Short Courses and Seminars - Fall 2021

Joint ICMS & IMSA Seminar

August Hozie
University of Miami & IMSA

Some Examples and Asymptotics of Irregular Connections of Singularities

Friday, December 10, 2021, 9:30am
Hybrid
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Abstract: A common feature of Noncommutative Hodge Structures, Frobenius Manifolds, and Irregular Hodge Structures is a connection on P1 which is Poincare rank 1 irregular at 0 and regular at infinity. In many cases we can compute local equations for these connections and investigate their asymptotics at the singular point. In this talk we'll look at some examples of the asymptotic spectra associated with these equations and some behavior under deformation.


Joint ICMS & IMSA Seminar

Dr. Ludmil Katzarkov
University of Miami

Multispectra

Monday, November 22, 2021, 5:00pm
Hybrid
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Abstract: We will describe some new birational invariants.


Joint ICMS & IMSA Seminar

Dr. Shaoyun Bai
Princeton University

On the Rouquier Dimension of Wrapped Fukaya Categories

Monday, November 22, 2021, 4:00pm
Hybrid
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Abstract: Given a triangulated category, its Rouquier dimension is defined to be the minimal generation time of all of its split-generators. I will explain how the Rouquier dimension of derived wrapped Fukaya categories of Weinstein manifolds/sectors is related to problems in symlectic topology of classical flavor, including quantitative intersection question of Lagrangian skeleta and estimating minimal numbers of critical points of symplectic Lefschetz fibrations. Moreover, using recent advances on symplectic flexibility (the arboreal program) and the local-to-global characterization of wrapped Fukaya categories, I will show how to resolve new cases of Orlov’s conjecture by bounding the Rouquier dimension of derived categories of algebraic varieties using homological mirror symmetry. This is joint work with Laurent Cote.


Joint ICMS & IMSA Seminar

Dr. Kyoung-Seog Lee
University of Miami & IMSA

Homological Mirror Symmetry for Hypersurface Singularities II

Friday, November 5, 2021, 9:30am
Hybrid
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Abstract: In this talk, I will continue to introduce homological mirror symmetry for singularities. I will explain the categories of graded matrix factorizations of invertible polynomials and discuss several ways to describe them. If time permits, I will discuss stability conditions on the categories of the graded matrix factorizations of weighted homogeneous polynomials constructed by Takahashi, Kajiura-Saito-Takahashi, Toda, Otani-Takahashi.


Joint ICMS & IMSA Seminar

Dr. Kyoung-Seog Lee
University of Miami & IMSA

Homological Mirror Symmetry for Hypersurface Singularities I

Friday, October 22, 2021, 9:30am
Hybrid
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Abstract: In this talk, I will briefly introduce homological mirror symmetry for certain hypersurface singularities. I will introduce basic definitions, Berglund-Hubsch duality of invertible polynomials, and some known results. Then I will discuss several examples in detail.


IMSA Seminar

Dr. Han-Bom Moon
Fordham & Stanford

Conformal Blocks in Algebraic Geometry Part III

Thursday, October 14, 2021, 2:15pm
Hybrid

Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.


IMSA Seminar

Dr. Han-Bom Moon
Fordham & Stanford

Conformal Blocks in Algebraic Geometry Part II

Thursday, October 14, 2021, 1:00pm
Hybrid

Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.


IMSA Seminar

Dr. Han-Bom Moon
Fordham & Stanford

Conformal Blocks in Algebraic Geometry Part I

Wednesday, October 13, 2021, 2:00pm
Hybrid

Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.


Joint ICMS & IMSA Seminar

August Hozie
University of Miami & IMSA

More Instances of Steenbrink Spectra in Singularity Categories

Friday, October 8, 2021, 9:50am
Hybrid
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Abstract: Now that we have set up the Frobenius Manifold of a singularity, we can see 2 more places where the steenbrink spectrum of a singularity appears in its singularity category, namely, the non-commutative mixed hodge structure of a singularity and the dimensional properties of the category. The former appearance is somehow natural, and the latter somewhat mysterious. In this talk we will conduct a surface level investigation of these appearances.


Joint ICMS & IMSA Seminar

Sebastian Torres
ICMS-Sofia

Windows and the BGMN Conjecture

Friday, October 8, 2021, 8:30am
Via Zoom
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Abstract: Let C be a smooth projective curve of genus at least 2, and let N be the moduli space of semistable rank-two vector bundles of odd degree on C. We construct a semi-orthogonal decomposition in the derived category of N conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form D(Cd) where Cd are d-th symmetric powers of C, and the semi-orthogonal complement to these blocks is conjecturally trivial.

In order to prove our result, we use the moduli spaces of stable pairs over C. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall.

This is a joint work with J. Tevelev.

This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.


Joint ICMS & IMSA Seminar

August Hozie
University of Miami & IMSA

Steenbrink Spectra in Singularity Categories

Friday, October 1, 2021, 9:30am
Hybrid
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Abstract: The spectrum of a singularity as defined by Steenbrink encodes important analytic information about a singularity and describes the Hodge filtration on the vanishing cycles of the singularity. In this talk we will explore a few different ways in which the spectral numbers show up in the triangulated singularity category associated with a singularity.


Joint ICMS & IMSA Seminar

Dr. Kyoung-Seog Lee
University of Miami & IMSA

Alexander Polynomials of Algebraic Links

Friday, September 24, 2021, 9:30am
Hybrid
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Abstract: The Alexander polynomial is one of the most well-known invariants in knot theory. In the first part of this talk, I will review basic definitions and examples of Alexander polynomials of algebraic links. Then I will survey several results expressing Alexander polynomials of algebraic links via tools of algebraic geometry. Then I will discuss briefly how the spectrum of a plane curve singularity is related to the Alexander polynomial of its algebraic link.


Joint ICMS & IMSA Seminar

Dr. Kyoung-Seog Lee
University of Miami & IMSA

Plane Curve Singularities and Spectrum 

Friday, September 17, 2021, 9:30am
Hybrid

Abstract: In the first part of this talk, I will review basic notions and results about plane curve singularities, e.g. blow-ups, resolution of singularities, Newton-Puiseux series, etc. Then I will explain how to compute the spectrum of a plane curve singularity via Newton-Puiseux series based on Morihiko Saito's work.


IMSA Seminar

Josef Svoboda
University of Miami & IMSA

Surface Singularities and Invariants of their Links

Friday, September 10, 2021, 9:30am
Hybrid
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Abstract: To an isolated surface singularity, we can assign a natural 3-manifold - the link of the singularity. It is an old question how many of the properties of the singularity can be recovered from this purely topological information. Based on the work of Lawrence-Zagier, Hikami and others, I will show how quantum topological invariants of the link are related to the spectrum in the case of Brieskorn spheres.


IMSA Seminar

Josef Svoboda
University of Miami & IMSA

Spectrum of Singularities

Friday, September 3, 2021, 9:30am
Hybrid
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Abstract: The spectrum is a strong invariant of a hypersurface singularity, defined originally by Arnold, Varchenko and Steenbrink. I will start with examples of singularities and their spectra and sketch the definition of the spectrum. Then I will concentrate on the most important properties of the spectrum such as the Thom-Sebastiani theorem and semicontinuity, which are very powerful in applications. Finally, I will talk about computational techniques to obtain the spectrum.


Joint ICMS & IMSA Seminar

Dr. Rodolfo Aguilar
ICMS

Quantum Representations of Fundamental Groups of Curves with Infinite Image

Friday, August 20, 2021, 10:30am (5:30pm Sofia)
Via Zoom

Abstract: We will report some results due to Koberda-Santharoubane showing an element of $\pi_1(\mathbb{P}^1\setminus \{3-\text{points}\})$ having infinite order under some quantum representations of the mapping class group.

This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.


Joint ICMS & IMSA Seminar

Dr. Rene Mboro
ICMS

On Determinantal Cubic Hypersurfaces (after Iliev-Manivel, Beauville,...)

Friday, August 20, 2021, 9:30am (4:30pm Sofia)
Via Zoom

Abstract: We give an account of the problem of writting an equation of a cubic (or other degree) hypersurface $X$ as a (kind of) determinant of a matrix with homogeneous entries. Expressing the equation of $X$ as a determinant is equivalent to produce a arithmetically Cohen-Macaulay vector bundle on $X$. The talk will focus on the cases of cubic hypersurfaces of dimension at most 8. 

This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.