Here Algebra and Number Theory (ANT) are interpreted broadly. If you work in related aspects of ANT and like to give a talk here, feel free to contact us: Thomas Bitoun (thomas.bitoun@ucalgary.ca) or Dang-Khoa Nguyen (dangkhoa.nguyen@ucalgary.ca).
For talks in previous years, click here.
These talks are online through Zoom Meeting. For the meeting id and password, please contact DKN. During Fall 2020, we aim to schedule talks given by "nearby" speakers on Tuesday at 1:30pm local time and talks given by "far away" speakers on Tuesday at 9am local time; however these might be changed when needed. This is a 1-hour seminar and the length of talks is usually around 45-50 minutes.
September 29 at 1:30pm local time in Calgary:
Random Fibonacci Sequences From Balancing Words by John Charles Saunders (UCalgary).
Abstract: click here. We are grateful to J.C. for allowing us to upload the recorded talk.
October 13 at 1:30pm local time in Calgary
Counting rational points and polynomial dynamics by Harry Schmidt (Basel).
Abstract: we give a short overview of recent applications of transcendence techniques to polynomial dynamics by Gareths Boxall and Jones as well as the speaker. We then discuss in a bit more detail, a more recent pre-print of the author in which non-archimedean transcendence-type methods are introduced to the study of polynomial dynamics.
We are grateful to Harry for allowing us to upload the recorded talk.
October 20 at 9 am local time in Calgary
Bornological D-modules on rigid analytic spaces by Andreas Bode (ENS Lyon).
Ardakov-Wadsley introduced p-adic D-cap-modules on rigid analytic spaces in order to study p-adic representations geometrically, in analogy to the theory of Beilinson-Bernstein localization over the complex numbers. In this talk, we report on an ongoing project to extend their framework to the (derived) category of complete bornological D-cap-modules, which allows us to define analogues of the usual six operations. We then consider a subcategory playing the role of D^b_coh(D) and prove a number of stability results.
We are grateful to Andreas for allowing us to share the recorded talk; passcode: YS=r4?7^
October 27 at 9 am local time in Calgary
Multiple zeta values in deformation quantization by Erik Panzer (Oxford).
In 1997, Kontsevich constructed a universal quantization of every Poisson manifold as a formal power series. Its coefficients are given as integrals over moduli spaces of marked holomorphic discs. In joint work with Peter Banks and Brent Pym, we show that these integrals always evaluate to multiple zeta values, which are interesting transcendental numbers that appear in several other contexts. I will give an introduction to deformation quantization, explain Kontsevich's formula and our result and briefly discuss some ideas of the proof.
We are grateful to Erik for allowing us to share the recorded talk; passcode: 684K@^e+
November 3 at 1:30pm local time in Calgary by Sacha Mangerel (U Montreal)
Monotone Chains of Fourier Coefficients of Cusp Forms by Sacha Mangerel (U Montreal)
Abstract: click here.
We are grateful to Sacha for allowing us to upload the recorded talk.
November 17 at 1:30 pm local time in Calgary
Configuration polynomials and their singularities by Uli Walther (Purdue)
A natural way to produce a polynomial from a (finite) graph is to form the Kirchhoff polynomial, by adding terms that correspond to spanning trees of the graph.
A more general construction arises from a hyperplane arrangement A (i.e., a realization of a matroid); the Kirchhoff case corresponds to the graphical arrangement with participating hyperplanes x_i=x_j for all edges (i,j) of the graph.
Kirchhoff polynomials appear in Feynman integrals that describe probabilities of the "interior workings" of particle scatterings. This makes their singularities pohysically interesting, since they are responsible for non-converging integrals that require "renormalization".
In this talk I will discuss what configuration polynomials are, study their factors, elaborate on the size of the singular locus. I will then consider the projective variety defined by the configuration polynomial and discuss a conjecture of Aluffi on the topology of its complement if the base field is CC.
I may have time to indicate what the relation is to physics.
This is joint work with Graham Denham, Delphine Pol and Mathias Schulze.
November 24 at 1:30 pm local time in Calgary
Category O for oriented matroids by Carl Mautner (UC Riverside)
(joint with Ethan Kowalenko) Category O of a complex semi-simple Lie algebra has rich structure and is connected to the algebraic geometry of the cotangent bundle of the associated flag variety. Braden-Licata-Proudfoot-Webster discovered that similarly rich representation theory, which they named hypertoric category O, can be extracted from the geometry of hypertoric (a.k.a. toric hyperkahler) varieties. Motivated by this discovery, they and others introduced and studied other `geometric’ categories O associated to more general symplectic resolutions. In the current work, we generalize their notion of hypertoric category O in a different direction, to the purely combinatorial setting of oriented matroids. We are motivated in part by earlier joint work with Braden on matroidal Schur algebras.
We thank Carl for letting us share the recording of his talk (Access Passcode: +Lpz1aR6).
December 1 at 1:30pm local time in Calgary
Unbounded Denominators for Non-Congruence Forms of Index 7 by Andrew Fiori (Lethbridge)
Abstract:
In this talk I will report on some recent joint work with Cameron Franc (McMaster) where we investigate the structure of modular forms for the non-congruence groups of minimal index.
In contrast to the case of congruence groups, little is known about the Fourier coefficients of even the Eisenstein series let alone more general forms. One way in which non-congruence forms differ from congruence forms is that when their Fourier coefficients are algebraic the denominators are conjectured to diverge to infinity. We will demonstrate that this conjecture holds for the index 7 subgroups of PSL2(Z) while highlighting some questions which remain open about the coefficients of Eisenstein series.
December 8 at 1:30pm local time in Calgary, talk by Avi Kulkarni (Dartmouth)
For this last talk, the speaker will try something different by presenting a "split talk":
Titles and abstract for the split talk.
Part I: An explicit family of cubic number fields whose class group contains $(ZZ/2ZZ)^8$.
Abstract
A uniquely trigonal curve is a smooth algebraic curve that has an essentially unique morphism to the projective line of degree 3. In part I, we construct an infinite family of cubic number fields whose class groups have many 2-torsion elements by constructing a particular uniquely trigonal genus 4 curve.
Part II: Deep learning Gauss-Manin connections
Abstract:
In this talk, I will discuss how machine learning can aid in the computation of the periods of projective hypersurfaces. I will also report on the results of our large scale computation to find the periods of all smooth quartics in P3 that are the sum of 5 monomial terms with unit coefficients. Joint work with Kathryn Heal and Emre Sertoz.
We are grateful to Avi for allowing us to upload the recorded talk.
January 19 at 3 pm local time in Calgary
Bernstein-Sato polynomials in positive characteristic by Eamon Quinlan (Michigan)
Over the complex numbers, the Bernstein-Sato polynomial of an ideal is an invariant that originated in complex analysis and with now strong applications in birational geometry and singularity theory. In this talk we present an analogue of this invariant in positive characteristic and discuss some of its basic properties, generalizing constructions of Bitoun and Mustaţă.
We are grateful to Eamon for allowing us to share the recorded talk, passcode: +k&X2kC=
January 26 at 1:30 pm local time in Calgary
Towards a Logarithmic Comparison Theorem for Quasi-free Divisors by Dan Bath (Purdue)
One way to compute the cohomology of the complement of a hypersurface {f=0} is to compute the cohomology of the de Rham complex of meromorphic forms with poles of arbitrary order along f. Sitting inside this complex is the logarithmic de Rham complex, consisting of certain forms with poles of order at most one. It is a long standing question to find necessary and sufficient conditions on the hypersurface ensuring that the natural inclusion of complexes is a quasi-isomorphism, that is, that f satisfies the Logarithmic Comparison Theorem. The modern approach utilizes D-module techniques and the "best" sufficient conditions require assuming that (among other things) the singular locus of f is, morally, as large as possible. To relax this assumption, we introduce a new variant of the Logarithmic Comparison Theorem where: vector fields tangent to {f=0} are replaced with a submodule closed under Lie brackets; the logarithmic de Rham complex is replaced with a new de Rham complex. Work in progress; joint with Luis Narvaez-Macarro and Francisco Castro-Jimenez.
We are grateful to Dan for allowing us to share the recorded talk, passcode: 4=%o&1?E
Bernstein-Sato polynomials in positive characteristic b