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Title: Turning lemmings into hogs: the Yoccoz-Birkeland (livestock) population model coupled with (random) price dynamics
Speaker: Stefano Marmi [Scuola Normale Superiore di Pisa]
Abstract: Around 1996 Yoccoz and Birkeland introduced an endogenous lagged integral equation to explain the
approximately periodic but chaotic lemming population fluctuations in the Arctic ecosystem.
The evidence of cyclical but unpredictable behavior of fluctuations is also a well-known feature of
livestock product prices (hog cycle) that has attracted the attention of economists for a long time.
I will discuss deterministic and random versions of the population-market model proposed
by Arlot, Marmi, and Papini in Arlot et al. (2019).
The model is obtained by coupling the Yoccoz–Birkeland integral equation
with a demand- and supply-dependent price dynamics as in Bélair and Mackey (1989).
In the random model, we introduce a stochastic component into the price equation
inspired by the Black-Scholes market model and prove the existence of a random attractor
and a random invariant measure.
We numerically compute the fractal dimension and the entropy of the
random attractor and prove its convergence to the deterministic one
when the volatility of the market equation tends to zero.
We also numerically analyze in detail the dependence of the attractor on the choice of the
temporal discretization parameter. We implement several statistical distances
to quantify the similarity between the attractors of the discretized systems and the original ones.
In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This distance is a discrete and penalized version
of the optimal transport distance between two measures, given a transportation cost matrix.
The work on the random dynamics and the investigation of the dependence on the discretization is joint with R. Ceccon and G. Livieri.
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Title: On free boundary problems
Speaker: Daniela De Silva [Columbia University]
Abstract: In this talk, we will present an overview of techniques and results concerning the regularity theory for Free Boundary Problems (FBP); that is, problems in which one must solve a PDE and along the way find out the region in which the PDE holds. FBPs naturally arise in a variety of applications and research in this area has been very fruitful and active for several decades. Using the Bernoulli one-phase problem as a basic elliptic model, we will highlight main contributions and open questions often originating from a striking resemblance with the regularity theory for minimal surfaces. We will further consider parabolic problems, including the classical Stefan problem. If time permits it, we will describe so-called thin free boundary problems, in which the free boundary occurs on a lower dimensional subspace, and that arise in connection with non-local phenomena.
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Title: Applied Group Theory in the Quantum and Artificial Intelligence Era
Speaker: Delaram Kahrobaei [City University of New York]
Abstract: In this talk I present an overview of the current state-of-the-art in post-quantum group-based cryptography. I describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. I then describe some applications of combinatorial algebra in fully homomorphic encryption, and in particular homomorphic machine learning. In the end I will discuss several open problems in this direction. (https://arxiv.org/abs/2202.05917 to appear in the Notices of AMS).
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For the full schedule of talks please visit: https://scgp.stonybrook.edu/archives/35037
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Title: 6D SCFTs, Long Quivers, and (Super-)Spin Chains
Abstract:
Nearly all 6D N=(1,0) SCFTs can be described in terms of one-dimensional spines of quiver nodes linked by conformal matter, a strongly-coupled generalisation of bifundamental hypermultiplets. These theories have a large-R-charge sector of operators for which anomalous dimensions are at leading order governed by an integrable open Heisenberg spin chain. Via superconformal symmetry, this can in turn be extended to an osp(2,6|1) super-spin chain, and 4D N=2 avatars can be obtained through compactification. I will review the construction of those SCFTs and how to build the large-charge operators from their conformal matter building blocks, their interpretation as a spin chain, and how to determine the integrable Hamiltonian capturing the operator mixing.
Title: Electronic fluidics and vortices
Title: No seminar: Fall Break
Abstract:
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Title: Graded Characters, Quantum Q-systems, and spherical DAHA
Abstract
The tensor products of finite-dimensional representations of affine algebras has a natural grading, e.g. as defined by Feigin and Loktev, reflecting the low lying physical spectrum of generalized Heisenberg spin chains. The combinatorial structure of the graded characters of these products, computed from the Bethe ansatz, is governed by the Q-systems, equations satisfied by characters of KR-modules. They are mutations in a cluster algebra, whose quantization is responsible for the grading. The quantum Q-system is an integrable discrete dynamical system for the non-commutative cluster variables, which generate the spherical DAHA in the q-Whittaker limit. The commuting Hamiltonians are quantizations of quantum Toda Hamiltonians, obtained from the Macdonald operators via duality in the q-Whittaker limit. The graded characters are generalized q-Whittaker functions and satisfy q-difference equations given by the generalized Toda Hamiltonians.
Title: A twist on holography
Title: Sections of families over CP1
Speaker: Alex Pieloch [Columbia University]
Abstract: We will discuss the existence of rational (multi)sections and unirulings for projective families over CP1 with at most two singular fibres. We obtain these rational curves by using techniques from symplectic geometry. In this talk, we will focus on (1) discussing the motivation for this work from Hodge theory and the higher dimensional Shafarevich problem and (2) presenting the geometric constructions/ideas involved in our proofs. No knowledge on symplectic geometry is required.
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Title: Sliceness and the rank of some knot homologies
Speaker: Marco Marengon [Alfréd Rényi Institute of Mathematics]
Abstract: A popular notion in knot theory is that of “sliceness”: a knot in S ^3 is called slice if it bounds a smooth disc in B^4. There are various reasons why this concept is so fundamental: for example, sliceness is at the core of a trendy strategy proposed to disprove the smooth 4-dimensional Poincaré conjecture, and it has recently been shown that a generalisation of this concept to 4-manifolds other than B^4 can detect exotic pairs, i.e. 4-manifolds that are homeomorphic but not diffeomorphic to each other.
We study whether sliceness poses any restrictions on the rank of certain homology theories associated to knots. We prove some results and formulate some curious conjectures. This is joint work with Hockenhull and Willis, and partly also with Dunfield and Gong.
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Title: From Koornwinder operators to cluster algebra: Proof of the Macdonald-Q-system conjecture
Abstract
We present various constructions of commuting difference operators for the theory of Koornwinder polynomials. We show how a specialization/limiting procedure produces a functional representation for quantum Q-system cluster algebras associated to affine and twisted types A,B,C,D, also interpreted as discrete algebraic quantum integrable systems. The correspondence uses Koornwinder duality and a suitable Fourier-Whittaker transform allowing to interpret Koornwinder polynomial Pieri rules as relativistic Toda systems. (Based on joint work with Rinat Kedem).
Title: Weathering the storm
Speaker: Stony Brook faculty [Stony Brook University]
Abstract: Professors Evita Nestoridi, Christian Schnell, Raanan Schul, and Jason Starr will briefly discuss their own career trajectories with an emphasis on "weathering the storm", meaning coping with problems big and small. Examples include dealing with mistakes, being stuck, handling difficult collaborators, discovering that a problem is much harder than previously thought, etc. Most of the time will, we hope, be devoted to answering questions from the audience.
(Please note the change in time: Friday, 12pm to 1pm)
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Title: TBA
Speaker: Hongming Nie [Stony Brook University]
Abstract: TBA
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