Weekly Meetings: Thursdays 3:45-5:00pm

Location:  Math Thesis Room 4214.03, CUNY Graduate Center

Organizers: Valera Sergeev (vsergeev at gradcenter dot cuny dot edu)

Connor Stewart (stewartconnor04 at gmail dot com) -- contact me to be added to the seminar mailing list

September 4

• Speaker: Valera Sergeev and Connor Stewart

• Title: Organizational Meeting

• Description:  We will discuss topics of interest for the semester and make a preliminary schedule.

September 11

• Speaker: Valera Sergeev

• Title:  Cohen-Macaulay Rings and Invariants of Finite Groups

• Description: We will review the basic notions of geometric invariant theory and use them to introduce Cohen-Macaulay rings.

September 18

• Speaker: Connor Stewart

• Title: du Val Singularities, Part I: Quotients of C^2 by Finite Subgroups of SL_2(C)

• Description: du Val singularities are an important class of surface singularities that have amazingly rich connections to several areas of mathematics. In invariant theory, du Val singularities arise as quotients of C^2 by finite subgroups of SL_2(C). In birational geometry, they are the canonical singularities in dimension 2, important for the minimal model program. Beyond this, du Val singularities are surprisingly classified by the simply laced "ADE" Dynkin diagrams, originally arising in Lie theory.

In this first talk, we will classify the finite subgroups of SL_2(C) and use this to derive normal forms for the equations of the du Val singularities up to isomorphism. We will then describe the exceptional divisor of the minimal resolution of a du Val singularity in terms of simply laced Dynkin diagrams. Finally, we will check this correspondence explicitly in the simple example of an A_1 singularity.

September 25

• Speaker: Connor Stewart

• Title: du Val Singularities, Part II: Canonical Singularities and McKay Correspondence

• Description: Last time, we defined du Val singularities (up to analytic isomorphism) as the singularities at the origin of the quotients C^2/G, where G is a finite subgroup of SL_2(C). We used this description to give a table of standard equations for the du Val singularities, from which one could check explicitly that the exceptional divisor of the minimal resolution is a tree of (-2)-curves whose dual graph is a simply-laced "ADE" Dynkin diagram.

In this talk, we will give an equivalent characterization of du Val singularities as canonical surface singularities, and show that this definition implies the same description of the exceptional divisor in terms of simply-laced Dynkin diagrams. If time permits, we will also discuss the McKay correspondence, which recovers the Dynkin diagram associated to the du Val singularity C^2/G from the representation theory of G.

October 2 - No meeting. CUNY holiday.

October 9

• Speaker: Kioshi Morosin

• Title: Toric Varieties Are Cohen-Macaulay

• Description: Results of Hochster in the 1970s on rings of invariants under the action of a torus had the consequence that the singularities of a normal toric variety are at worst Cohen-Macaulay. We will explore a different, more geometric proof of the same fact due to Ishida.

October 16

• Speaker: Valera Sergeev

• Title: Logarithmic Forms and Singularities

• Description: We will discuss several examples of Iitaka's philosophy, which will naturally lead to logarithmic forms and the study of singularities of the minimal model program. If time permits we will also discuss relationships with singularities in characteristic p defined using the Frobenius map.

October 23 - No meeting this week.

October 30

• Speaker: Valera Sergeev

• Title: Logarithmic Forms and Singularities, Part II

• Description: We will introduce the notions of (strongly) F-regular, F-split, F-rational, and F-injective local rings in positive characteristic and discuss the known and conjectured relationships between these notions and those of log terminal, log canonical, rational, and du Bois singularities in characteristic 0.

November 6

• Speaker: James Austin Myer

• Title: Local Parameters à la Conrad, Edixhoven, & Stein: a Temptress Toward Calculation…

• Description: We seek an appropriate regular model of the projective line whose normalization in the function field of a hyperelliptic curve (over a complete, discretely-valued field of characteristic 0 whose residue field is algebraically closed of characteristic 2) is a regular model of the hyperelliptic curve. Lemma 2.3.2 of “J_1(p) has Connected Fibers” guarantees local parameters in a hyperlocal neighborhood of a nil-semistable point. Unfortunately, we may not assume each point of our regular model of the projective line is nil-semistable, only that the reduced scheme associated to it is (nil-)semistable. However, I’ve been assuming the existence of these local parameters throughout my thesis project with Andrew Obus! Watch my entire thesis project fall apart… or not!

November 13 - ONLINE MEETING

• Speaker: Bhargavi Parthasarathy (Syracuse University)

• Title: Matrix Factorizations: An Incomplete Summary

• Description: Matrix factorizations are a tool introduced by Eisenbud that helped understand (and indeed classify) the Maximal Cohen-Macaulay modules (MCM) over a hypersurface ring. In this talk, I will define matrix factorizations, discuss how the category of matrix factorizations and category of MCM modules are related, why this makes them a useful tool for classification of MCM modules, when is such a classification finite and some cases where it is not.

Note: This talk will be online over Zoom. Contact organizer Connor Stewart (stewartconnor04 at gmail dot com)  for the meeting link.

November 20 - No meeting this week. Math department colloquium.

November 27 - No meeting this week. Thanksgiving.

December 4

• Speaker: Sevan Bharathan

• Title: Sites: Definition(s), First Examples, and Applications

• Description: We will discuss the definition of a sheaf on a topological space, its history, and the inadequacy of traditional sheaf cohomology to yield interesting invariants in algebraic geometry (with respect to the Zariski topology). We will then introduce Grothendieck (pre-)topologies and sites and look at several important sites used in algebraic geometry today.

December 11

• Speaker: Connor Stewart

• Title: Stacks and Moduli Spaces

• Description: Last week, we generalized the notion of sheaves of sets on topological spaces to sheaves on categories equipped with a Grothendieck topology. This week, we will take a further step by generalizing from sheaves of sets to sheaves of groupoids, or stacks. We will define stacks and related concepts, including their fiber categories, points, associated spaces, and fiber products. Motivated by the example of the moduli of smooth curves of genus g, we will then discuss how the category of stacks often provides a natural setting to seek universal objects that fail to exist in the category of schemes.

LAST TALK OF THE SEMESTER. See you in the spring! 

Weekly Meetings: Tuesdays 10:00-11:15am

Location: Room 5417, CUNY Graduate Center

Organizers: Valera Sergeev (vsergeev at gradcenter dot cuny dot edu)

Connor Stewart (cstewart3 at gradcenter dot cuny dot edu)

February 4

• Speaker: James Myer

• Title:  A geometric construction of a blowup of any variety at finitely many points that is a family of curves over a projective space of one less dimension whose generic fiber is regular, etc., and Semistable Reduction à la de Jong

• Description: Link to full abstract. We will discuss a result of de Jong stating that every variety admits a blowup at finitely many points that is fibered by curves over a projective space of one less dimension. We will connect this result to the Semistable Reduction Theorem and the existence of regular alterations.

February 11

• Speaker: James Myer

• Title:  Part II: Semistable Reduction à la de Jong, and the Final Frontier for the Problem of Resolution of Singularities: the Wild West

• Description: I'll briefly review the geometric construction of a blowup of any variety fibered by curves over a projective space of one less dimension whose generic fiber is regular, such that the smooth locus is dense in each fiber (and whose restriction to a divisor is finite & étale) via a geometric construction via generic projections & a Lefschetz pencil. Afterward, I'll explain how the generalization of the Semistable Reduction Theorem à la de Jong, i.e. the existence of a regular alteration of any variety, affords us a slick proof of the existence of a resolution of the singularities of any variety if the characterisitic of the base field does not divide the degree of the alteration, e.g. if the characteristic of the base field is zero. 

February 18 - No meeting. CUNY Monday schedule.

February 25

• Speaker: Connor Stewart

• Title: Divisors and Line Bundles

• Description: We will review Weil and Cartier divisors, the Picard group, and the canonical sheaf in preparation for next week's talk on the Riemann-Roch Theorem. We will look at these concepts in a concrete example with an elliptic curve.

March 4

• Speaker: Valera Sergeev

• Title:  Applications of the Riemann-Roch Theorem, Part I

• Description: We will state the Riemann-Roch Theorem and discuss several applications including: computing the canonical divisor of a genus 1 curve; showing P^1 is the only smooth proper curve of genus 0; and showing every smooth proper curve over a field can be embedded in P^3. We will discuss when a divisor gives rise to a projective embedding and work out the details of the O(2)-embedding of P^1 as a conic in P^2.

March 11

• Speaker: Valera Sergeev

• Title:  Applications of the Riemann-Roch Theorem, Part II

• Description: We will apply the Riemann-Roch Theorem to study the canonical divisor and automorphism group of curves of genus >=2 in the hyperelliptic and non-hyperelliptic cases.

March 18 - Joint meeting with MoPA seminar 10:30-11:30am in Room 4213.04

• Speaker: Alf Dolich

• Title:  Scott Sets in Algebraic Setting

• Description: TBA

March 25

• Speaker: Connor Stewart

• Title:  Explicit Computations Related to the Riemann-Roch Theorem

• Description: We will unpack the definitions of the canonical divisor and Riemann-Roch spaces L(D) to write equations for the canonical embedding of a smooth plane quintic.

April 1

• Speaker: Sevan Bharathan

• Title:  The Etale Fundamental Group: An Introduction to Grothendieck Topologies, Part I

• Description: For varieties in the Zariski topology, the usual fundamental group from algebraic topology is well-defined but fails to give an interesting invariant.  To see what goes wrong, it is helpful to adopt the perspective of covering space theory: the fundamental group can be viewed as the group of automorphisms of the fiber functor taking a topological cover of a space X to the preimage of a fixed base-point x; since all topological covers of an irreducible variety are trivial, so is the fundamental group. Amazingly, the construction of the fundamental group for a variety X can be repaired by enlarging the category of "covers" of X to include not only topological covers but more general etale covers. The etale fundamental group, or the automorphism group of a "geometric fiber" functor on this larger category, recovers much of the geometric information we would hope to obtain from the classical fundamental group. In this talk, we will introduce these ideas and compute them in several simple examples.

April 8

• Speaker: Sevan Bharathan

• Title:  The Etale Fundamental Group: An Introduction to Grothendieck Topologies, Part II

• Description: We will outline the proof of the Riemann Existence Theorem over C and apply it to show that in characteristic 0, the etale fundamental group is the profinite completion of the usual topological fundamental group. We will then consider an example of Serre of two "conjugate" schemes over C with the same etale fundamental group but different topological fundamental groups. Finally, we will introduce the notion of a Grothendieck topology on a category with fiber products.

April 15 - No meeting. CUNY Spring Break.

April 22

• Speaker: James Myer

• Title:  Recovering Algebraic Topology from Algebraic Geometry, Part I

• Description: We will show that every closed surface can be realized via a sequence of real blow-ups and blow-downs starting from S^2=P^1(C), and discuss a theorem of Kollar generalizing this result to 3-manifolds.  We will then go through an example of using Macaulay 2 to compute the singular cohomology of the analytic space of a complex projective variety.

April 29

• Speaker: James Myer

• Title:  Recovering Algebraic Topology from Algebraic Geometry, Part II

• Description: We will attempt to compute the first singular homology (Z/2Z) of an Enriques surface using only the algebraic data of its defining equation and Macaulay2.

May 6

• Speaker: Valera Sergeev

• Title:  Valera's Birthday Talk - Intersection Theory on Surfaces

• Description: We will develop intersection theory on a smooth projective surface over an algebraically closed field and build up to he famous result that a smooth cubic surface in P^3 contains exactly 27 lines (in honor of Valera's 27th birthday).

May 13

• Speaker: Connor Stewart

• Title:  Root Systems and Lines on del Pezzo Surfaces

• Description: We will introduce root systems, Weyl groups, and Dynkin diagrams via the simple example of A_2. We will then look at the classification of del Pezzo surfaces and describe the set of lines E_r on the del Pezzo surface X_r obtained by blowing up P^2 at 1 ≤ r ≤ 8 points in general position. Finally, we will identify the group of intersection-preserving symmetries of E_r, 3 ≤ r ≤ 8, with the Weyl group of a root system constructed from Pic(X_r). We will compute the Dynkin diagram associated to this root system in the case of a smooth cubic surface.

LAST TALK OF THE SEMESTER. See you in the fall! 

Weekly Meetings: Mondays 12:30-1:45pm 

Location: Math Thesis Room 4214.03, CUNY Graduate Center

Organizers: Valera Sergeev (vsergeev at gradcenter dot cuny dot edu)

Connor Stewart (cstewart3 at gradcenter dot cuny dot edu)

September 16

• Speaker: Valera Sergeev

• Title: First Meeting -- Review of Commutative Algebra

• Description: We will discuss topics/scheduling for the semester and review facts from commutative algebra, motivated by applications to invariant theory.

September 23

• Speaker: Connor Stewart

• Title: Normalization from a Geometric Perspective

• Description: We will discuss normalization morphisms, their importance in algebraic geometry, and how they can be viewed as "ungluings" of subschemes or tangent spaces.

Link to Zoom Recording 

September 30

• Speaker: Kioshi Morosin

• Title: Homological Algebra via the Tohoku Paper

• Description:  TBA

October 7

• Speaker: Valera Sergeev

• Title: Regular Rings

• Description:  TBA

Tuesday, October 15 (CUNY Conversion Day - Monday Schedule)

• Speaker: Nathaniel Kingsbury

• Title: p-Adic Numbers and the Weil Conjectures I: Background and Cohomology

• Description:  We will state the Weil Conjectures and their history, with a focus on how most of them "should" be immediate corollaries of a correct cohomology theory for varieties in positive characteristic. We will then make some simple reductions of the first conjecture (rationality) and briefly discuss the structure of Dwork's proof. Finally, we will introduce the p-adic numbers, which will be our principle tool in future lectures.

Link to Zoom Recording (passcode: 7n3Aqp+6)

October 21

• Speaker: Nathaniel Kingsbury

• Title: p-Adic Numbers and the Weil Conjectures II:  The Dwork Trace Formula and Meromorphicity

• Description:  We will briefly introduce the p-adic numbers, and use them to prove that the zeta function is p-adically meromorphic. In order to do so, we will introduce a certain operator on an infinite dimensional space of power series associated to an affine hypersurface whose trace encodes the counts of the rational points of the hypersurface.

Link to Zoom Recording (passcode: ^2Cf?4XA)

October 28

• Speaker: Nathaniel Kingsbury

• Title: p-adic Numbers and the Weil Conjectures III: The Dwork Trace Formula and Conclusion of the Proof

• Description:  We will finish deriving the Dwork trace formula, and use it to establish p-adic meromorphicity of the zeta function. The zeta function of an (affine) algebraic hypersurface also has a positive radius of convergence over C. Time permitting, we will show how by playing these facts off of one another we may show that it is in fact a rational function.

Link to Zoom Recording (passcode: qrs=vG52)

November 4

• Speaker: Connor Stewart

• Title: Models of Curves and Mac Lane Valuations, Part I

• Description:  We will discuss blow-ups and embedded resolutions for curves on a surface. We will see how blow-ups on a surface work in the arithmetic case and apply this to computing a regular Z_7-model of a curve over Q.  

November 11 

• NO MEETING (AGNES @ Dartmouth)

November 18

• Speaker: Valera Sergeev

• Title: Injective Modules and Matlis Duality

• Description: We will review the definition and properties of injective modules and define the injective hull of a module. We will then go over the main results of Matlis Duality and see an application to elliptic curves.

November 25

• Speaker: Connor Stewart

• Title: Models of Curves and Mac Lane Valuations, Part II

• Description: We will discuss divisors on an arithmetic surface. We will then compute a regular Z_7-model of a curve over Q in a second way, by considering the curve as a cover of P^1 and resolving the branch divisor. 

Link to Zoom Recording 

December 2

• Speaker: Mac Mccormick

• Title: Schur-Weyl Duality, Part I

• Description: We will discuss the statement of Schur-Weyl Duality and review necessary background from representation theory, including Maschke's Theorem, class functions, Young Diagrams, and Specht modules.

December 9 

• Speaker: Kioshi Morosin

• Title: Gorenstein and non-Gorenstein Singularities

• Description: TBA

LAST TALK OF THE SEMESTER. See you in the spring!