Mon, 01 May 2023
14:15
L4

Morse theory on moduli spaces of pairs and the Bogomolov-Miyaoka-Yau inequality

Paul Feehan
(Rutgers University)
Abstract

We describe an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

Mon, 08 Mar 2021

15:45 - 16:45
Virtual

Conformal blocks for vertex operator algebras, sewing and factorization.

Bin Gui
(Rutgers University)
Abstract

In rational conformal field theory, the sewing and factorization properties are probably the most important properties that conformal blocks satisfy. For special examples such as Weiss-Zumino-Witten models and minimal models, these two properties were proved decades ago (assuming the genus is ≤1 for the sewing theorem). But for general (strongly) rational vertex operator algebras (VOAs), their proofs were finished only very recently. In this talk, I will first motivate the definition of conformal blocks and VOAs using Segal's picture of CFT. I will then explain the importance of Sewing and Factorization Theorem in the construction of full rational conformal field theory.

Thu, 21 May 2020

16:00 - 17:00

An Equilibrium Model of the Limit Order Book: a Mean-field Game approach

EunJung NOH
(Rutgers University)
Abstract

 

We study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. We will see that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers.
With a detailed analysis on the N-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller.
We then validate the dynamic programming principle (DPP) and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.
We argue that the value function can be used to obtain the equilibrium density function of the LOB. (Joint work with Jin Ma)

Thu, 18 Oct 2018

16:00 - 17:30
L4

Incomplete Equilibrium with a Stochastic Annuity

Kim Weston
(Rutgers University)
Abstract

In this talk, I will present an incomplete equilibrium model to determine the price of an annuity.  A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity.  The novelty of this model is its ability to handle running consumption and general income streams.  In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium.  The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs.  This work is joint with Gordan Zitkovic.

Mon, 13 Nov 2017
15:45
L6

A Lie group analog for the Monster Lie algebra

Lisa Carbone
(Rutgers University)
Abstract

The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of M. We associate the analog of a Lie group G(m) to the Monster Lie algebra m. We give generators for large free subgroups and we describe relations in G(m).

Fri, 07 Mar 2014

17:00 - 18:00
L3

Icosahedral clusters: the stem cell of the solid state?

Jean Taylor
(Rutgers University)
Abstract

Recent experimental work has determined the atomic structure of a quasicrystalline Cd-Yb alloy. It highlights the elegant role of polyhedra with icosahedral symmetry. Other work suggests that while chunks of periodic crystals and disordered glass predominate in the solid state, there are many hints of icosahedral clusters. This talk is based on a recent Mathematical Intelligencer article on quasicrystals with Marjorie Senechal.


The seminar will be followed by a drinks reception and forms part of a longer PDE and CoV related Workshop.


To register for the seminar and drinks reception go to http://doodle.com/acw6bbsp9dt5bcwb

Thu, 13 Dec 2012

12:00 - 13:00
Gibson 1st Floor SR

Two nonlinear wave equations with conformal invariance

Po Lam Yung
(Rutgers University)
Abstract

In this talk, we will look at two non-linear wave equations in 2+1 dimensions, whose elliptic parts exhibit conformal invariance.

These equations have their origins in prescribing the Gaussian and mean curvatures respectively, and the goal is to understand well-posedness, blow-up and bubbling for these equations.

This is a joint work with Sagun Chanillo.

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