Title: Intersections of Local Cohomology and Hodge Theory
Speaker: Scott Hiatt [Purdue University]
Abstract: Introduced by Grothendieck, Local Cohomology has become a valuable tool in commutative algebra, and has many applications in Algebraic Geometry. In this talk, we will use the theories of Local Cohomology with Hodge Theory to study Hodge structures and differential forms on singular and quasi-projective varieties.
View Details
Speakers: Tere Seara, Pau Martin and Rafael Ramirez-Ros (Universitata Politecnica de Catalunya, Spain)
Title: Exponentially small phenomena in analytic convex billiards
Title: TBA
Speaker: Abhishek Mallick [Rutgers University]
Abstract:
View Details
Last day for Fall/Winter degree candidates to apply for graduation via SOLAR. Applications submitted after 11/9/22 will not appear in the commencement publication.
Speaker: Marco Lenci (Università degli Studi di Bologna, Italy)
Title: Exactness and K-mixing for extensions of dynamical systems, with application to periodic Lorentz gases
Abstract: Exactness and the K-property (a.k.a. K-mixing) are strong ergodic properties whose significance is that a dynamical system progressively loses all information about its initial conditions. These properties were the subject of intense investigation in the second half of last century before being somewhat superseded by the stronger Bernoulli property, which was at the time proved for a number of popular systems, including many billiards. But the Bernoulli property can only be formulated for dynamical systems preserving a finite measure, while exactness and the K-property work equally well in infinite-measure contexts, which partly explains a revived interest in them.
I will present some recent general theorems on the exact or K decomposition of extensions of exact or K dynamical systems. One immediate application is that, in large generality, a recurrent periodic Lorentz gas whose finite-measure factor is proved to be hyperbolic is K-mixing. This includes virtually all previously studied 2-dimensional periodic Lorentz gases and d-dimensional periodic Lorentz tubes, as well as many d-dimensional periodic Lorentz slabs.
Joint work with Daniele Galli.
Title: Real bimodal quadratic rational maps: moduli space and entropy (with K. Filom and S. Kang)
Speaker: Kevin Pilgrim [Indiana University Bloomington]
Abstract: Bruin-van Strien and Kozlovski showed that for multimodal
self-maps $f$ of the unit interval, the function $f \mapsto h(f)$
sending $f$ to its topological entropy is monotone. K. Filom and I
showed that for interval maps arising from real bimodal quadratic
rational maps, this monotonicity fails. A key ingredient in our proof
is an analysis of a family $f_{p/q}, p/q \in \mathbb{Q}/\mathbb{Z}$ of
critically finite maps on which the dynamics on the postcritical set
is conjugate to the rotation $x \mapsto x+p b \mod q$ on
$\mathbb{R}/\mathbb{Z}$, where $x=0$ and $x=1$ correspond to the two
critical points. The recent PhD thesis of S. Kang constructs a
piecewise-linear (PL) copy of the well-known Farey tree whose vertices are expanding PL quotients of the $f_{p/q}$'s. This PL model, conjecturally, sheds light on the moduli space of the real quadratic bimodal family, and on the variation of entropy among such maps.
View Details
Title: Invariants of Real Vector Bundles
Speaker: Jiahao Hu [Stony Brook University]
Abstract: For a compact smooth manifold with corners (or finite CW-complex) X, we can prescribe a finite set of spin or $spin^h$ manifolds (possibly with boundary) mapping into it so that every real vector bundle over X is determined, up to stable equivalence, by the Dirac indices of the real vector bundle when pulled-back onto those prescribed spin or $spin^h$ manifolds. Our proof features a thorough study of indices of Dirac operators on $spin^h$ manifolds and a general duality between cycles and cocycles. Dissertation Advisor: Dennis Sullivan
View Details
Title: TBD
Speaker: Tarek Elgindi [Duke University]
Abstract: Please note the special day and time
View Details
Title: TBA
Speaker: Chris Bishop [Stony Brook University]
Abstract:
View Details
Speaker: Alexey Glutsuyk (CNRS & ENS Lyon)
Title: On algebraically integrable planar, dual and projective billiards
Title: On the Formation of Coherent Structures in Ideal Fluids
Abstract: We will discuss recent ideas and results related to the formation and dynamics of coherent structures in ideal fluids. The study of such structures has had a profound impact on many fundamental problems in the study of the incompressible Euler equation. As a first example, we will discuss self-similar solutions; these are three-dimensional flows that exactly self-enhance all the way to a finite-time singularity. Self-similar solutions have been shown to attract nearby solutions to become singular as well. We will also discuss coherent structures in two dimensions, which play an important role in the infinite-time relaxation of solutions despite the conservative nature of the equations.
Mini-course by Gregory Falkovich
September 26 - November 28, 2023 on Tuesdays at 2:30pm in room 313
Title: Introduction to Information Theory
Selected Subjects:
Measuring Uncertainty
Mutual Information and Entanglement Entropy, Redundancy of Language and Genetic Code
Information is Life and Money
Theory of Mind – Active Inference
Information in Quantum Mechanics and Statistics
Title: ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow
Speaker: Paula Burkhardt-Guim [NYU]
Abstract: We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.
View Details
Title: TBA
Speaker: Insung Park [Stony Brook University]
Abstract:
View Details
Title: Complexes of stable birational invariants
Speaker: James Hotchkiss [Columbia University]
Abstract: In the study of the rationality problem, a common situation is that one has a smooth variety degenerating to a singular one, and one wishes to find computable invariants of the special fiber which obstruct the rationality of the general fiber. I will explain a general procedure for constructing such invariants. As an application, A^1-connectedness specializes in smooth families. Joint with David Stapleton.
View Details
Speaker: Alexey Glutsuyk (CNRS & ENS Lyon)
Title: On algebraically integrable planar, dual and projective billiards
Title: TBA
Speaker: Vukasin Stojisavljevic [CRM, Universite de Montreal]
Abstract:
View Details