There are no events at the Simons Center today. Here are the events for this week

Tuesday, January 21st, 2020

Gauged linear sigma model and gauged Witten equation

Time: 1:00 PM - 2:30 PM

Location: 313

**Abstract:**(Review of symplectic vortex equation) In this talk, I will explain the basic theory of the symplectic vortex equation as the first part of the preparation for the symplectic geometry construction of the gauged linear sigma model (GLSM). If V is a symplectic manifold with a Hamiltonian group action, then analogous to the holomorphic curve equation, one has a first order elliptic system over a Riemann surface. Depending on different choices of domain metrics, one can (in principle) either define Gromov--Witten type invariants on the equivariant cohomology or the cohomology of the quotient space.
Wednesday, January 22nd, 2020

Neural Networks Program Seminar: Luca Mazzucato

Time: 11:00 AM - 12:00 PM

Location:

I will discuss recent experimental and theoretical findings on metastable activity; its potential role for representing internal states as well as relevant task variables; and how it may arise in biologically realistic models. We will introduce the basic concepts in the theory of attractor networks of spiking neurons as a model of metastable activity.

I will show how to simulate recurrent networks of spiking neurons with clustered connectivity generating attractor dynamics. I will provide a demo for students to play with and discover how these networks respond to stimuli and other external perturbations. We will introduce hidden Markov models (HMM) as a tool to analyze attractor dynamics in spiking networks.

This tutorial is based on code freely available at https://github.com/mazzulab/contamineuro_2019_spiking_net .

**Title:**Tutorial on modeling and simulating neural population dynamics**Abstract:**table brain dynamics are characterized by abrupt, jump-like modulations so that the neural activity in single trials appears to unfold as a sequence of discrete, quasi-stationary ‘states’. Evidence that cortical neural activity unfolds as a sequence of metastable states is accumulating at fast pace. Metastable activity occurs both in response to an external stimulus and during ongoing, self-initiated activity. Metastable states are increasingly found to support internal representations that are not locked to external triggers, including states of deliberations, attention and expectation. Decoding stimuli or decisions via metastable states can be carried out trial-by-trial, shifting our perspective from traditional concepts based on trial-averaging to models based on dynamic ensemble representations.I will discuss recent experimental and theoretical findings on metastable activity; its potential role for representing internal states as well as relevant task variables; and how it may arise in biologically realistic models. We will introduce the basic concepts in the theory of attractor networks of spiking neurons as a model of metastable activity.

I will show how to simulate recurrent networks of spiking neurons with clustered connectivity generating attractor dynamics. I will provide a demo for students to play with and discover how these networks respond to stimuli and other external perturbations. We will introduce hidden Markov models (HMM) as a tool to analyze attractor dynamics in spiking networks.

This tutorial is based on code freely available at https://github.com/mazzulab/

Gauged linear sigma model and gauged Witten equation

Time: 1:00 PM - 2:30 PM

Location: 313

**Abstract:**(Review of FJRW theory) In this talk, I will recall the detailed construction of FJRW theory (following Fan-Jarvis-Ruan) as the second part of the preparation for the symplectic geometry construction of the gauged linear sigma model (GLSM). The FJRW theory, also called the orbifold Landau--Ginzburg A-model theory, is a cohomological field theory associated to a pair $(W, G)$, where $W$ is a nondegenerate quasihomogeneous polynomial and $G$ is a symmetry group. The correlation functions are defined by counts of solutions to the Witten equation over higher spin curves in a way analogous to the construction of Gromov-Witten invariants. I will explain how to set up the Witten equation, how to properly perturb it, and if time permits, how to construct the virtual fundamental cycle and prove axioms of FJRW invariants.
Thursday, January 23rd, 2020

Gauged linear sigma model and gauged Witten equation

Time: 1:00 PM - 2:30 PM

Location: 313

**Abstract:**(A mathematical theory of gauged linear sigma model in geometric phase) In this talk, I will overview the symplectic geometric construction of Witten’s gauged linear sigma model (in a geometric phase). The construction is based on the analysis of the gauged Witten equation, which is a combination of the Witten equation as in FJRW theory and the vortex equation in gauged Gromov-Witten theory. The upshot is that the counts of solutions to the gauged Witten equation defines a cohomological field theory on the cohomology of the classical vacuum (e.g. the quintic threefold). If time permits, I will briefly explain how to prove the relation between GLSM correlation functions and Gromov--Witten invariants. This is a joint work with Gang Tian.
Friday, January 24th, 2020

Gauged linear sigma model and gauged Witten equation

Time: 1:00 PM - 2:30 PM

Location: 313

**Abstract:**(Adiabatic limit of gauged Witten equation and mirror map) In this talk, I will review the adiabatic limit analysis of both the vortex equation and the gauged Witten equation. If one rescales the metric on the domain curves by a large constant, then as proved by Gaio-Salamon, vortices should approximate holomorphic curves in the symplectic quotient, modulo bubbling of “point-like instantons.” This picture leads to an isomorphism between the gauged Gromov-Witten theory and the usual Gromov-Witten theory of the quotient up to a change of variable (the quantum Kirwan map). I will show you how to extend this picture to the gauged Witten equation and how to obtain the precise relation between the GLSM CohFT and the Gromov-Witten CohFT of the classical vacuum. This is a joint work in progress with Gang Tian.