Abstract motivic homotopy theory
Motivic homotopy theory is a fusion of homotopy theory and algebraic
Grothendieck ring of varieties and motivic measures
This will be mostly a survey talk of the subject. I will discuss some interesting measures that can be used to study the Grothendieck ring of varieties. Applications to rationality of the motivic zeta function will be explained.
Isocrystals associated to Arithmetic Jet Spaces
The aim of the talk is to discuss the construction of an isocrystal associated to abelian schemes over p-adic fields using the arithmetic jet space theory. Isocrystals are objects in p-adic Hodge theory that lead to Galois representations. A well-known example of isocrystals coming from geometry are the ones obtained from the de Rham cohomology of the scheme. We will also talk about the interaction of our object with the first deRham cohomology of the abelian scheme. This is a joint work with Jim Borger.
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Twisted Blanchfield pairings and Casson-Gordon invariants
In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted signature functions for knots, and describe their relation to the Casson-Gordon invariants. If time permits, we will describe how this can be used to obstruct the sliceness of some algebraic knots. This is joint work with Maciej Borodzik and Wojciech Politarczyk.
What is 3-dimensional hyperbolic geometry?
3-dimensional hyperbolic geometry is a classical subject.
We will give an introduction emphasizing concrete questions,
computations and answers, as well as hints to quantum aspects
of this beautiful theory. Please bring concrete questions and you may be
lucky and get some answers.
Demazure flags: A combinatorial journey
In this talk, we will briefly review the theory of characters of Demazure modules which are modules for the standard maximal parabolic subalgebra of an affine Lie algebra. The discussion will be followed by some connections that I have discovered(with collaborators) in my own research between algebraic combinatorics and other areas of mathematics such as representation theory and number theory. For instance, we show that the graded multiplicities of higher level Demazure modules in Demazure flags can be expressed in terms Dyck paths. I will describe some results and further questions in this direction.
Descartes, Newton and hyperfields
Descartes bounds the number of positive roots of a polynomial with real coefficients by the number of sign changes in the coefficients, Newton bounds the number of roots of a polynomial over a nonarchimedean field in terms of its Newton polygon.
In this talk, we review these two clasical theorems and explain how they follow by the same principle from assertions about roots of polynomials over the hyperfield of signs and the tropical hyperfield, respectively. This is joint work with Matthew Baker.
Talks in the Seminar on Topological insulators
Additive combinatorial problems of Number Theory
We give a survey on problems from Number Theory that can be solved or were solved by the methods of Additive
Combinatorics such as Green--Tao theorem, Ostmann conjecture, the distribution of Fermat quotients, exponential sums over multiplicative subgroups, combinatorial problems with continued fractions and so on.
Mixed derived category on affine Grassmannians and coherent sheaves on Springer resolution
Computing p-canonical bases
We discuss a diagrammatic presentation of the Hecke category and its cellular structure. Using this, we will compute decomposition numbers in Hecke categories and hence determine some tilting characters.
Topological Hopf algebra of Feynman graphons and its application to Quantum Field Theory, III
This mini-course is on the basis of the speaker's monograph "a mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory" (The MPIM-preprint series 2018-65). I plan to explain the construction of a new Hopf algebraic formalism on graphons derived from the Kreimer's renormalization coproduct. Then I will show the application of this new mathematical object in dealing with non-perturbative parameters.
Topological Hopf algebra of Feynman graphons and its application to Quantum Field Theory, II
This mini-course is on the basis of the speaker's monograph "a mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory" (The MPIM-preprint series 2018-65). I plan to explain the construction of a new Hopf algebraic formalism on graphons derived from the Kreimer's renormalization coproduct. Then I will show the application of this new mathematical object in dealing with non-perturbative parameters.
Topological Hopf algebra of Feynman graphons and its application to Quantum Field Theory, I
This mini-course is on the basis of the speaker's monograph "a mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory" (The MPIM-preprint series 2018-65). I plan to explain the construction of a new Hopf algebraic formalism on graphons derived from the Kreimer's renormalization coproduct. Then I will show the application of this new mathematical object in dealing with non-perturbative parameters.
New guests at the MPIM
Preprojective cohomological Hall algebras and Yangians
Khovanov homotopy types
I'll begin by discussing the construction of the Khovanov
homology groups from a link. I'll then discuss a framework that allows
the construction of similar objects in many contexts, in particular
recovering Lipshitz and Sarkar's refinement to an object in stable
homotopy whose homology groups recover Khovanov homology.
Elliptic analogues of cyclotomic multiple zeta values
We will report on ongoing joint work with Martin Gonzalez (MPIM) about analogues of cyclotomic multiple zeta values for complex elliptic curves with level structure. These are holomorphic functions on the upper half plane which degenerate to cyclotomic multiple zeta values at cusps. One of the motivations for their study is the existence of several modular phenomena for cyclotomic multiple zeta values. For example, in the simplest case when the level structure is trivial, these elliptic analogues give a new perspective on relations between double zeta values coming from period polynomials of modular forms which were originally found by Gangl--Kaneko--Zagier.
An algebraic characterization of the Kronecker function
The Kronecker function is a kind of generalization of the classical Eisenstein series which lies at the heart of Kronecker's theory of elliptic functions. Nowadays the Kronecker function is also known to play important roles in the geometry of configuration spaces of elliptic curves as well as periods of modular forms for PSL2(Z). In both of these contexts a key role is played by a certain three-term functional equation satisfied by the Kronecker function, the Fay identity. The goal of this talk is to explain the significance of this identity and how the Kronecker function is (up to mild ambiguity) uniquely determined by it.
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