Noncommutative Geometry

Speaker: Sean Fitzpatrick (Western)

"Localization in equivariant cohomology for non-abelian group actions"

Time: 14:30 - 16:00

Room: MC 108

The Duistermaat-Heckman exact stationary phase approximation was shown by Berline-Vergne (and independently by Atiyah-Bott) to be a consequence of a localization theorem for equivariant differential forms. This result, although stated in the setting of a Hamiltonian action of a compact Lie group on a symplectic manifold, is really a result about circle actions, since it relies on first choosing a Hamiltonian vector field with isolated zeros and periodic flow. In his paper "Two dimensional gauge theories revisited", Witten proposed a "not necessarily abelian" version of localization for compact Lie groups, and investigated some of the properties of such a localization. His results were motivated by physics, and not fully rigorous from a mathematical point of view. His ideas were subsequently explored by Jeffrey and Kirwan, and later by Paradan, and put on a sound mathematical footing. I will explain Witten's ideas, and briefly explore the approach of Jeffrey and Kirwan before outlining Paradan's approach to the problem, and how his results can be applied to obtain a cohomological formula for the index of transversally elliptic operators.

Geometry and Topology

Speaker: Emily Riehl (Harvard)

"Homotopy coherent adjunctions, monads, and algebras"

Time: 15:30 - 16:30

Room: MC 107

A monad is a device for encoding algebraic structure. Conversely, if an adjunction is monadic (i.e., encodes a category of algebras), this implies several useful categorical properties.

This talk describes joint work with Dominic Verity to develop this theory for quasi-categories (aka infinity-categories). We introduce the free homotopy coherent adjunction, demonstrate that any adjunction of quasi-categories gives rise to such, and give a formal proof of the monadicity theorem that is directly applicable in other contexts.

Homotopy Theory

Speaker: Ivan Kobyzev (Western)

"Triangulated dg-categories"

Time: 14:30 - 15:30

Room: MC 107

Distinguished Lecture

Speaker: Carlos Simpson (Nice)

"Nonabelian Hodge theory---a panorama, I"

Time: 15:30 - 16:30

Room: MC 107

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.

Analysis Seminar

Speaker: Wayne Grey (Western)

"Inclusions among mixed-norm $L^P$ spaces"

Time: 11:30 - 12:30

Room: MC 108

Mixed-norm $L^P$ spaces, generalizing the Lebesgue space, have been studied for over half a century, with various applications in pure and applied mathematics. For classical Lebesgue spaces, given exponents $p$ and $q$ and $\sigma$-finite measures $\mu$ and $\nu$ on the same measurable space, there are well-known conditions for when $L^p(\mu)$ is contained in $L^q(\nu)$. This talk presents a mostly complete solution describing when two (permuted) mixed-norm spaces, again with different exponents and measures, have such an inclusion.

The only non-trivial situation is when the mixed norms integrate over their variables in differing orders, as seen in Minkowski's integral inequality. J.J.F Fournier called these "permuted mixed norms" and developed a generalization of Minkowski's integral inequality which is key to this solution.

A full solution is given when no measure is purely atomic. This turns out to depend only on the necessary one-variable inclusions and a condition derived from Minkowski's integral inequality. When purely atomic measures are allowed, there are still partial solutions, but the situation is substantially more complicated. Solutions in some cases turn out to involve optimization problems in weighted $l^p$.

Distinguished Lecture

Speaker: Carlos Simpson (Nice)

"Nonabelian Hodge theory---a panorama, II"

Time: 15:30 - 16:30

Room: MC 107

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.

Distinguished Lecture

Speaker: Carlos Simpson (Nice)

"Nonabelian Hodge theory---a panorama, III"

Time: 15:30 - 16:30

Room: MC 107

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"Dirac Operators and Geodesic Metric on the Sierpinski Gasket"

Time: 14:30 - 16:00

Room: MC 108

In this talk I will report some parts of a paper of Michel L. Lapidus and Jonathan J. Sarhad titled "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets". First the Sierpinski gasket will be introduced as the unique fixed point of a certain contraction on the compact subsets of the Euclidean plane. Then, by defining the graph approximation of the Sierpinski gasket and also using a spectral triple on the unit circle, a Dirac operator and a spectral triple for the Sierpinski gasket will be constructed. Next, it will be shown that the spectral dimension of this spectral triple is bigger than or equal to 1. Connes' distance formula of noncommutative geometry provides a natural metric on this fractal. Finally, we shall see that Connes' metric is the same as the geodesic metric on the Sierpinski gasket.

Geometry and Topology

Speaker: Andrew Salch (Wayne State)

"Computing all model structures on a given category"

Time: 15:30 - 16:30

Room: MC 107

Suppose C is a category. We put a (quasi-)metric on the collection of all model structures on C, such that two model structures are more distant from one another if it takes a longer chain of Bousfield localizations and co-localizations to arrive at one from the other. We then compute the closed unit ball centered at the discrete model structure in this (quasi-)metric space.

Then we develop some methods for computing the entire (quasi-)metric space of all model structures on C! Our methods are actually categorical generalizations of constructions from classical linear algebra, namely Smith normal form. We describe categorical Smith normal form and its implications for the collection of model structures on a given category, and then we use these methods for some example computations: we explicitly compute all model structures and all their homotopy categories, their associated algebraic K-theories, and all their localizations and co-localizations, for the category of vector spaces over a field and then also for the category of modules over the tangent neighborhood of a regular closed point in a 1-dimensional normal scheme (e.g. Z/p^2-modules or k[x]/x^2-modules). Finally we do some commutative algebra and exploit our methods described above to prove the following amusing result: suppose R is a principal ideal ring whose modules admit generalized Smith normal form, that is, every indecomposable morphism of R-modules has indecomposable domain and indecomposable codomain (during the talk we will characterize exactly which principal ideal rings have this property). Then there exist exactly 5^A 11^B model structures on the category of R-modules, where A is the number of points in Spec R with reduced stalk, and B is the number of points in Spec R with non-reduced stalk.

Time allowing, we will actually say how to do the necessary commutative algebra to prove some of these results, which involves some work (for example, a cohomological solution to the compatible splitting problem: given a map from one split short exact sequence to another split short exact sequence, when does there exists a splitting of each short exact sequence which is compatible with the map?), and leads to a natural conjecture about how to continue the work into more sophisticated rings, by relating categorical Smith normal forms to edges in the Auslander-Reiten quiver; and we will sketch how one would go about giving an explicit classification of all model structures on torsion quasicoherent modules over a genus 0 nonsingular algebraic curve, if one had a proof of this conjecture.

PhD Thesis Defence

Speaker: Fatemeh Bagherzadeh (Western)

"W-groups of Pythagorean Formally Real Fields"

Time: 13:00 - 14:00

Room: MC 108

In this work we consider the Galois point of view in determining the structure of a space of orderings of fields via considering small Galois quotients of absolute Galois groups G_F of Pythagorean formally real fields. Galois theoretic, group theoretic and combinatorial arguments are used to reduce the structure of W-groups.

Homotopy Theory

Speaker:

"Talk canceled"

Time: 14:30 - 15:30

Room:

We will resume next week.

Index Theory Seminar

Speaker: Matthias Franz (Western)

"The infinitesimal index"

Time: 12:00 - 13:30

Room: MC 107

We discuss the definition and basic properties of the infinitesimal index as introduced be De Concini-Procesi-Vergne. Unlike the index of transversally elliptic operators, its domain is the equivariant cohomology with compact supports of the zeroes of the moment map. We also look at the special case of the cotangent bundle of a representation space, where connections to splines appear.

Colloquium

Speaker: Chuck Weibel (Rutgers)

"Co-operations for motivic K-theory"

Time: 15:30 - 16:30

Room: MC 107

It is classical that the topological K-theory KU(X) of a space X agrees with maps from X to KU, and that cohomology operations correspond to maps from KU to itself. Dual to this is the structure of "co-operations", i.e., the KU-homology of KU relative to the ring KU(point). This data has a structure, dubbed Hopf algebroid, which is related to combinatorics and numerical polynomials. In joint work with Pelaez, we determine the analogous structure for algebraic K-theory KGL, regarded as a motivic object. Applying the motivic slice filtration, we solve a problem of Voevodsky.

Geometry and Topology

Speaker: Aaron Adcock (Stanford)

"Tree-like structure in social and information networks"

Time: 14:30 - 15:30

Room: MC 107

Although large social and information networks are often thought of as having hierarchical or tree-like structure, this assumption is rarely tested. We have performed a detailed empirical analysis of the tree-like properties of realistic informatics graphs using two very different notions of tree-likeness: Gromov's δ-hyperbolicity, which is a notion from geometric group theory that measures how tree-like a graph is in terms of its metric structure; and tree decompositions, tools from structural graph theory which measure how tree-like a graph is in terms of its cut structure. Although realistic informatics graphs often do not have meaningful tree-like structure when viewed with respect to the simplest and most popular metrics, e.g., the value of δ or the treewidth, we conclude that many such graphs do have meaningful tree-like structure when viewed with respect to more refined metrics, e.g., a size-resolved notion of δ or a closer analysis of the tree decompositions. We also show that, although these two rigorous notions of tree-likeness capture very different tree- like structures in the worst-case, for realistic informatics graphs they empirically identify surprisingly similar structure. We interpret this tree-like structure in terms of the recently-characterized "nested core-periphery" property of large informatics graphs; and we show that the fast and scalable k-core heuristic can be used to identify this tree-like structure.

Geometry and Combinatorics

Speaker: Zsuzsanna Dancso (University of Toronto)

"A categorical realisation of the cut and flow lattices of graphs"

Time: 15:30 - 16:30

Room: MC 107

I will introduce some fundamental concepts of lattice theory (unimodular lattices, lattice gluing) and explain why we expect them to naturally appear on a homological algebra level. We will discuss the example of the cut and flow lattices of a graph, and a categorical realisation which serves as an example for lifting lattice theoretic concepts as mentioned above. We end with a number of open questions and directions. Joint work with Anthony Licata.

Geometry and Topology

Speaker: Agnes Beaudry (Univ. of Chicago)

"Finite Resolutions and K(2)-Local Computations"

Time: 15:30 - 16:30

Room: MC 107

The chromatic filtration of stable homotopy breaks calculations one prime at a time and one level at a time. At chromatic level 2, finite resolutions of the K(2)-local sphere, have been a successful tool for computations. I will explain how to use such resolutions to do computations at the prime 2.

Geometry and Topology

Speaker: Victor Snaith (Sheffield)

"The bar-monomial resolution and applications to automorphic representations"

Time: 13:30 - 14:30

Room: MC 108

Admissible representations of locally $p$-adic Lie groups and automorphic representations of adelic Lie groups are important ingredients in modern number theory. This is because of their deep relationship to modular forms, L-series and Galois representations. This talk will explain how to construct these representations as objects (monomial resolutions) in a (non-abelian) derived category. This construction applies to representations defined over an algebraically closed field of any characteristic. The intended advantage of this point of view lies in the fact that in this derived category L-series and base base change have elementary descriptions.

Analysis Seminar

Speaker: Vassili Nestoridis (University of Athens)

"Universality and regularity of the integration operator"

Time: 15:30 - 16:30

Room: MC 108

Let $Y$ denote the space of holomorphic functions in a planar domain $\Omega$, such that the derivatives of all orders extend continuously to the closure of $\Omega$ in the plane $\mathbb{C}$. We endow $Y$ with its natural topology and let $X$ denote the closure in $Y$ of all rational functions with poles off the closure of $\Omega$. Some universality results concerning Taylor series or Pade approximants are generic in $X$. In order to strengthen the above results we give a sufficient condition of geometric nature assuring that $X$=$Y$. In addition to this, if a Jordan domain $\Omega$ satisfies the above condition, then the primitive $F$ of a holomorphic function $f$ in $\Omega$ is at least as smooth on the boundary as $f$, even if the boundary of $\Omega$ has infinite length. This led us to construct a Jordan domain $\Omega$ supporting a holomorphic function $f$ which extends continuously on the closure of $\Omega$, such that its primitive $F$ is even not bounded in $\Omega$. Finally we extend the last result in generic form to more general Volterra operators.

This is based on a joint work with Ilias Zadik.

Homotopy Theory

Speaker: Mike Misamore (Western)

"Applications of dg-categories"

Time: 14:30 - 15:30

Room: MC 107

Analysis Seminar

Speaker: Nadya Askaripour (University of Cincinnati)

"Spaces of automorphic functions and projections on these spaces"

Time: 11:30 - 12:30

Room: MC 108

This will be a survey talk. Let R be a Riemann surface, I will talk about some spaces of automorphic functions on R. Poincare series map is one way to construct automorphic forms, but it might not be convergent when we have an automorphic function. Some projections on automorphic functions will be introduced, also I will show few applications of these projections.

Index Theory Seminar

Speaker: Matthias Franz (Western)

"The infinitesimal index II"

Time: 12:00 - 13:30

Room: MC 107

This time we look at the cotangent bundle of a representation M of a torus. We discuss how the infinitesimal index gives an isomorphism between equivariant cohomology modules induced by the orbit filtration of M and certain spaces of splines which we are going to define.

Colloquium

Speaker: Uli Walther (Purdue)

"Local cohomology (of determinantal ideals)"

Time: 15:30 - 16:30

Room: MC 107

We give an introduction to the theory of local cohomology by pointing out examples and its connections to topology, differential equations, and algebraic geometry. We discuss, in the presence of a field, some techniques for investigating local cohomology modules, focussing on the study of finiteness conditions. We show how the absence of a field, or singularities in the ambient space, complicate things significantly, and work out the case of the variety consisting of matrices of bounded rank over every field and over the integers.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"The Dirac Operator and Gravitation"

Time: 14:30 - 16:00

Room: MC 107

In 1992, Connes conjectured that the Wodzicki residue of the inverse square of the Atiyah-Singer-Lichnerowicz Dirac operator D equals the Einstein- Hilbert functional of general relativity, up to a constant multiple. After giving a description of the fundamental actions in Mathematics and Physics, including the Yang-Mills action, Polyakov action and Einstein-Hilbert action, I will describe the fundamental steps involved in the purely computational proof given by Daniel Kastler in 1994. In addition, I will present a more straightforward and direct proof given by Coiai and Spera in 2000 that only involves an application of the Riemann-Zeta function and the second Seeley de-Witt coefficient of the heat kernel expansion of $\Delta = D^{2}$.

Colloquium

Speaker: Jason Bell (Waterloo)

"Linear recurrences, automorphisms, and finite-state machines"

Time: 15:30 - 16:30

Room: MC 107

The Skolem-Mahler-Lech theorem is a beautiful result in number theory, which asserts that if a complex-valued sequence f(n) satisfies a linear recurrence (e.g., the Fibonacci numbers) then the set of natural numbers n for which f(n)=0 is a finite union of arithmetic progressions along with a finite set.  We'll show that this has a geometric analogue in which one has a complex variety $X$, an automorphism $g\colon X\to X$, and a point $x$ in $X$ and one wishes to know when $g^n(x)$ lies in some fixed subvariety $Y$.  We'll then discuss the positive characteristic case.  In positive characteristic, the conclusion to the Skolem-Mahler-Lech theorem need not hold and we'll talk about work of Harm Derksen, which shows that one can express the zero sets of linear recurrence using what are called finite-state machines, and we'll ask whether Derksen's result has a similar geometric analogue.

Analysis Seminar

Speaker: Vassili Nestoridis (University of Athens)

"Some universality results concerning harmonic functions on trees"

Time: 15:30 - 16:30

Room: MC 108

We present some universality results concerning harmonic functions on trees. One of these results relates to approximation by martingales on the boundary of the tree. We discuss topological genericity, algebraic genericity and spaceability.

Homotopy Theory

Speaker: Martin Frankland (Western)

"More applications of dg-categories"

Time: 14:30 - 15:30

Room: MC 107

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket"

Time: 14:30 - 16:00

Room: MC 108

This is a report on a paper by Lapidus and Sarhad titled "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets". First I will define the graph approximation of the Sierpinski gasket. Then I will talk about Kusuoka's measurable Riemannian geometry on Sierpinski gasket and introduce counterparts of Riemannian volume, Riemannian metric and Riemannian energy in that setting. Thereafter harmonic functions on Sierpinski gasket will be introduced as energy minimizing functions. Using those functions we can define harmonic gasket. I will also talk about Kigami's geodesic metric on harmonic gasket. Then I will construct two spectral triples on harmonic gasket and we will see that those two triples induce the same spectral metric as Kigami's geodesic metric.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Multiplicities formula for the equivariant index"

Time: 12:00 - 13:30

Room: MC 107

For both elliptic and transversally elliptic operators, we have seen that the equivariant index defines a virtual $G$-representation, where $G$ is a compact Lie group. This representation can be expressed as a sum of irreducible representations with multiplicities. In the elliptic case, this sum is finite. In the transversally elliptic case, the sum is infinite, but the index still defines a distributional character on $G$.

The aim of this talk is to give an overview of how the de Concini-Procesi-Vergne machinery (associated to the infinitesimal index) can be used to give a formula for the multiplicities of the irreducible representations within the equivariant index. This will be more of a ``big picture'' talk that attempts to tie together some of the particular results we've encountered so far, without getting too much into the details. The main reference will be Vergne's paper on the Euler-Maclaurin formula for the multiplicity function (arXiv:1211.5547).

Geometry and Combinatorics

Speaker: Graham Denham (Western)

"intersection-theoretic characteristic polynomial formulas"

Time: 15:30 - 16:30

Room: MC 107

I will describe a project with June Huh on Chern-Schwarz-MacPherson classes of some varieties associated with matroids, and combinatorial inequalities that result.

Geometry and Topology

Speaker: Hugh Thomas (UNB)

"Monodromy for the quintic mirror"

Time: 15:30 - 16:30

Room: MC 107

The mirror to the quintic in P^4 is a family X_p of Calabi-Yau 3-folds over a thrice-punctured sphere. As p moves in a loop around each of the three punctures, we can parallel transport classes in H^3(X_p), and observe the monodromy. H^3(X_p) is four-dimensional, and the monodromy can be expressed by matrices in Sp(4,Z). These matrices generate a subgroup which is dense in Sp(4,Z), but it was not known whether or not it was of finite index. We showed that the subgroup is isomorphic to the free product Z/5 * Z, from which it follows that it cannot be of finite index. The mirror quintic family is one of 14 similar families of CY 3-folds; our methods establish similar results for 7 of the 14 families. For the other 7, it has recently been shown that the monodromy is of finite index, so our result is best possible. This talk is based on joint work with Chris Brav, arXiv:1210.0523.