Geometry and Topology

Speaker: Hiro Tanaka (Northwestern)

"Factorization homology and link invariants"

Time: 15:30

Room: MC 107

Homology is easy to compute, thanks to excision, but it isn't very sensitive. It only detects homotopy types. In this talk I'd like to give one answer to the question: Is there a notion of homology theory for manifolds that's sensitive to more? I will present the definition of factorization homology, which Lurie has also called topological chiral homology. Factorization homology generalizes usual Eilenberg-Steenrod homology, and is and invariant of manifolds and stratifications on them. The main result will be a classification of all homology theories, namely by giving an equivalence between the category of homology theories and the category of certain kinds of algebras. I will explain how the theorem in turn gives candidates for new sources of invariants of embedding spaces (and in particular, link invariants). If time allows, I can discuss connections to topological field theories and to Koszul duality. This is joint work with David Ayala and John Francis.

Analysis Seminar

Speaker: Ilya Kossovkiy (Western)

"Analytic Continuation of Holomorphic Mappings From Non-Minimal Hypersurfaces"

Time: 14:30

Room: MC 107

The classical result of H.Poincare states that a local biholomorphic mapping of an open piece of the 3-sphere in $\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. This theorem was generalized by H.Alexander to the case of a sphere in an arbitrary $\mathbb{C}^n,\,n\geq 2$, then later by S.Pinchuk for the case of strictly pseudoconvex hypersurface in the preimage and a sphere in the image, and finally by R.Shafikov and D.Hill for the case of an essentially finite hypersurface in the preimage and a quadric in the image. In this joint work with R.Shafikov we consider the - essentially new - case when a hypersurface $M$ in the preimage contains a complex hypersurface. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends to a punctured neighborhood of the complex hypersurface, lying in $M$, as a multiple-valued locally biholomorphic mapping.

Pizza Seminar

Speaker: Rasul Shafikov (Western)

"Introduction to Continued Fractions"

Time: 16:30

Room: MC 107

In this elementary talk I will discuss the definition and basic properties of continued fractions, a simple and in many respects a convenient way to represent real numbers. I will also give some applications.

Ph.D. Presentation

Speaker: Chris Plyley (Western)

"Group-Graded Algebras, Polynomial Identities, and The Duality Theorem"

Time: 13:00

Room: MC 107

In polynomial identity theory, when an associative algebra A has the additional structure of an (associative) group-grading or a G-action, one can often relate the identities of A to the more general graded-identities and G-identities. This technique has proved a powerful method, for example, in discovering a bounded version of Amitsur's celebrated theorem regarding algebras with involution. In this talk we describe several alternate ways to endow a grading on A, namely by considering the induced Lie and Jordan algebras. Moreover, one of these new gradings is used to extend the well known duality between the associative-G-gradings and the G-actions (by automorphisms) of A to include actions by anti-autopmorphisms. We call this new graded structure a Lie-Jordan-G-graded algebra, and mention some of the applications it has to Shirshov bases, polynomial identities, and other topics.

Geometry and Topology

Speaker: Inna Zakharevich (MIT)

"Scissors congruence as K-theory"

Time: 15:30

Room: MC 107

Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. In this talk we will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

Analysis Seminar

Speaker: Elizabeth Mansfield (University of Kent, UK)

"Moment maps for smooth Hamiltonian systems and their discrete symplectic analogues"

Time: 14:40

Room: MC 107

Discrete symplectic mappings are not called discrete Hamiltonian systems by the cognoscenti for a variety of reasons, and my aim with this project was to explore the problems involved. In this talk I first show how conservation laws for high order Lagrangian systems transfer to their equivalent Hamiltonian systems with a strikingly beautiful formula. This formula transfers mutatis mutandis to the discrete case. However, analogues of other results do not transfer. I present a series of examples that illustrate the various difficulties, and end with some conjectures and possible ways forward that could involve specialist analytic techniques. The talk assumes no specialist knowledge of the topic.

Graduate Seminar

Speaker: Mehdi Mousavi (Western)

"Pascal's Matrices"

Time: 17:10

Room: MC 107

Pascal's matrices are constructed from Pascal's triangle and they are encountered frequently in probability theory. In this talk we will discuss some of their striking properties and their application to polynomials.

(Note: lecture will start at 5:15)

Colloquium

Speaker: Elizabeth Mansfield (University of Kent)

"Discrete Moving Frames"

Time: 15:30

Room: MC 107

Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. In this talk we discuss moving frames and some applications. We then show how what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains, offers significant computational advantages over a single moving frame for some applications, in particular to discrete integrable systems. The talk assumes no specialist knowledge.

Geometry and Topology

Speaker: Victor Turchin (Kansas State University)

"Context free manifold calculus of functors and the operad of framed discs"

Time: 15:30

Room: MC 107

Manifold calculus of functors was introduced and developed by T. Goodwillie and M. Weiss in order to study spaces of embeddings. In a few words the goal of their method is to understand how from the spaces Emb(U,N) of smaller open subsets U of M we can describe the space Emb(M,N) of embeddings of the entire manifold M into N. Naively it is sometimes called "patching method". I will describe briefly the ideas of this theory and also explain some recent advances which gives a connection with the theory of operads.

Pizza Seminar

Speaker: Seymour Ditor (Western)

"Infinite Exponentials"

Time: 16:30

Room: MC 107

When does an "infinite tower of exponentials" converge? To clarify, for positive real numbers $a,b, \ldots$ let us set $E_a(x) = a^x$, and $E(a,b, \ldots, c) = E_a \circ E_b \circ \cdots \circ E_c (1)$, so $E(a) = a$, $E(a,b) = a^b$, $E(a,b,c) = a^{b^c}$. The question then is: for what sequences $\{a_n\}$ of positive real numbers does the sequence $\{E(a_1, \ldots, a_n)\}$ converge?

Algebra Seminar

Speaker: Lex Renner (Western)

"The generic point of a group action"

Time: 14:40

Room: MC 107

Starting with an action $G\times X\to X$ we analyze the maximal $G$-rational subalgebra $\mathscr{O}_K$ of $k(X)$ and use it to obtain the action $G_K\times U_K\to U_K$ where $K = k(X)^G$, and $U_K$ is a certain quasi-affine variety over $K$ with $\mathscr{O}(U_K) = \mathscr{O}_K$. This gives us a generic "homogeneous" picture of the original action.

We also analyze the maximal $G$-rational subalgebra of $k[X]_\mathfrak{p}$, where $\mathfrak{p}$ is a height-one $G$-prime of $k[X]$. We use these results to assess the behavior of the canonical map $\pi : U\to U/G$ for a sufficiently small $G$-invariant, open subset $U$ of $X$.

Finally we use ${\textit{observable}}$ $G$-actions over $k$ to construct the functor $K\mapsto H^1(K,G/H)$, from finitely generated fields over $k$ to ${\textit{Sets}}$. From there we define the ${\textit{essential dimension}}$ of a homogeneous space $G/H$, whenever $H\subset G$ is a pair of connected, reductive groups.

Geometry and Topology

Speaker: Jordan Watts (U Toronto)

"Differential Forms on Symplectic Quotients"

Time: 15:30

Room: MC 107

While a symplectic quotient coming from a Hamiltonian action of a compact Lie group is generally not a manifold (it is a stratified space), one can still define a notion of differential form on it. Indeed, one can obtain a de Rham Theorem, Poincaré Lemma, and a version of Stokes' Theorem using this de Rham complex of forms. I will show how these forms are defined, and then explore the question of intrinsicality of the complex. This question leads into a discussion of different definitions of a smooth structure on the quotient, and the pros and cons of each

Analysis Seminar

Speaker: Steven Rayan (University of Toronto)

"Poincare series for the Higgs moduli space on $P^1$ from operations on quivers"

Time: 14:30

Room: MC 107

In this talk, I will highlight some differences between the moduli space of Higgs bundles (in the sense of Hitchin) on a curve of positive genus and the the moduli space of "twisted" Higgs bundles at genus 0. The Betti numbers of both spaces can be determined by a localization calculation, with respect to an $S^1$ action. This was exactly Hitchin's method for obtaining the Betti numbers of the rank-2 instance of the usual Higgs moduli space. The $S^1$ fixed points are what are called "holomorphic chains": these are similar to complexes of vector bundles, but the differential (the Higgs field itself) is nilpotent with order equal to the length of the complex. I will show how the localization calculation can be made very combinatorial in the genus 0 case. The appropriate language for organizing this data is that of quivers, which we use to represent (and construct) families of chains.

Graduate Seminar

Speaker: Girish Kulkarni (Western)

"Introduction to Category Theory"

Time: 16:30

Room: MC 107

In this introductory talk I will start with basic definitions and examples. After defining natural transformations I will prove the Yoneda lemma which is a fundamental result in category theory. It will indeed be a good opportunity for the beginners to befriend the theory and refresher for the others.

Geometry and Combinatorics

Speaker: Mehdi Garrousian (Western/Windsor)

"Tropical Geometry learning seminar I"

Time: 14:30

Room: MC 105C

this will be the first talk in a short series looking over some basics of tropical geometry.

Colloquium

Speaker: Greg Arone (University of Virginia)

"On the structure of polynomial functors in topology"

Time: 15:30

Room: MC 108

Let f be a function. The two most basic ways to approximate f with a polynomial function are, probably, the interpolation polynomial and the Taylor polynomial. The interpolation polynomial (of degree n) is determined by the n+1 numbers f(0), f(1), ..., f(n). The Taylor polynomial is determined by a different set of n+1 numbers - the first n+1 derivatives of f (at 0 say).

In the talk we will explore the analogues of these two constructions for functors that arise in topology. It turns out that while a polynomial function is determined by a sequence of numbers, a polynomial functor is determined by a (truncated) symmetric sequence with an extra structure. The extra structure can be expressed in terms of operads and their modules. The relationship between the interpolation and the Taylor polynomial can be understood in terms of (a version of) Koszul duality between operads.

A good example to test the theory on is the mapping bi-functor that sends a pair of topological spaces (X, Y) to the space of maps F(X, Y). An equally interesting example is the functor that sends a pair ofsmooth manifolds (M, N) to the space of smooth embeddings Emb(M, N). We will use these functors, and others related to them, to illustrate the general theory.

Algebra Seminar

Speaker: Sergey Rybakov (Moscow Institute of Information Transmission Problems)

"Coherent DG-modules over de Rham complex"

Time: 14:40

Room: MC 107

Recently Positselski proved that an unbounded derived category of quasi-coherent D-modules on a smooth algebraic variety X is equivalent to a so-called coderived category of quasi-coherent DG-modules over the de Rham algebra of X. I will explain how to work with this coderived category.

Graduate Seminar

Speaker: Enxin Wu (Western)

"Some geometries on the irrational torus"

Time: 16:30

Room: MC 107

In differential geometry, we know that an irrational torus is a pathological space rather than a smooth manifold. However, we can still do some kind of differential geometry by using a new categorical viewpoint towards geometry. In this talk, basic geometry, homotopy groups, smooth bundles, and some new work on tangent spaces for irrational torii will be introduced. If time permits, we will talk about differential forms, de Rham cohomology and foliation theory on irrational torii as well.

Colloquium

Speaker: Stefan Gille (University of Alberta)

"Chow motives of surfaces"

Time: 15:30

Room: MC 107

After recalling the definition of Chow motives I will present a proof of Rost nilpotence for surfaces and some three folds, and discuss some applications of this result.

Algebra Seminar

Speaker: Stefan Gille (University of Alberta)

"The Brauer group of a semisimple algebraic group"

Time: 14:40

Room: MC 107

Let k be a field and G be an algebraic group. If char k=0 Birger Iversen showed in 1976 that the pull-back Br(k)$\to$ Br (G) is an isomorphism if $G$ is simply connected. If fact, he proved this using topological methods for $k$ the field of complex numbers from which the general case follows by Galois cohomology. In my talk I will present one or two (if time permits) algebraic proofs of Iversen's result which show more: If $G$ is simply connected and $k$ arbitrary then the pull-back $n$-torsion Br(k)$\to$ $n$-torsion Br (G) is an isomorphism as long as $n$ is prime to the characteristic of $k$. If $k$ is not perfect of char $p$ I will show then that the pull-back $p$-torsion Br(k)$\to$ $p$-torsion Br(G) is not surjective for any semisimple isotropic connected linear algebraic group over $k$.