

Sati and Westerland have showed recently that Morava K-theroy admits twists by Eilenberg-MacLane spaces. In this talk we will first introduce the framework for describing twists of generalized cohomology theories via generalized Thom spectra and show how the classical twisted cohomologies like twisted K-theory can be interpreted in that context Then we will describe some properties of twisted Morava K-theory, including twisted Atiyah-Hirzebruch spectral sequence and a universal coefficient theorem that relates twisted Morava K-theory to the untwisted ones, and discuss some computations in twisted Morava homology of connected covers of BO.

Applied Topology Abstracts

Justin Curry

Realization Problems in Persistence

In this talk I will discuss an inverse problem in persistence. In particular, I will present a combinatorial result, which counts equivalence classes of functions from $[0,1]$ to $\mathbb{R}$ that realize a given barcode, which leads to a notion of entropy of a barcode. Depending on time, I will present other special cases of the realization problem in persistence.

David Damiano

A topological analysis of SPECT images of murine tumors

In this talk we employ computational topology methods to quantify heterogeneous uptake behavior across time series of single-photon emission computed tomography (SPECT) images of murine tumors. This behavior cannot be captured by standard aggregate measures such as percent injected dose per gram or tumor-to-heart ratio. Inspired by Morse theory, we analyze critical points of each tumor image. To quantify the uptake behavior in neighborhoods of local maxima, we utilize a modified form of zeroth order persistence diagrams as well as develop the novel concept of childhood diagrams. Statistical methods are applied to time series persistence and childhood diagrams to detect heterogeneity of uptake within and across study groups in two studies. This behavior is explained in terms of the underlying biological mechanisms.

Boris Goldfarb

Singular persistent homology and its applications

Persistent homology of a finite metric space is usually defined as the homology of Rips simplicial nerves of the space. This talk will introduce a counterpart, the singular persistent homology, where the perspective is different. The data set is left stationary while the parameter is allowed to change in the form of the size of singular simplices. Because of this nature, coverings of the data set are easier to handle than in other attempts to parallelize the computation of persistent homology. I will illustrate this and other applications of the singular theory.

Michael Lesnick

Universality of the Homotopy Interleaving Distance: Towards an "Approximate Homotopy Theory" Foundation for TDA

We introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in topological data analysis to articulate stability and inference theorems. Whereas ordinary interleavings can be interpreted as pairs of "approximate isomorphisms" between filtered spaces, homotopy interleavings can be viewed as pairs of "approximate weak equivalences." Our main results are that homotopy interleavings induce an extended pseudometric $d_{HI}$ on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also show that $d_{HI}$ (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. This is joint work with Andrew Blumberg.

Rachel Levanger

A comparison framework for interleaved persistence modules

We give a new result in the form of a generalized algebraic stability theorem for persistence modules. We show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance through two examples: one in image analysis and one for computations of persistence diagrams of large point clouds.

Elizabeth Munch

The interleaving distance for posets

The interleaving distance has been shown to be an incredibly powerful tool in Topological Data Analysis (TDA), as various incarnations provide distances between persistence diagrams, Reeb graphs, and merge trees. In this work, we have extended the notion of interleavings to encompass cosheaves arising from a functor defined on a poset. One result of this machinery is that it provides a definition for the interleaving distance for the mapper graph. In addition, we can approximate the Reeb graph interleaving distance by the interleaving distance for its mapper which opens the door to computational possibilities. This work is joint with Magnus Botnan and Justin Curry.

Donald Sheehy

Persistent Nerves Revisited

The nerve of a cover of a space is a simplicial complex whose simplifies are all families of cover sets with nonempty intersection. Nerves are used in many computational problems in both geometry and topology as a way to discretize spaces. The Persistent Nerve Lemma of Chazal and Oudot translates this classic construction to persistent homology and has found use in topological reconstruction, clustering, and sensor networks. In this talk, I'll talk about recent work on generalizing the Persistent Nerve Lemma to give theoretical guarantees even under much weaker assumptions about the cover. Specifically, we replace the standard "good cover" condition with one that only requires to be "good in a persistent sense".

Ann Sizemore

Closures and Cavities in the Human Connectome

Persistent homology has taken root in big data as a tool for analyzing global features of point clouds. Yet in the biological sciences, networks -- as opposed to point clouds -- often more aptly describe observed systems. Here we study white matter networks in the human brain, searching for features that support cognition. The distribution of cliques in these networks and how they arrange into cycles provides us with new understanding of the mesoscale organization underlying local and distributed neural computation.

Anastasios Stefanou

Interleavings on categories with lax $[0,\infty)$-action and the hom tree functor

The interleaving distance is a powerful tool in TDA which has been shown to provide a metric for such topological signatures as persistence diagrams and Reeb graphs. In this talk we generalize the idea of interleavings to a broader class of objects, namely categories with a lax $[0,\infty)$-action. This allows us to show that many commonly used distances, such as the $L_\infty$ and Hausdorff metrics, are in fact special cases of interleaving distances. In addition, there is a natural way to define morphisms between these categories that generalizes the stability results of TDA to a broad class of objects by showing that the morphisms are 1-Lipschitz. As an application of this result, we will give an example of such a morphism, known as the hom-tree functor, which provides a new bound on the Reeb graph interleaving distance.

Mikael Vejdemo-Johansson

A topos foundation for persistent homology

Persistent homology is a core component of applied algebraic topology and topological data analysis. Since its first introduction by Edelsbrunner, Letscher and Zomorodian, numerous algebraic formalisms have been introduced, invariably opening the field to more applications and more algorithms. A topos is a category with enough structure for set theory and logic; one large family of topoi is given by categories of set-valued sheaves over topological spaces -- or over a specific weakening of topological spaces called a Heyting algebra. In collaboration with Primoz Skraba and João Pita Costa, we have established that after fixing a Heyting algebra that captures the structure of a specific persistence theory, the structures and methods of the various formalisms for persistent homology emerge as the natural combinatorial algebraic topology internal to the topos determined by sheaves over that Heyting algebra. In our work we have established Heyting algebras for classical persistent homology, for zig-zag and multidimensional persistent homology, and for the circular persistent homology of Dey and Burghelea.

Sara Kalisnik Verovsek

Tropical Coordinates on the Space of Persistence Barcodes

In the last two decades applied topologists have developed numerous methods for "measuring" and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not well-adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes and these outputs can then be used as input to standard algorithms. I will talk about max-plus polynomials and tropical rational functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to the standard distance functions (bottleneck, Wasserstein) used on the barcode space.

Category Theory Abstracts

Evangelia Aleiferi

Cartesian double categories

A double category $\mathbb{D}$ is said to be Cartesian if the diagonal double functor $\Delta:\mathbb{D} \to \mathbb{D}\times \mathbb{D}$ and the unique double functor $!: \mathbb{D} \to \mathbb{1}$ have right adjoints. Some of the examples of Cartesian double categories are ones whose horizontal bicategory inherits this Cartesian structure in the bicategorical sense, as it was introduced by Carboni, Kelly, Walters and Wood. However this is not necessarily the case. In this talk we will give examples of such double categories and in particular, we will study the double category of monads and modules over a Cartesian double category. We will also talk about concepts introduced by Street in ``The formal theory of monads", extended to double categories.

Kristine Bauer

Abelian functor calculus and a higher order chain rule

The calculus of functors, first established by Goodwillie for homotopy functors of spaces or spectra, provides a Taylor tower of polynomial approximations for functors in the same way that the sequence of Taylor polynomials provides approximations for functions. There are many analogies between functor calculus and undergraduate calculus, not the least of which is the existence of a Faa di Bruno style chain rule in both contexts (due to work of Arone-Ching for functors). In many cases, calculus for functions has provided useful clues regarding the available structure of functor calculus. In the 1990's, Johnson and MCarthy established a variation of Goodwille's homotopy calculus for functors of abelian categories. Earlier this year, a team of mathematicians (B., Johnson, Osborne, Riehl and Tebbe) set out to find a higher order chain rule for abelian functor calculus. Motivated by work of Huang-Marcantognini-Young, we established a higher order chain rule for the directional derivatives of a functor. In the process, we discovered underlying structure which explains many of the analogies between the calculus of functors and undergraduate calculus. In short, abelian functor calculus is an example of a cartesian differential category in the sense of Blute-Cockett-Seely. I will explain the chain rule and this categorical structure in this talk.

Richard Blute

Robin Cockett

Secant theories

A classical way to obtain a differential uses the fact that the secant of a function is unique: the differential is then obtained as the secant of length zero. This led Kock and Dubuc to study``Fermat theories" in which this property is taken as an axiom. Their approach enshrined the idea that a differential is somehow unique. However, all is not as it seems. This talk will discuss secant theories, a generalization of Fermat theories in which the secant is not necessarily unique. Secant theories always give rise to Cartesian differential categories -- by setting the``length" of the secant to zero. They (together with Fermat theories) provide a very useful gateway into many of the classical examples of Cartesian differential categories.

Darien Dewolf

Restriction monads and algebras

A restriction monad in a double category $\mathbb{D}$ is a monad equipped with structure reminiscent of that in a restriction category. Indeed, the fact that monads in the double category $\mathrm{Span}(\mathbf{Set})$ are small categories generalizes immediately to restriction monads in $\mathrm{Span}(\mathbf{Set})$ being small restriction categories. We introduce also a notion of restriction algebra for a monad. If $X$ is a restriction category and $R(X)$ its corresponding restriction monad in $\mathrm{Span}(\mathbf{Set}),$ then the restriction algebras for $R(X)$ are right $X$-modules. Moreover, the modules arising in this way can be naturally given a suitable restriction structure. We will then organize these structures in a double category $\mathrm{rMod}(\mathbb{D})$ of restriction monads, which gives an example of a so-called double restriction category, all of whose vertical arrows are total.

Nelson Martins Ferreira

Duality between algebra and geometry

In this talk we will consider a new category whose objects may be seen either as algebras or as spaces. The morphisms are pairs of maps running in opposite directions which gives rise for two possible composition laws. It is the choice for one composition law that determines whether the behaviour of an object is geometrical or algebraic. The main goal of this construction is the possibility of working on the duality between algebra and geometry without having to change the objects or the morphisms in our category, we only need to change the formula for the composition law. Some applications into Physics are expected, namely on certain dualities as well as on a possible unification of General Relativity (geometry) and Quantum Mechanics (algebra). The main structure is derived from the notion of fibrous preorder introduced in [1].

[1] N. Martins-Ferreira, From A-spaces to arbitrary spaces via spatial fibrous preorders, Math. Texts 46 (Categorical methods in algebra and topology) (2014) 221--235.

Jonas Frey

Realizability toposes as homotopy categories

We give a definition of a category Set<P> of fibrant objects for any tripos $P : Set^op\to Ord$, and show that its homotopy category is the topos Set[P]. As special cases we obtain ways to view realizability toposes, but also localic toposes as homotopy categories (since both can be constructed from triposes). On the other hand, the construction generalizes to existential hyperdoctrines $A : C^op \to Ord$ (i.e. indexed preorders having fiberwise finite meets and left adjoints to reindexing satisfying Beck-Chevalley and Frobenius conditions) on finite limit categories $C$.

Jonathon Funk

Commutative objects in a topos

An isotropy automorphism of an object $X$ of a topos $\cal E$ is an automorphism of the geometric morphism ${\cal E}/X \to {\cal E}$ associated with $X$. Such an automorphism is said to be central if it factors through the identity ${\cal E}/X \to {\cal E}/X$. Let us say that an object $X$ of a topos ${\cal E}$ is commutative if every one of its isotropy automorphisms is central. I will discuss some of the properties of commutative objects, and explore the particular example of the group classifier.

Pieter Hofstra

Partial lambda calculus revisited

Motivated by considerations of partial computable functions, Moggi's PhD thesis considered various notions of partial lambda calculus and their semantics in partial CCCs. Since then, several people have suggested improvements and have investigated variations on this theme, and have established (or at least claimed) generalizations of classical results to the partial setting. However, we submit that they still leave several things to be desired, both on the syntactical and on the categorical level. In this talk, I will first focus on cartesian closed restriction categories, as well as a weak version of those. These categories form the intended semantics, and are more general (and convenient) than the classes of categories considered hitherto. The main results concern idempotent splittings of such categories, as well as a generalized Scott-Koymans theorem. Next, I will discuss a substructural approach to the partial lambda calculus as a potential solution to the uncomfortable fact that under the intended interpretation, syntactic substitution does not agree with composition. Based on joint work with Robin Cockett (Calgary) and Jerome Fortier (Ottawa).

Michael Lambert

Representation embeddings of algebraic theories

Guided by definitions in the representation theory of associative algebras over an algebraically closed field, a representation embedding between categories of models of algebraic theories can be defined to be a functor between the categories of models that preserves indecomposibility and projectivity and that reflects epics when restricted to the full subcategory of indecomposable projective models. The main result is that such a representation embedding preserves undecidability of theories; that is, if $$E \colon {\mathbb T}-Mod({\bf Set}) \to {\mathbb T'}-Mod({\bf Set})$$ is a representation embedding of algebraic theories in this sense, then if $\mathbb{T}$ is undecidable, so is $\mathbb{T}'$ This result is applied to obtain a near-resolution in the affirmative to a conjecture of M. Prest [1] that every "wild" associative algebra over an algebraically closed field has an undecidable theory of modules.

[1] M. Prest. Model Theory and Modules. Cambridge UP, Cambridge, 1988.

JS Lemay

Introduction to tensor integral categories*

In 2005, based on Ehrhard and Regnier's differential lambda calculus, Blute, Cockett and Seely introduced tensor differential categories [1], which formalized the concept of pure differentiation by axiomatizing a differential structure on a symmetric monoidal category. These axioms include that the derivative of a linear function is a constant, the derivative of a constant is zero, the product rule, the chain rule and the independence of order of differentiation. While tensor differential categories have now been extensively studied, with many examples and applications, not much attention has been given to the inverse process of differentiation: integration.

In this talk, we will introduce tensor integral categories. The integral axioms include the integral of a constant is a linear map; the Rota-Baxter rule [2], which is an expression of the integration by parts using only integrals and no derivatives; and the independence of order of integration. Consequences of the compatible interaction between integral and differential structures include antiderivatives, Poincaré's lemma and the fundamental theorems of calculus [3]. We have defined settings in which these occur as "tensor calculus categories". Furthermore, we will also present sufficient conditions for when a differential category is a calculus category.

*Joint work with Richard Blute, Robin Cockett and Kristine Bauer

[1] R. Blute, R. Cockett, R. Seely, Differential categories, Mathematical Structures in Computer Science Vol. 1616, pp 1049-1083, 2006.
[2] L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, Volume IV, 2012.
[3] T. Ehrhard An introduction to Differential Linear Logic: proof-nets, models and antiderivatives arXiv:1606.01642, 2016.

Ernie Manes

Tight categories

By a tight category, we mean a full reflective subcategory of the category of algebras of a monad in the category of sets which contains all free algebras. Certain tight categories of interest are full subcategories of topological spaces and continuous maps. The conundrum of explaining tight categories to topologists in a twenty-minute talk has found a solution in that arbitrary tight categories can be captured by four axioms which are easy to explain by properties familiar in topology. To the extent that time permits, we will show how dynamical notions such as proximal and equicontinuous can be formulated in tight categories.

Jeffrey Morton

Transformation structures for 2-group actions

2-groups, also known as categorical groups, are higher dimensional algebraic structures which generalize groups, and are useful for describing the symmetries of categories. I will describe 2-groups and their actions, and describe how similar generalizations of the transformation groupoids associated to group actions show up in this context. I will outline a motivating example for this work coming from the geometry of gerbes. Based on joint work with Roger Picken (IST, Lisbon).

Keith O'Neill

Smoothness in codifferential categories

Differential categories arise as models for differential linear logic. As such, they are equipped with the structure necessary for a broad investigation of algebraic smoothness, including noncommutative notions of algebraic smoothness. Motivated by the Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of a smooth algebra to its differential structure, we consider characterisations of smoothness relative to varying monads. To facilitate this, Andre-Quillen homology is developed specifically for the context of a codifferential category.

Angélica Osorno

Dorette Pronk

Homs in bicategories of fractions as pseudo colimits

Given a bicategory ${\mathcal B}$, we present a weaker set of conditions on a class of arrows $W$ in ${\mathcal B}$ to give rise to a bicategory of fractions ${\mathcal B}(W^{-1})$ than what was originally presented in the author's 1996-paper. We will then show that the hom-category ${\mathcal B}(W^{-1})(A,B)$ can be viewed as a conical pseudo colimit of the hom-functor ${\mathcal B}(-,B)$ on the diagram $W/B$ (suitably defined). This colimit can itself be viewed as a category of fractions. We will discuss two ways to derive this result, presenting closely related work by Matias Data, Eduardo Dubuc, and Ross Street. This is joint work with Laura Scull.

Lucius Schoenbaum

A generalization of the Curry-Howard correspondence

We present a variant of the calculus of deductive systems and use it to give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving cartesian morphisms. Along the way we show that, motivated by the effect of generalization on the category of contexts construction, cartesian closed structure on an ordinary category can be modified to include mappings on the set of objects that recover the base and the power of an exponential. By making this modification, we widen further the class of admissible functors (for some purposes relevant to categorical logic and type theory) to include arbitrary cartesian functors. Our result shows that generalized cartesian closed categories are equivalent to a lambda calculus in which types are algebraically structured, and the existence of a notion of generalized elementary topos in which essentially the same calculus exists as in ordinary elementary toposes.

Richard Wood

Totally distributive categories

A locally small category ${\cal T}$ is said to be total(ly cocomplete) if the Yoneda functor $Y:{\cal T}\rightarrow {\bf set}^{{\cal T}^{\rm op}}$ has a left adjoint. A total category is totally distributive if the defining left adjoint has a left adjoint. It is natural to hope that this very compact definition can provide results with the same ease and clarity as that of CCD lattices, which it obviously follows. However, progress has been slower than one would like because of difficulties with ``size'' problems. The talk will describe the current state of knowledge about these categories and how some size problems have been met, together with recent thoughts on how to solve another one. This is joint work with Ross Street.

Differential Geometry and Geometric Analysis Abstracts

Amir Aazami

Symplectic 4-manifolds via Lorentzian geometry

On a noncompact 4-manifold endowed with a Lorentzian metric, we give a condition under which an exact symplectic form may be constructed using a null vector field with geodesic flow; this construction parallels the association of a contact form to ``twisted'' vector fields on 3-manifolds. The roles played by both the curvature and the causal structure of the Lorentzian manifold are discussed. The second half is joint work with Gideon Maschler in which we discuss attempts to use this symplectic form to construct a Kähler structure on the manifold.

Hubert Bray

Flatly Foliated Relativity: Gravity without Gravitational Waves

In this talk, I will describe a new theory which is 2/3 of the way from Special Relativity to General Relativity. The key difference from General Relativity is that spacetime is assume to be foliated by flat 3-dimensional Euclidean spaces. This results in a model of spacetime where gravity similar to general relativity still exists, but travels infinitely quickly and without gravitational waves.

David Calderbank

The geometry of dispersionless integrable PDEs (Joint work with Boris Kruglikov)

A scalar PDE on a manifold M is integrable by a dispersionless Lax pair if it arises as the integrability condition for rank 2 distribution on a rank 1 bundle over M. By estabilishing that any such Lax pair is characteristic on solutions, we show that, in the second order case, where the symbol of the PDE induces a conformal structure on M, the PDE is dispersionless integrable if and only if the conformal structure is self-dual (for n=4) or Einstein-Weyl (for n=3) on any solution. These results unify and extend work of E. Ferapontov et al.

Caner Koca

Einstein-Maxwell Metrics on Ruled Surfaces

In Riemannian geometry, the Einstein-Maxwell Equations, which originate from physics, can be thought of as a geometric PDE for Riemannian metrics on oriented 4-manifolds. Einstein metrics and constant-scalar-curvature-Kähler metrics are among the (trivial) solutions of this PDE. In this talk, we will construct families of non-trivial solutions on complex higher-genera minimal ruled surfaces. These solutions are non-Kähler, but conformally Kähler. This is a joint work with Christina Tonnesen-Friedman.

Sajjad Lakzian

Characterization of low dimensional $RCD^*(K,N)$ spaces

We give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with $Ric \geq K$ and Hausdorff dimension $N$ and the class of $RCD^*(K,N)$ spaces coincide for $N<2$ (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is, roughly speaking, a converse to the Lévy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest. This is a joint work with Yu Kitabeppu.

Gideon Maschler

Equivariant index and eta invariant and distinguished metrics

The talk will describe an extension to the equivariant case of the Atiyah-Patodi-Singer index formula for Dirac operators on manifolds with boundary arising from Riemannian metrics which are not product near the boundary. The non-equivariant case was initiated by Gilkey, while the equivariant product case was developed by Donnelly and Goette. The characteristic and transgression forms appearing in the formula are computed for the class of SKR metrics on a four-manifold with boundary, which includes many Kähler conformally Einstein metrics. This is joint work with Maxim Braverman.

Frank Morgan

Isoperimetry with Density

The latest generalization of the log-convex density theorem by Kenigsberg et al. says that for distinct perimeter and volume densities $r^k$ and $r^m$ on $R^n$, spheres about the origin are isoperimetric if and only if they are stable. We'll discuss such results and open problems.

Stefan Müller

Lagrangian and symplectic rigidity

Symplectic structures first arose in classical mechanics as the laws of motion (they relate the energy of a system with the motion of particles in phase space). By definition, symplectic geometry is a priori a smooth theory (in the sense that the notions of interest, such as symplectic diffeomorphisms, are smooth). This talk is concerned with some of the surprising topological phenomena that are abundant in symplectic topology. Lagrangian submanifolds play a central role in symplectic topology. We will show that if a submanifold (of half dimension) is not Lagrangian, then it has a neighborhood that does not contain any Lagrangian submanifolds of a fixed topological type. As a consequence, we characterize symplectic embeddings in terms of invariants that do not involve derivatives and are preserved by uniform limits.

Corbett Redden

Differential Equivariant Cohomology

Given a manifold equipped with a smooth Lie group action, I will consider a version of the quotient stack that is defined using principal bundles with connection. I will then explain how the differential cohomology of this quotient stack provides a natural home for equivariant Chern-Weil theory.

Henri Roesch

Proof of a Null Penrose conjecture using a new Quasi-local mass

We define an explicit quasi-local mass functional which is nondecreasing along all null foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the Null Penrose Conjecture under fairly generic conditions.

Ioana Suvaina

ALE Kähler manifolds

The talk presents a characterization of ALE Kähler manifolds. As a consequence, the number of diffeomorphism types of the underlying smooth structure of minimal ALE Kähler surfaces with prescribed asymptotic is finite. The special case of ALE Ricci flat Kähler surfaces is also described.

Number Theory Abstracts

Saikat Biswas

Capitulation, unit groups, and the cohomology of $S_{\infty}$-idèle classes.

For a finite cyclic extension $L/K$ of number fields with Galois group $G$, we relate the $G$-cohomology of the $S_{\infty}$-idèle classes of $L$ to the capitulation map for $L/K$ as well as to the $G$-cohomology of the unit group $U_L$.

Michael Chou

Growth of torsion on elliptic curves from $\mathbb{Q}$ to the maximal abelian extension of $\mathbb{Q}$

Torsion of an elliptic curve over a number field is finite due to the Mordell-Weil theorem. However, even in certain infinite extensions of $\mathbb{Q}$ we have that torsion is finite. Ribet proved that, when base extended to the maximal abelian extension of $\mathbb{Q}$, the torsion of an elliptic curve over $\mathbb{Q}$ is finite. In this talk, we show that the size of such torsion subgroups is in fact uniformly bounded as we range over all curves $E/\mathbb{Q}$. Further, we give a classification of all possible torsion structures appearing in this way.

Zhenguang Gao

The Central Binomial Coefficient in (a+b)^(2^n) (mod 8)

We explore the properties of Catalan numbers by using an interesting the central binomial formula.

Alden Gassert

Index divisibility in dynamical sequences and cyclic orbits modulo $p$

Let $\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(\phi^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $\phi$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \mid \phi^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes is finite when $d$ is even, and conjecture that it is infinite when $d$ is odd.

Haydee Lindo

Trace ideals and Endomorphism rings

In this talk we will present some results concerning when the center of the endomorphism ring of a module can be realized as an endomorphism ring of a trace ideal. We include some applications.

Yuan Liu

Realizability Problems with Inertia Conditions

We consider the inverse Galois problem with described inertia behavior: for a finite group $G$, one of its subgroups $I$ and a prime integer $p$, we ask whether or not $G$ and $I$ can be realized as the Galois group and the inertia subgroup at $p$ for an extension over $\mathbb{Q}$. In this talk, we first discuss the result when $G$ is an abelian group. Then, in the case when $|G|$ and $p$ are odd, Neukirch shows that there exists such an extension over $\mathbb{Q}$ if and only if the given local condition is realizable over $\mathbb{Q}_p$, from which we derive the answer to our realizability problem by studying the structure of the maximal pro-$p$ extension over $\mathbb{Q}_p$ and applying techniques from embedding problems. Lastly, we show partial results and several interesting examples in the cases when $G$ is GL$_2(\mathbb{F}_p)$ and when $G$ is a 2-group, which relate to modular forms and Shafarevich's theorem on solvable groups as Galois groups respectively.

Dmitry Malinin

On the arithmetic os integral representations

We consider the arithmetic background of integral representations of finite groups. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite $p$-groups over the rings of integers of number fields with the extra congruence conditions are constructed. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. We investigate related problems concerning globally irreducible representations, primitive representations of the Galois groups of local fields, finite arithmetic groups, Galois action and Galois cohomology.

Tianyi Mao

The Distribution of Integers in a Totally Real Cubic Field

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Grössencharacter $L$-functions.

Ken McMurdy

Elliptic Curves with Non-abelian Entanglements

Let $K$ be a number field. An elliptic curve $E/K$ is said to have a non-abelian entanglement if there are relatively prime positive integers, $m_1$ and $m_2$, such that $K(E[m_1])\cap K(E[m_2])$ is a non-abelian Galois extension of K. The elliptic curves over $K$ exhibiting any one particular type of non-abelian entanglement correspond to $K$-rational points on some modular curve. In this talk, we discuss methods for finding explicit equations for that modular curve, and share a few examples. (Joint work with Nathan Jones, UIC)

Daniel Nichols

Zeta functions of a family of quaternion extensions of $\mathbb{F}_p(t)$

Field extensions with Galois group $Q_8$ are interesting because this is the smallest group with a self-dual representation. I describe a partial catalog of function fields $K\mathbb{F}_p(t)$ with this Galois group and describe an infinite family of such fields which have zeta functions that vanish at the point $s = 1/2$ at a higher order than that required by the functional equation.

Caleb Shor

Compound sequences, numerical semigroups, and power sums

Suppose $A=(a_1,\dots,a_k)$ and $B=(b_1,\dots,b_k)$ are $k$-tuples of natural numbers. A compound sequence is a sequence of the form $G(A,B)=\{a_1\cdots a_i b_{i+1}\cdots b_k : 0\leq i\leq k \}.$ Such a sequence can be thought of as a generalization of a geometric sequence. In this talk, we will consider the set $NR(A,B)$, the set of positive integers which are not representable as non-negative linear combinations of elements of $G(A,B)$. Under certain conditions, $NR(A,B)$ is a finite set, and we can investigate its cardinality (also called the genus), largest element (also called the Frobenius number or the largest non-McNugget number), and sum of the $m$th powers of elements. Our primary tool is the generalization of an identity that Tuenter found in 2006 which corresponds to case where $k=1$. We will use our identity to analyze $NR(A,B)$, generalizing a result of Rødseth for the sum of the $m$th powers of elements. Time permitting, we will see how one can apply the results here to compute higher-order Weierstrass weights of points in certain towers of algebraic curves. This is joint work with T. Alden Gassert.

Long Tran

To be announced

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