Noncommutative Geometry

Speaker: (Western)

"An algebraic approach to characteristic classes"

Time: 11:30

Room: MC 107

We shall review the Chern-Weil theory cast in the language of finitely generated projective modules. This algebraic approach has the advantage that it can be extended to noncommutative settings.

Homotopy Theory

Speaker:

"Organizational meeting"

Time: 13:30

Room: MC 107

Analysis Seminar

Speaker: Purvi Gupta (Western)

"Some new analogies between convex and complex analysis - Part I"

Time: 15:30

Room: MC 107

Convexity has played an important role in the development of the theory of several complex variables. In this first of a series of talks, I will give a brief introduction to the various notions of convexity that complex analysts have found useful. We will then define an (equi)affine-invariant surface area measure for convex bodies, and discuss some related problems.The subsequent talk(s) will focus on its relatively less explored complex analog --- the Fefferman hypersurface area measure. This talk will be accessible to graduate students.

Noncommutative Geometry

Speaker: (Western)

"Maurer-Cartan forms and odd Chern character"

Time: 11:30

Room: MC 107

In this talk, we shall exhibit explicit formulas for the odd Chern character map.

Homotopy Theory

Speaker: Chris Kapulkin (Western)

"Properties of the Identity Type (part 1)"

Time: 13:30

Room: MC 107

Following Chapter 2 of the HoTT Book, I will present the main properties of the Identity Type.

Analysis Seminar

Speaker: Purvi Gupta (Western)

"Some new analogies between convex and complex analysis - Part II"

Time: 15:30

Room: MC 107

We will discuss the convex floating body method of extending Blaschke's affine surface area measure to all convex bodies. We will then introduce the Fefferman hypersurface area measure on the boundary of a strongly pseudoconvex domain. This measure transforms well under biholomorphisms, and we will discuss why this can be useful. Taking tube domains as the motivating case, we will propose a method for extending this measure to general pseudoconvex domains.

Basic Notions Seminar

Speaker: Rick Jardine (Western)

"What is a ring spectrum?"

Time: 15:30

Room: MC 107

This talk is a leisurely introduction to spectra, symmetric spectra and cup products in stable homotopy theory. Ring spectra are discussed as examples, in various contexts.

Geometry and Topology

Speaker: Nicole Lemire (Western)

"Pushforwards of Tilting Sheaves"

Time: 15:30

Room: MC 107

In joint work with A. Dhillon and Y. Yan, we investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf.

Noncommutative Geometry

Speaker: (Western)

"Yang-Mills equations I"

Time: 11:30

Room: MC 107

I shall give a quick introduction to Yang-Mills equations and self-duality as they appear both in mathematics and high energy physics.

Homotopy Theory

Speaker: Chris Kapulkin (Western)

"Properties of the Identity Type (part 2)"

Time: 13:30

Room: MC 107

Following Chapter 2 of the HoTT Book, I will continue presenting the main properties of the Identity Type.

Analysis Seminar

Speaker: Purvi Gupta (Western)

"Some new analogies between convex and complex analysis - Part III"

Time: 15:30

Room: MC 107

I will (re)state and prove estimates that relate the super-level sets of the diagonal Bergman kernel on a pseudoconvex tube domain with the floating bodies of its convex base.

Graduate Seminar

Speaker: Dinesh Valluri (Western)

"Riemann-Roch theorem and consequences"

Time: 13:30

Room: MC 108

In this talk we will introduce the notions of divisor, meromorphic functions and meromorphic forms on a Compact Riemann Surface. We will state the Riemann Roch theorem and derive several interesting consequences of it. For example this gives us the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. We will interpret the theorem as a statement about Euler characteristic and explore possible generalizations of this theorem in the context of algebraic geometry. If time permits we shall see a sketch of a proof of Riemann-Roch.

Basic Notions Seminar

Speaker: Graham Denham (Western)

"Geometric approaches to matroid inequalities"

Time: 15:30

Room: MC 107

Abstract: Newton showed that, if a polynomial \(p(t)=\sum_{i=0}^n a_i t^i\) has only real roots, then the coefficient sequence \((a_0,a_1,\ldots,a_n)\) satisfies the inequalities \(a_i^2\geq a_{i-1}a_{i+1}\). This implies, in particular, that the sequence is (up to sign) unimodal.

In 1968, Ronald Read conjectured that the coefficients of the chromatic polynomial of a graph form a (sign-alternating) unimodal sequence. Soon afterwards, Rota, Heron and Welsh proposed a much more daring conjecture: that the coefficients of the characteristic polynomial of a matroid form a sign-alternating log-concave sequence.

In a sequence of recent papers, June Huh, then Huh with Eric Katz, and finally Huh, Katz and Karim Adiprasito proved the Rota-Heron-Welsh conjecture. First for matroids realizable in characteristic zero, then over any field, and most recently for matroids without linear realizations. The methods in each case make use of or are inspired by inequalities in algebraic geometry.

My objective is to give a gentle introduction to their program.

Algebra Seminar

Speaker: June Huh (Princeton)

"Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries"

Time: 14:30

Room: Kresge K106

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids.

Joint work with Karim Adiprasito and Eric Katz.

Noncommutative Geometry

Speaker: (Western)

"Yang-Mills equations and (anti-) self duality"

Time: 11:30

Room: MC 107

We shall explore consequences of (anti-) self-duality for solutions of Yang-Mills equations in dimension 4.

Homotopy Theory

Speaker: Mitchell Riley (Western)

"Homotopy n-Types (part 1)"

Time: 13:30

Room: MC 107

Following Chapter 7 of the HoTT Book, I will present some basic properties of Homotopy n-Types.

Geometry and Topology

Speaker: Spiro Karigiannis (Waterloo)

"Partial classification of twisted austere 3-folds"

Time: 14:30

Room: MC 107

Calibrated submanifolds are special kinds of minimal submanifolds (vanishing mean curvature) that are defined by first order conditions on the immersion. The most studied examples are complex submanifolds of Kahler manifolds, special Lagrangian submanifolds in Calabi-Yau manifolds, and certain special types of submanifolds in $G2$ and $Spin(7)$ manifolds. By imposing a certain amount of symmetry, one can sometimes reduce the nonlinear elliptic first order equations defining such submanifolds to simpler equations on lower-dimensional manifolds. For example, a result of Harvey-Lawson is that the conormal bundle of an austere submanifold of $\mathbb{R}^n$ is special Lagrangian in $\mathbb{C}^n$. This "bundle construction" was generalized in 2004 by Ionel-Karigiannis-Min-Oo to other calibrations, and then extended in 2012 by Karigiannis-Leung to a "twisted" version. Thus, in particular, we obtain many more examples of special Lagrangian submanifolds of $\mathbb{C}^n$ by considering "twisted austere submanifolds" of $\mathbb{R}^n$. I will describe these constructions and review the earlier results. Then I will state several theorems that give a partial classification of twisted austere submanifolds of dimension 3. These new theorems are joint work with Tom Ivey at Charleston College.

Analysis Seminar

Speaker: Debraj Chakrabarti (Central Michigan University)

"$L^2$- Dolbeault cohomology of annuli"

Time: 15:30

Room: MC 107

By an annulus we mean a domain in $\mathbb{C}^n$ obtained by removing a compact set from a pseudoconvex domain. We study when the $L^2$ $\overline\partial$ operator has closed range from functions to $(0,1)$-forms. In particular, we show that the Chinese Coin problem, i.e. to prove $L^2$-estimates on a domain in $\mathbb{C}^2$ obtained by removing a bidisc from a ball, has a positive solution.

Pizza Seminar

Speaker: Matthias Franz (Western)

"Numbers"

Time: 17:30

Room: MC 108

The natural numbers are the first thing one learns in mathematics. Because they lack some desirable properties, one soon extends them to the integers, the rational, real and complex numbers. In this talk I want to focus on other numbers systems that are less often encountered in undergraduate mathematics courses, for instance $p$-adic numbers, quaternions, octonions and cardinal numbers.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Volume Quantization from Spin Geometry"

Time: 11:00

Room: MC 107

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Comprehensive Exam Presentation

Speaker: Shahab Azarfar (Western)

"A Report on Quanta of Geometry"

Time: 16:00

Room: MC 108

We try to investigate a generalization of the Heisenberg commutation re- lation [p; q] = ô€€€i}, introduced by Chamseddine, Connes and Mukhanov as \the one-sided and the two-sided quantization equations", which captures the geometry. The momentum variable p is encoded by the Dirac operator and the analogue of the position variable q is the Feynman slash of real scalar elds over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a dis- connected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the rened version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Yang-Mills equations III: BPST instantons"

Time: 11:30

Room: MC 107

I shall give a detailed construction of the BPST instantons.

Homotopy Theory

Speaker: Mitchell Riley (Western)

"Homotopy n-Types (part 2)"

Time: 13:30

Room: MC 107

Continuing the previous talk, I will present some properties of (-1)- and 0-types; propositions and sets.

Analysis Seminar

Speaker: Octavian Mitrea (Western)

"Polynomial Convexity"

Time: 15:30

Room: MC 107

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem. This talk is given in fulfillment of the requirements for Part II of the PhD comprehensive examination.

Comprehensive Exam Presentation

Speaker: Octavian Mitrea (Western)

"Polynomial Convexity"

Time: 15:30

Room: MC 107

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Volume Quantization from Spin Geometry II"

Time: 11:00

Room: MC 107

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Graduate Seminar

Speaker: Mitchell Riley (Western)

"Combinatorial Games"

Time: 13:30

Room: MC 108

In this talk we will introduce the theory combinatorial games, a simple mathematical structure with incredibly rich algebraic properties. As well as containing all real numbers, the class of games contains all ordinals, a collection of infinitesimals and plenty in between. The study of combinatorial games can be applied directly to the analysis of actual strategy games, including Chess and Go. If time permits, we will use the techniques of the talk to analyse a curious chess endgame.

Comprehensive Exam Presentation

Speaker: Ahmed Ashraf (Western)

"Characterizing f-vector"

Time: 14:30

Room: MC 108

Given a d-dimensional convex polytope, its k-th face number is the number of (k 􀀀-1)-dimensional faces it has. The f-vector of a polytope is the sequence of its face numbers. Beside Euler's formula, these numbers satisfy further equalities and inequalities. Characterization of f-vector of d-dimensional convex polytope is already known for d ≤ 3 . For d ≥ 3, we do not have a complete answer, but g-theorem gives us a characterization for simplicial (and dually simple) case. Here we review g-theorem and its various proofs.