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29 Comprehensive Oral Presentation
Comprehensive Oral Presentation Speaker: Udit Mavinkurve (Western) "TBA" Time: 15:30 Room: MC 107 |
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4 Equivariant Topology Seminar
Equivariant Topology Seminar Speaker: Rafael Gomes (Western) "The Borel construction and the equivariant cohomology of a $G$-space" Time: 09:30 Room: WSC 184 & online Through the universal $G$-space $EG$, once can replace a $G$-space $X$ by a homotopy equivalent space in which the $G$-action is free. the orbit space of this action is the Borel construction. This Borel construction is then a better model to study the $G$-action, as not only can we use tools of algebraic topology such as cohomology (which leads to equivariant cohomology) and spectral sequences, but also computations using these tools are often way nicer. Meeting ID: 997 4840 9440
Passcode: 911104 Geometry and Combinatorics
Geometry and Combinatorics Speaker: Brett Nasserden (Western) "Surjective endomorphisms of projective bundles over an elliptic curve" Time: 15:30 Room: MC 107 We will discuss how to compute dynamical invariants of a surjective endomorphism of a projective bundle over an elliptic curve. This has applications to arithmetic dynamics and the Kawaguchi-Silverman conjecture. In particular, we will discuss how to use the theory of Schur functors to compute the Iitaka dimension of certain projective bundles. This extends work done by Atiyah in classifying vector bundles on elliptic curves. |
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7 Analysis Seminar
Analysis Seminar Speaker: Blake Boudreaux (Western) "Weighted Bergman Kernels on Domains in $\mathbb{C}^n$" Time: 10:30 Room: MC 108 Given a domain $\Omega\subseteq\mathbb{C}^n$, the space of square-integrable holomorphic functions on $\Omega$ is a Hilbert space with the standard inner product. This space is denoted by $L^2_h(\Omega)$ and is known as the Bergman space of $\Omega$. It can be shown that the evaluation functionals $E_z:L^2_h(\Omega)\to\mathbb{C}$ given by $E_z(f)=f(z)$ are continuous on $L^2_h(\Omega)$, and hence via the Riesz representation theorem there exists a $K(\,\cdot\,,z)\in L^2_h(\Omega)$ that reproduces square-integrable holomorphic functions on $\Omega$. This function (on $\Omega\times\Omega$) is known as the Bergman kernel of $\Omega$, and has had a profound impact on the theory of holomorphic functions of several complex variables. This theory can also be generalized to weighted $L^2$-spaces, given that the weight function is sufficiently "nice".
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This will be a mostly expository talk on Bergman kernel, with an emphasis on weighted Bergman kernels. Time allowing I will sketch some work I have done regarding the zeroes of weighted Bergman kernels. |
8 Algebra Seminar
Algebra Seminar Speaker: Felix Baril Boudreau (Western) "Computing an L-function modulo a prime" Time: 14:30 Room: 968 6609 0477 Let $E$ be an elliptic curve with non-constant $j$-invariant over a function field $K$ with constant field of size an odd prime power $q$. Its $L$-function $L(T,E/K)$ belongs to $1 + T\mathbb{Z}[T]$. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we propose an approach to compute $L(T,E/K)$. The idea is to compute, for sufficiently many primes $\ell$ invertible in $K$, the reduction $L(T,E/K) \bmod{\ell}$. The $L$-function is then recovered via the Chinese remainder theorem. When $E(K)$ has a subgroup of order $N \geq 2$ coprime with $q$, Chris Hall showed how to explicitly calculate $L(T,E/K) \bmod{N}$. We present novel theorems going beyond Hall's. |
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12 Equivariant Topology Seminar
Equivariant Topology Seminar Speaker: Kumar Sannidhya Shukla "The Weil model for the equivariant de Rham cohomology of $G$-manifolds" Time: 09:30 Room: online Last week, equivariant cohomology for $G$-spaces was defined. In this talk, we will define equivariant cohomology in the setting of smooth manifolds under Lie group action. We will first construct an algebraic model for the universal $G$-bundle (called the Weil algebra). Using this we shall define an equivariant version of de Rham complex. Lastly, we will work out the Weil algebra for circle actions. Meeting ID: 997 4840 9440
Passcode: 911104 |
13 Analysis Seminar
Analysis Seminar Speaker: Aftab Patel (Western) "Departmental PhD Exam" Time: 15:30 Room: MC 108 TBA |
14 GAP Seminar
GAP Seminar Speaker: Blake J. Boudreaux (Western) "Weighted Bergman Kernels on Domains in $\mathbb{C}^n$: Part 2" Time: 10:30 Room: MC 108 Given a domain $\Omega\subseteq\mathbb{C}^n$, the space of square-integrable holomorphic functions on $\Omega$ is a Hilbert space with the standard inner product. This space is denoted by $L^2_h(\Omega)$ and is known as the Bergman space of $\Omega$. It can be shown that the evaluation functionals $E_z:L^2_h(\Omega)\to\mathbb{C}$ given by $E_z(f)=f(z)$ are continuous on $L^2_h(\Omega)$, and hence via the Riesz representation theorem there exists a $K(\,\cdot\,,z)\in L^2_h(\Omega)$ that reproduces square-integrable holomorphic functions on $\Omega$. This function (on $\Omega\times\Omega$) is known as the Bergman kernel of $\Omega$, and has had a profound impact on the theory of holomorphic functions of several complex variables. This theory can also be generalized to weighted $L^2$-spaces, given that the weight function is sufficiently "nice".
$$
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This will be a mostly expository talk on Bergman kernel, with an emphasis on weighted Bergman kernels. Time allowing I will sketch some work I have done regarding the zeroes of weighted Bergman kernels. |
15 Algebra Seminar
Algebra Seminar Speaker: Senate meeting - no Algebra Seminar "No talk" Time: 14:30 Room: |
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19 Equivariant Topology Seminar
Equivariant Topology Seminar Speaker: Kumar Sannidhya Shukla "The Weil model for the equivariant de Rham cohomology of $G$-manifolds, II" Time: 09:30 Room: online We will revisit two important operations on differential forms,
namely, contraction and Lie derivative. In particular, if M is a $G$-manifold, then contraction and Lie derivative (both with respect to the fundamental vector fields), make the de Rham complex $\Omega(M)$ a $\mathfrak{g}$-DGA. Next, we will return to the Weil model for circle actions and show that it is isomorphic to the polynomial ring with circle-invariant forms on $M$ as the coefficients. Meeting ID: 997 4840 9440
Passcode: 911104 |
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21 GAP Seminar
GAP Seminar Speaker: Avi Steiner (Western) "D-Modules for Analysts" Time: 10:30 Room: MC 108 The main idea of D-modules (that is, of algebraic analysis) is to re-interpret systems of linear PDEs as modules over a ring of differential operators. You then prove things about these modules using the tools of algebra and algebraic geometry, and then translate the results back into PDE language. An archetypal example of such a result is the Cauchy-Kovalevskaya-Kashiwara theorem, which is a vast generalization of the Cauchy-Kovalevskaya theorem from the classical study of PDEs. I will give a sketch of the ideas that are needed to digest this theorem. Along the way, you'll be introduced to D-modules (of course!) and their characteristic varieties. Colloquium
Colloquium Speaker: Matteo Smerlak (Max Planck Institute for Mathematics in the Sciences, Leipzig) "Evolutionary landscapes and retrospective processes" Time: 15:30 Room: MC 108 Many systems across the sciences evolve through the interaction of multiplicative growth and diffusive transport. In the presence of disorder, these opposing forces can generate localized structures and bursty dynamics, a phenomenon known as "intermittency" in non-equilibrium physics and as "punctuated equilibrium" in evolutionary theory. This behaviour is difficult to forecast; in particular there is no general principle to locate the regions where the system will settle, how long it will stay there, or where it will jump next. In this talk I will introduce a Markovian representation of growth-transport dynamics that closes these gaps. This retrospective view of evolution unifies the concepts of linear intermittency and metastability, and provides a generally applicable method to reduce, and predict, the dynamics of disordered linear systems. Applications range from Zeld'dovich's parabolic Anderson model to Eigen's quasispecies model of molecular evolution. |
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26 Random Matrix Theory Seminar
Random Matrix Theory Seminar Speaker: Masoud Khalkhali (Western) "Topological expansion for matrix integrals III" Time: 14:30 Room: MC 106 Feynman rules, 1-particle irreducible graphs, effective action and Legendre transform. |
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28 Transformation Groups Seminar
Transformation Groups Seminar Speaker: Rafael Gomes (Western) "Equivariant cohomology by example" Time: 09:30 Room: WSC 184 & online This is a follow-up talk from my initial talk, where we introduced the definition and some properties of the Borel construction and the equivariant cohomology of $G$-spaces.
In this talk, we aim to highlight several interesting properties of the equivariant cohomology of $G$-spaces through examples. It will also be discussed a couple of additional features for the case of a connected compact Lie group $G$. Meeting ID: 997 4840 9440
Passcode: 911104 GAP Seminar
GAP Seminar Speaker: Michael Francis (Western) "Groupoids and Algebras of Foliations: Part II" Time: 10:30 Room: MC 108 Last time, we defined (possibly singular) foliations to be certain collections of vector fields. It was emphasized that, defined this way, singular foliations are not uniquely determined their leaves. In this sequel talk, I will discuss a class of singular foliations I considered in my PhD thesis. These foliations have only two or three leaves total: a closed hypersurface (the singular leaf) and the components of its complement. Depending which vector fields gave the partition, however, interesting holonomy can result along the singular leaf. It turns out this holonomy can be used to completely classify such foliations (localized around the singular leaf). If time permits, I will talk about a question I am currently thinking about: under a suitable orientation hypothesis, is the "fundamental class" of these foliations always nontrivial? The answer to this question hinges on a rather concrete index calculation. |
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1 Geometry and Combinatorics
Geometry and Combinatorics Speaker: Johannes Hofscheier (University of Nottingham) "Experimenting with polytopes: exploring the landscape of lattice polytopes" Time: 13:30 Room: Zoom Algebraic geometry and combinatorics have a long-established culture of producing and interrogating classification datasets. These datasets can be at the limit of current computing resources, e.g., the Kreuzer-Skarke classification has almost half-a-billion entries. The time is ripe to explore the landscape of algebra-geometric datasets by means of data science techniques. In this talk, I will report on initial joint work with Bao, He, Hirst, Kasprzyk, and Majumder investigating the effectiveness of machine learning (ML) methods for predicting properties of Hilbert series and lattice polytopes. Our results support the idea that in many cases a mathematical theorem underlies ML predictions of high accuracy. Furthermore, our observations show that ML can also provide hints to new exciting and unexpected relations. |
2 Random Matrix Theory Seminar
Random Matrix Theory Seminar Speaker: Masoud Khalkhali (Western) "Topological expansion for matrix integrals IV" Time: 14:30 Room: MC 106 Feynman rules, 1-particle irreducible graphs, effective action and Legendre transform. |
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5 Algebra Seminar
Algebra Seminar Speaker: Khanh Nguyen Duc (Otto-von-Guericke-University Magdeburg) "The relations between Littlewood-Richardson coefficients and its shifted version" Time: 14:30 Room: ZOOM (968 6609 0477) We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$. More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and $g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that $g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid. We present an approach using Fomin diagrams and Viennot's geometric construction for RSK correspondence to attack the conjecture. |
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Mathematics Calendar 
October 2021
