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For more information and registration: https://www.pims.math.ca/seminars/PIMSPDF.

ABSTRACT:
In this talk, I will describe how certain non-linear exponential tilts of the Discrete Gaussian Free Field yield invariant measures for a discretized PDE, the focusing Non Linear Schrödinger equation. I will briefly talk about how this tilted measure relates to self-intersections of the random walk on $\mathbb{Z}^{d}$. I will then describe joint work with Partha Dey and Kay Kirkpatrick on how this measure undergoes a phase transition in the thermodynamic limit with general $d\geq 3$ and non-linearity $p$. I will conclude with some new results with Gourab Ray, on the weak convergence of the random field sampled with respect to this measure to the massive discrete Gaussian Free Field.

SPEAKER BIO:
Kesav is a PIMS postdoctoral fellow at the University of Victoria who works with Gourab Ray. Kesav’s academic trajectory began as an undergraduate student in physics at the University of Delhi. After realizing that mathematical methods (specifically probability) excited him more, Kesav switched track and completed his masters in mathematics at the Indian Statistical Institute, Bangalore. As a PhD student at the University of Illinois, Urbana-Champaign, Kesav focused on problems in probability that have origins in statistical and mathematical physics. He continues to work in this field, specifically on phase transitions in lattice field theories and disordered models in statistical mechanics. He is thrilled to have gotten the opportunity to spend two years in the Pacific Northwest and explore its spectacular natural beauty.

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Abstract:
We will introduce the research area of complex dynamics by studying the family of (complex) quadratic polynomials, $z->z^2+c$. Historically, mathematicians first studied the dynamics of {\em real} quadratic polynomials and then moved to the complex realm. Recently, there has been a resurgence of mathematical activity surrounding real quadratic polynomials. We will explore some well known features and some rather new features of real polynomials in contrast with those enjoyed by their complex counterparts.

Speaker Biography:
Sarah Koch is a an Arthur F. Thurnau Professor in the Department of Mathematics at the University of Michigan. Her research incorporates topology, algebraic geometry, complex analysis, and Teichmueller theory to better understand complex dynamical systems (in one and several variables) and their associated dynamical moduli spaces from both analytic and algebraic points of view.

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Description: We consider metric tensors on undirected weighted graphs G, which allows us to treat P(G), the set of probability vectors on G, as a length space. On defines a divergence operator div_\mu(G) for mu in P(G), in such a way that we can use control vectors m to define paths s:[0,T] \to P(G), satisfying the system of ODEs: d\sigma/dt + div_G(m) + \hbar div_\sigma(\nabla_G log \sigma)=0. These paths serve as characteristics for Hamilton-Jacobi equations involving graph-individual noise operators. We propose a well posedness theory on P(G). (This talk is based on a joint work with C. Mou and A. Swiech)

Other Information: Time: All network wide colloquia take place at 1:30pm Pacific Time with a few exceptions.

Registration:
Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.