Talks hosted at KIAS
21.11.2018 17:20--18:20 at 1424
Using category theory for non-commutative probability
Accessible to graduate students
Prof. Arthur Parzygnat from University of Connecticut in Storrs CT, USA
Classical conditional probabilities and disintegrations can be formulated diagrammatically in a category of stochastic matrices. Combining this with the functor taking stochastic matrices to positive operators on C*-algebras, a notion of non-commutative disintegration and conditional probability can be made for states on C*-algebras. The existence and uniqueness of conditional probabilities is an important fact in probability theory. However, unlike the classical case, the existence of non-commutative disintegrations is not guaranteed even on finite-dimensional matrix algebras. We will state some general existence and uniqueness results as well as illuminating examples. This is joint work with Benjamin P. Russo (Farmingdale State College SUNY).
21.11.2018 16:00--17:00 at 1424
Probability monads
Accessible to graduate students
Prof. Arthur Parzygnat from University of Connecticut in Storrs CT, USA
What does it mean for a theory to admit a probabilistic analogue? We will explore the Giry/Radon monads on categories of measurable/topological spaces and measurable/continuous maps. Then, we will construct the associated Kleisli category of the monad. This category defines a probabilistic analogue of the original category. No knowledge of probability theory, monads, and Kleisli categories will be assumed in this talk.
28.08.2018 16:00--17:00 and 17:20-- 18:20 at 8101
Equivariant connections and quotient stacks I, II Accessible to graduate students
Prof. Corbett Redden from Long Island University C. W. Post Campus in Brookville NY, USA
Suppose a (compact) Lie group G acts on a smooth manifold M. In these two talks, I will explain how the language of stacks (on the site of smooth manifolds) provides a natural framework for encoding and classifying certain G-equivariant structures on M. In particular, there is a natural refinement of the equivariant Chern-Weil homomorphism that lives in an equivariant extension of Deligne cohomology. Furthermore, in joint work with Byungdo Park (KIAS) we prove that equivariant gerbe connections are classified by equivariant Deligne cohomology in degree 3.
13.07.2018 13:00--14:00 at 1424
Immersed crosscap number and Z_2 Thurston norm
Dr. Seungwon Kim from National Institute for Mathematical Sciences in Daejeon, Korea
In this talk, we will talk about 3-dimensional immersed crosscap number of a knot, which is a non-orientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial (2p,q)-torus or (2p, q)-cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the 4-dimensional crosscap number. We then use these constructions to find an infinite family of 3-manifolds such that one of its second Z2 homology class can be represented by an immersed real projective plane but any embedded representative of it has a component with Euler characteristic strictly less than 1.
29.05.2018 10:30--11:30 at 8101
Fibrations of R^3 by oriented lines Accessible to graduate students
Dr. Michael Harrison from Lehigh University in Bethlehem PA, USA.
Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel? More precisely, does there exist a fibration of $\R^3$ by pairwise skew lines? We give some examples and provide a complete topological classification of such objects, by exhibiting a deformation retract from the space of skew fibrations of R^3 to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations. Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss when such objects induce contact structures.
04.05.2018 16:00--17:00 and 17:20--18:20 at 8101
Twisted Differential cohomology, the twisted Atiyah-Hirzebruch spectral sequence and applications I, II
Dr. Daniel Grady from New York University in Abu Dhabi, UAE.
This talk closely parallels the untwisted case. The first part of the talk will begin with motivation, continue with a discussion on topological twisted theories (loosely following Ando, Blumberg, Gepner [Hopkins,Rezk]), and end with a discussion on the differential refinement (following Bunke and Nikolaus). The second part of the talk will be on joint work with Hisham Sati in which we construct an AHSS for twisted differential cohomology theories. This portion will begin with a review of the classical twisted AHSS, continue with our construction in the differential case and end with examples and applications.
02.05.2018 13:00--14:00 and 14:20--15:20 at 8101
Differential cohomology, the Atiyah-Hirzebruch spectral sequence and applications I, II Accessible to graduate students
Dr. Daniel Grady from New York University in Abu Dhabi, UAE.
The aim of this talk is to provide an introduction to a geometric refinement of cohomology called differential cohomology and discuss some new constructions and applications for the theory. The first part of the talk will begin with motivation, continue with an introduction to ordinary differential cohomology, and end with generalized differential cohomology theories (closely following Bunke, Nikolaus, and Völkl). The second part of the talk will be on joint work with Hisham Sati in which we construct an Atiyah-Hirzebruch spectral sequence for differential cohomology theories. This portion will begin with a review of the classical AHSS, continue with our construction in the differential case and end with examples and applications.
06.04.2018 16:00 -- 17:00 at 8101
Augmented semi-regular biperiodic links in the thickened torus are hyperbolic Accessible to graduate students
Ms. Alice Kwon from CUNY Graduate Center in New York NY, USA
A hyperbolic ideal tetrahedron is determined by its three dihedral angles. In the 1990s, Andrew Casson and Igor Rivin discovered a technique for solving Thurston’s gluing equations using angle structures on an ideal triangulation for a hyperbolic 3-manifold. Using angle structures the gluing equations separate into a linear part and a non-linear part. The solutions to the linear system of equations form a convex polytope, and the solutions to the non-linear part is a critical point of a certain volume functional on this polytope. Abhijit Champanerkar, Ilya Kofman and Jessica Purcell used this idea to prove that biperiodic alternating links in the thickened torus with certain conditions are hyperbolic. I will be showing that semi-regular biperiodic alternating links with augmentation is hyperbolic.
13.12.2017 17:30 -- 18:30 at 8101
Algebraic K-theory of higher algebra
Dr. Mariko Ohara from University of Shinshu in Matsumoto-City, Japan
I explain a devissage theorem for, a variant of the algebraic K-theory of an E-infinity ring introduced by Elmendorf, Kriz, Mandell and May. They use the category of finite cell modules, I use the retract of finite cell modules. A main point is the correspondence of the two filtrations of subcategories in the model category of module spectra and of sub infinity subcategories in the infinity category of (higher) modules. I mention the calculation of the mapping space and the correspondence of the algebraic K-theories between the two filtrations.
13.12.2017 16:00 -- 17:00 at 8101
A brief review of the algebraic K-theory Accessible to graduate students
Dr. Mariko Ohara from University of Shinshu in Matsumoto-City, Japan
I talk about the background of the algebraic K-theory of an algebraic variety, from the Grothendieck group for a scheme to the higher algebraic K-theory space introduced by Quillen. I mention cycles and regulator maps, which play an important role of Tate conjecture.
01.12.2017 17:30 -- 18:30 at 8101
Field theories and orbifold cohomology
Dr. Augusto Stoffel from Max-Planck-Institut für Mathematik in Bonn, Germany
I will describe a dimensional reduction procedure for 1-dimensional supersymmetric Euclidean field theories over an orbifold. As a basic application, we obtain a geometric interpretation of the equivariant Chern character for finite group actions. I will also explain how twisted K-theory fits into this picture: a gerbe with connection gives rise to a 1-dimensional twist, and 0-dimensional field theories twisted by the dimensionally reduced twist represent classes in (twisted) complexified K-theory.
01.12.2017 16:00 -- 17:00 at 8101
Introduction to supermanifolds Accessible to graduate students
Dr. Augusto Stoffel from Max-Planck-Institut für Mathematik in Bonn, Germany
I will give an introduction to the theory of supermanifolds. This is a generalization of usual differential geometry allowing for local coordinates that anticommute with each other, and is used in the definition of supersymmetric (functorial) field theory.
24.11.2017 16:00 -- 17:00 at 1424
Link diagrams with low Turaev genus
Dr. Seungwon Kim from National Institute for Mathematical Sciences in Daejeon, Korea
In this talk, we will discuss classifications of link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating. If time remains, we will discuss recent progress based on this result and open problems.
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