Coherent sheaves in log geometry
I will present joint work with Hannah Dell, Xianyu Hu, Patrick Kennedy-Hunt and Kabeer Manali Rahul on a new notion of (quasi-)coherent sheaves in log geometry that aims to give a foundation for the theory of moduli spaces of sheaves on degenerations. This talk will go through our basic definitions and introduce the notion of "static" sheaves, which enables us to show that the category of log coherent sheaves is abelian and perform explicit computations in it. Time permitting, we will also discuss the resulting derived category of log coherent sheaves and its smoothness.
Analysis of Stability
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IMPRS Thementag: Short lectures by new and advanced IMPRS students
On Heat kernel on compact manifolds
In this talk we will prove existence of heat kernel manifolds which in turn allows us to prove spectral theorem. The construction will give Manikshisundaram-Pleijel asymptotic expansion which encodes certain curvature expressions of the manifold.
Mahler Measure and Numerical Semigroups
In this talk, we first review some classical results in the theory of the Mahler measure. Using classical results and assuming Lehmer’s conjecture, we then provide a proof of a conjecture due to P. Moree and A. Herrera-Poyatos.
We conclude by presenting some open questions concerning potential connections between Lehmer’s problem and the Mahler measure of polynomials arising from numerical semigroups.
Metrics and approximability for triangulated categories
Course Description: Neeman has recently introduced certain new techniques in the study of triangulated categories. These have been used to prove many important results, and have settled multiple open problems in algebraic geometry. In this course, we will discuss both these abstract techniques, and the growing body of works related to them along with several applications. In particular, we will introduce good metrics, completions, approximable triangulated categories, and see their various properties. Further, we will discuss applications to the non-existence of bounded t-structures, Rouquier dimension, semiorthogonal decompositions, representability results and so on. Most of the examples and applications in the course will be to algebraic geometry, but we note here that there are wider applications of the theory to representation theory and homotopy theory. See https://kabeermr.github.io/CourseOutline.pdf for the course outline.
Prerequisites: Some familiarity with derived and triangulated categories, and rudimentary knowledge of algebraic geometry.
Synthetic equivariant spectra and complex bordism
I’ll discuss and motivate a theory of synthetic equivariant spectra in which equivariant stable homotopy types are suitably resolved by even-dimensional cells. This recovers the Adams—Novikov spectral sequence based on equivariant complex bordism and additionally admits an interpretation in terms of motivic homotopy theory over the complex numbers. If time permits, I’ll discuss how these equivariant statements (at abelian compact Lie groups) are quite natural extensions of the corresponding nonequivariant statements that reflect the structure of a globally equivariant “decompletion" of the standard Quillen formal group on algebraic cobordism.
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A Selberg-type integral from an irrational question
The Hankel determinants built on moment sequences and their connection with transfinite diameter show their usefulness in proving irrationality of meaningful mathematical constants (think of the values of logarithm and $\pi$). They can be also viewed as particular generalisations of the celebrated Selberg integral, a closed-form period evaluation possessing a symmetry of root system. In my talk I will take a detour and discuss a general case of such Selberg-type integrals which a priory has nothing to do with irrationality. What are those creatures?! In our ongoing joint work with Ole Warnaar, we get an answer for a particular (interesting) instance - and it is surprisingly irrational, though a general case remains mysterious.
Mahler Measures and Dirichlet L-Functions: New Results on Chinburg’s Conjectures
This talk concerns Chinburg’s conjectures, which propose a connection
between two a priori different objects: Mahler measures and certain special
values of Dirichlet L-functions associated with odd quadratic characters.
The Mahler measure of a polynomial is the arithmetic mean of log|P| over
the unit torus. Chinburg’s conjecture (1984) states that, for each odd
quadratic Dirichlet character, there exists an integral bivariate rational
function (or, in its strongest form, an integral polynomial) whose Mahler
measure is equal to a rational multiple of the derivative at −1 of the
corresponding L-function. This relationship is currently known only in a
limited number of cases (18 values of the conductor of the Dirichlet
character).
I will present recent results obtained in collaboration with David Hokken
and Berend Ringeling, in which we construct new examples for previously
unknown conductors, thereby doubling the number of verified cases. Finally,
we establish a special case of the conjecture when coefficients in a
cyclotomic extension are allowed.
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques II: Matrix models and applications
Counting techniques I: Combinatorics and algebraic geometry
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