Wednesdays, 2:00-3:00PM in H425

Until Spring Breaak, the Algebra Seminar met on Wednesdays at 2:00-3:00PM in the Hill Center, on Busch Campus of Rutgers University. After Spring Break, the seminar moved on-line. A more comprehensive listing of all Math Department seminars is available.
Here is a link to the algebra seminars in previous semesters

Spring 2020 Seminars (Wednesdays at 2:00 in H425)
22 Jan Shira Gilat (Bar-Ilan U.) "Higher norm principles for norm varieties"
12 Feb Chuck Weibel (Rutgers) "The K'-theory of monoid sets"
19 Feb Linhui Shen (Michigan State) "Quantum geometry of moduli spaces of local system"
26 Feb Lev Borisov  Rutgers    "Six explicit pairs of fake projective planes"
 4 Mar Saurabh Gosavi (Rutgers) "Generalized Brauer dimension"
18 Mar  no seminar      ------------------- Spring Break ----------
25 Mar  CANCELLED  -- Future seminars moved on-line
 1 Apr Joaquin Moraga (Princeton) "On the Jordan property for local fundamental groups"
 8 Apr Ian Coley (Rutgers)      "Higher K-theory via generators and relations"
15 Apr Franco Rota (Rutgers)    "Moduli spaces on the Kuznetsov component of Fano threefolds of index 2"
22 Apr James Cameron (UCLA)     "Group cohomology rings via equivariant cohomology"
29 Apr Luca Schaffler (U. Mass) "Compactifications of moduli of points and lines in the projective plane"
 6 May Angela Gibney (Rutgers)    CANCELLED 

Spring 2020 classes start Tuesday January 22 and end Monday May 4

Fall 2019 Seminars (Wednesdays at 2:00 in H525)
18 Sep Sándor Kovács (U.Washington) "Rational singularities and their cousins in arbitrary characteristics"
20 Sep Jacob Lurie (IAS) "On Makkai's Strong Conceptual Completeness Theorem"
25 Sep Saurabh Gosavi (Rutgers) "Generalized Brauer dimension and other arithmetic invariants of semi-global fields"
 2 Oct Danny Krashen (Rutgers)  "The arithmetic of semiglobal fields via combinatorial topology"
 9 Oct Ian Coley (Rutgers)      "What is a derivator?" 
16 Oct Sumit Chandra Mishra (Emory U.) "Local-global principle for norm over semi-global fields"
23 Oct Chengxi Wang (Rutgers)   "strong exceptional collections of line bundles" 
30 Oct Angela Gibney (Rutgers)  "Vertex algebras of CohFT-type"
 6 Nov Robert Laugwitz (Nottingham, UK) "dg categories and their actions"
13 Nov Volodia Retakh (Rutgers)  "Noncommutative Laurent Phenomenon: two examples"
20 Nov Carl Lian  (Columbia)     "Enumerating pencils with moving ramification on curves"
27 Nov --- no seminar ---        Thanksgiving is Nov. 28; Friday class schedule 
11 Dec Diane Mclagan (U.Warwick) "Tropical scheme theory"
Fall 2019 classes begin Tuesday September 3 and end Wednesday Dec.11. Finals are December 14-21, 2019

Spring 2019 Seminars (Wednesdays at 2:00 in SERC 206)
Note: First 3 seminars were in H-005; 4th in H705; all others in SERC-206
23 Jan Patrick Brosnan  U.Maryland "Palindromicity and the local invariant cycle theorem"
30 Jan Khashayar Sartipi  UIUC "Paschke Categories, K-homology and the Riemann-Roch Transformation"
 6 Feb Chuck Weibel   Rutgers  "The Real graded Brauer group"
13 Feb Volodia Retakh Rutgers "An analogue of mapping class groups and noncommutative triangulated surfaces"
20 Feb Dawei Chen Boston College&IAS "Volumes and intersection theory on moduli spaces of abelian differentials"
 6 Mar no seminar  
13 Mar Jeanne Duflot Colorado State U. "A Degree Formula for Equivariant Cohomology"
20 Mar  no seminar      ------------------- Spring Break ----------
27 Mar Louis Rowen  Bar-Ilan Univ  "The algebraic theory of systems"
 3 Apr Iulia Gheorghita Boston College  "Effective divisors in the Hodge bundle"
10 Apr Gabriel Navarro  U.Valencia "Character Tables and Sylow Subgroups of Finite Groups"
17 Apr John Sheridan Stony Brook "Continuous families of divisors on symmetric powers of curves"
24 Apr Yaim Cooper  IAS  "Severi degrees via representation theory"
 1 May Dave Anderson  Ohio State  "Schubert calculus and the Satake correspondence"
Classes end Monday May 6; Finals are May 9-15, 2019

Fall 2018 Seminars (Wednesdays at 2:00 in H525)
19 Sep Nicola Tarasca  Rutgers  "Geometry and Combinatorics of moduli spaces of curves" 
26 Sep Angela Gibney  Rutgers  "Basepoint free loci on $M_{0,n}$-bar from Gromov-Witten theory of smooth homogeneous varieties"
 5 Oct(FRI) Michael Larsen Indiana U  "Irrationality of Motivic Zeta Functions"
    *** Friday at 10:00 AM in Hill 005 ***
10 Oct Yotam Hendel  Weizmann Inst. "On singularity properties of convolutions of algebraic morphisms"
17 Oct Qixiao Ma    Columbia Univ.  "Brauer class over the Picard scheme of curves"
24 Oct Sandra Di Rocco KTH-Sweden   "Generalized Polar Geometry"
31 Oct Igor Rapinchuk  Michigan State "Algebraic groups with good reduction and unramified cohomology" 
 7 Nov Isabel Vogt     MIT          "Low degree points on curves"
14 Nov Bob Guralnick   USC         "Low Degree Cohomology" 
21 Nov --- no seminar ---        Thanksgiving is Nov. 22; Friday class schedule 
28 Nov Julie Bergner  U.Virginia  "2-Segal spaces and algebraic K-theory"
 5 Dec Chengxi Wang     Rutgers    "Quantum Cohomology of Grassmannians"
12 Dec Patrick Brosnan  U.Maryland  POSTPONED
Classes end Wednesday Dec. 12; Finals begin Dec. 15, 2018
Spring 2018 Seminars (Wednesdays at 2:00 in H425)
24 Jan Aurélien Sagnier EP-Palaiseau "An arithmetic site of Connes-Consani type for the Gaussian integers"
31 Jan Jozsef Beck   Rutgers   "An annoying question about eigenvalues"
 7 Feb Lev Borisov  Rutgers    "Equations of Cartwright-Steger surface"
21 Feb Dhruv Ranganathan  MIT  "Curves, maps, and singularities in genus one"
28 Feb Rohini Ramadas  Harvard "Algebraic dynamics from topological and holomorphic dynamics"
15 Mar Nicola Pagani U.Liverpool "The indeterminacy of universal Abel-Jacobi sections" 
28 Mar Ana-Maria Castravet Northeastern U. "Derived categories of moduli spaces of stable rational curves"
11 Apr Chiara Damiolini Rutgers "Conformal blocks attached to twisted groups" 
18 Apr Joe Waldron  Princeton  "Singularities of general fibers in positive characteristic"
 2 May Ben Bakker  U. Georgia  "Hodge theory and o-minimal geometry"
 9 May Antonella Grassi U. Penn "Singularities in geometry, topology and strings"  (in H705)
Classes end Monday, April 30; Final Exams are May 4-9, 2018
Fall 2017 Seminars (Wednesdays at 2:00 in H525)
13 Sep Louis Rowen  Bar-Ilan Univ "A general algebraic structure theory for tropical mathematics"
20 Sep Nicola Tarasca Rutgers   "K-classes of Brill-Noether loci and a determinantal formula"
27 Sep Pham Huu Tiep  Rutgers   "Character levels and character bounds"
4 Oct Dave Jensen     Yale      "Linear Systems on General Curves of Fixed Gonality"
11 Oct Gernot Stroth  Martin-Luther Univ.  "On the Thompson Subgroup"
18 Oct Han-Bom Moon   IAS      "Birational geometry of moduli spaces of parabolic bundles"
 1 Nov Danny Krashen  Rutgers   "Extremely indecomposable division algebras"
 8 Nov Lev Borisov    Rutgers   "Explicit equations of a fake projective plane"
15 Nov Julia Hartmann U. Penn.  "Local-global principles for rational points and zero-cycles"
22 Nov --- no seminar ---      Thanksgiving is Nov. 23; Friday class schedule 
29 Nov Chuck Weibel   Rutgers   "K-theory of line bundles and smooth varieties"
 6 Dec Seth Baldwin  N.Carolina "Equivariant K-theory associated to Kac-Moody groups"
13 Dec Brooke Ullery  Harvard   "Gonality of complete intersection curves"
Classes end December 13; Final Exams are December 15-22, 2017

Here is a link to the algebra seminars in previous semesters

Abstracts of seminar talks

Spring 2020

Compactifications of moduli of points and lines in the projective plane (Luca Schaffler, April 29, 2020):
Projective duality identifies the moduli space Bn parametrizing configurations of n general points in projective plane with X(3,n), parametrizing configurations of n general lines in the dual plane. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces.
In this work, we consider Gerritzen-Piwek's compactification Bn and Kapranov's Chow quotient compactification X(3,n), and we show they have isomorphic normalizations.
We prove that Bn does not admit a modular interpretation claimed by Gerritzen and Piwek, namely a family of n-pointed central fibers of Mustafin joins associated to one-parameter degenerations of n points in the plane. We construct the correct compactification of Bn which admits such a family, and we describe it for n=5,6. This is joint work in progress with Jenia Tevelev.

Group cohomology rings via equivariant cohomology (James Cameron, April 22, 2020):
The cohomology rings of finite groups are typically very complicated, but their geometric properties are often tractable and retain representation theoretic information. These geometric properties become more clear once one considers group cohomology rings in the context of equivariant cohomology. In this talk I will discuss how to use techniques involving flag varieties dating back to Quillen and a filtration of equivariant cohomology rings due to Duflot to study the associated primes and local cohomology modules of group cohomology rings.
This talk will be online, using webex

Moduli spaces on the Kuznetsov component of Fano threefolds of index 2 (Franco Rota, April 15, 2020):
The derived category of a Fano threefold Y of Picard rank 1 and index 2 admits a semiorthogonal decomposition. This defines a non-trivial subcategory Ku(Y) called the Kuznetsov component, which encodes most of the geometry of Y.
I will present a joint work with M. Altavilla and M. Petkovic, in which we describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the stability conditions constructed by Bayer, Macri, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli space. As an application in the case of degree d=2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.
This talk will be online, using webex

Higher K-theory via generators and relations (Ian Coley, April 8, 2020):
K0 (the Grothendieck group) of an exact category has a nice description in terms of generators and relations. Nenashev (after Quillen and Gillet-Grayson) proved that K1 can also be described in terms of generators and relations, and Grayson extended that argument to all higher K-groups. I will sketch Grayson's argument and (ideally) show some advantages of the generators and relations approach.
This talk will be online, using webex

On the Jordan property for local fundamental groups (Joaquin Moraga, April 1, 2020):
We discuss the Jordan property for the local fundamental group of klt singularities. We also show how the existence of a large Abelian subgroup of such a group reflects on the geometry of the singularity. Finally, we show a characterization theorem for klt 3-fold singularities with large local fundamental group.
This talk will be online, using webex

Six explicit pairs of fake projective planes (Lev Borisov, February 26, 2020):
I will briefly review the history of fake projective planes and will talk about my latest work on the subject, joint with Enrico Fatighenti.

Quantum geometry of moduli spaces of local system (Linhui Shen, February 19, 2020):
Let G be a split semi-simple algebraic group over Q. We introduce a natural cluster structure on moduli spaces of G-local systems over surfaces with marked points. As a consequence, the moduli spaces of G-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the q-deformed Toda systems, quantum groups, and the modular functor conjecture for such representations. This talk will mainly be based on joint work with A.B. Goncharov.

The K'-theory of monoid sets (Chuck Weibel, February 5, 2020):
There are three flavors of K-theory for a pointed abelian monoid A; they depend on the A-sets one allows. This talk considers the well-behaved family of partially cancellative (pc) A-sets, and its K-theory. For example, if A is the natural numbers, then pc A-sets are just rooted trees.

Higher norm principles for norm varieties (Shira Gilat, January 22, 2020):
The norm principle for a division algebra states that the image of the reduced norm is an invariant of its Brauer-equivalence class. This can be generalized to symbols in the Milnor K-group KMn(F). We prove a generalized norm principle for symbols in KMn(F) for a prime-to-p closed field F of characteristic zero (for some prime p).  We also give a new proof for the norm principle for division algebras, using the decomposition theorem for (noncommutative) polynomials over the algebra.

Fall 2019

Tropical scheme theory (Diane Mclagan, December 11, 2019):
Tropical geometry can be viewed as algebraic geometry over the tropical semiring (R union infinity, with operations min and +). This perspective has proved surprisingly effective over the last decade, but has so far has mostly been restricted to the study of varieties and cycles. I will discuss a program to construct a scheme theory for tropical geometry. This builds on schemes over semirings, but also introduces concepts from matroid theory. This is joint work with Felipe Rincon, involving also work of Jeff and Noah Giansiracusa and others.

Enumerating pencils with moving ramification on curves, (Carl Lian, November 20, 2019):
We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E→ P1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

Noncommutative Laurent Phenomena: two examples (Volodia Retakh, November 13, 2019):
We discuss two examples when iterations of the noncommutative rational map are given by noncommutative Laurent polynomials. The first example is related to noncommutative triangulation of surfaces. The second example, which leads to a noncommutative version of the Catalan numbers, is related to solutions of determinant-like equations. The talk is based on joint papers with A. Berenstein from U. of Oregon.

Vertex algebras of CohFT-type (Angela Gibney, October 30, 2019):
Finitely generated admissible modules over "vertex algebras of CohFT-type" can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves. In this talk I will say what vertex algebras of CohFT-type are, and explain how such bundles define semisimple cohomological field theories.
As an application, one can give an expression for their total Chern character in terms of the fusion rules. I'll give some examples.

Strong exceptional collections of line bundles (Chengxi Wang, October 23, 2019):
We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of the stack, as long as the number of elements in the collection equals the rank of the (Grothendieck) K-theory group of the stack.
The problem reduces to an interesting combinatorial problem and is solved by combinatorial means.

Local-global principle for norm over semi-global fields, Sumit Chandra Mishra, Oct. 16, 2019):
Let K be a complete discretely valued field with residue field κ. Let F be a function field in one variable over K and X a regular proper model of F with reduced special fibre X a union of regular curves with normal crossings. Suppose that the graph associated to X is a tree (e.g. F = K(t)). Let L/F be a Galois extension of degree n with Galois group Gand n coprime to char(κ). Suppose that κ is algebraically closed field or a finite field containing a primitive nth root of unity. Then we show that an element in F* is a norm from the extension L/F if it is a norm from the extensions L⊗FFν (i.e., $L\otimes_F F_\nu/F_\nu$) for all discrete valuations ν of F.

What is a derivator? (Ian Coley, October 9, 2019):
Derivators were introduced in the 90s by Grothendieck, Heller, and Franke (independently) to generalize triangulated categories and answer questions in homotopy theory and algebraic geometry using a more abstract framework. Since then, applications to modular representation theory, tensor triangulated geometry, tilting theory, K-theory, equivariant homotopy theory, and more have been developed by scores of mathematicians.
This talk will give the basic definition of a derivator, motivated by the initial question of enhancing a triangulated category, describe some of these useful applications to the "real world" away from category theory. We assume a priori the listener's interest in triangulated category theory and one or more of the above disciplines. In particular, no knowledge of infinity/quasicategories is required!

Generalized Brauer dimension and other arithmetic invariants of semi-global fields (Saurabh Gosavi, October 2, 19):
Given a finite set of Brauer classes B of a fixed period ℓ, we define ind(B) to be the gcd of degrees of field extensions L/F such that α⊗FL=0 for every α in B. We provide upper-bounds for ind(B) which depends upon arithmetic invariants of fields of lower arithmetic complexity. As a simple application of our result, we will obtain upper-bounds for the splitting index of quadratic forms and finiteness of symbol length for function fields of curves over higher-local fields.

Rational singularities and their cousins in arbitrary characteristics (Sándor Kovacs, Sept. 18, 2019):
I will discuss several results about rational and closely related singularities in arbitrary characteristics. The results concern various properties of these singularities including their behavior with respect to deformations and degenerations, and applications to moduli theory.

On Makkai's Strong Conceptual Completeness Theorem (Jacob Lurie, Sept. 20, 2019):
One of the most fundamental results of mathematical logic is the celebrated Godel completeness theorem, which asserts that every consistent first-order theory T admits a model. In the 1980s, Makkai proved a much sharper result: any first-order theory T can be recovered, up to a suitable notion of equivalence, from its category of models Mod(T) together with some additional structure (supplied by the theory of ultraproducts). In this talk, I'll explain the statement of Makkai's theorem and sketch a new proof of it, inspired by the theory of "pro-etale sheaves" studied by Scholze and Bhatt-Scholze.

Spring 2019

Severi degrees via representation theory (Dave Anderson, May 1, 2019):
As a vector space, the cohomology of the Grassmannian Gr(k,n) is isomorphic to the k-th exterior power of C^n. The geometric Satake correspondence explains how to naturally upgrade this isomorphism to one of $gl_n$-representations. Inspired by work of Golyshev and Manivel from 2011, we use these ideas to find new proofs of Giambelli formulas for ordinary and orthogonal Grassmannians, as well as rim-hook rules for quantum cohomology. This is joint with Antonio Nigro.

Severi degrees via representation theory (Yaim Cooper, April 24, 2019):
The Severi degrees of $P^1$ x $P^1$ can be computed in terms of an explicit operator on the Fock space $F[P^1]$. We will discuss this and variations on this theme. We will explain how to use this approach to compute the relative Gromov-Witten theory of other surfaces, such as Hirzebruch surfaces and Ex$P^1$. We will also discuss operators for calculating descendants. Joint with R. Pandharipande.

Continuous families of divisors on symmetric powers of curves (John Sheridan, April 17, 2019):
For X a smooth projective variety, we consider its set of effective divisors in a fixed cohomology class. This set naturally forms a projective scheme and if X is a curve, this scheme is a smooth, irreducible variety (fibered in linear systems over the Picard variety). However, when X is of higher dimension, this scheme can be singular and reducible. We study its structure explicitly when X is a symmetric power of a curve.

Character Tables and Sylow Subgroups of Finite Groups (Gabriel Navarro, April 10, 2019):
Brauer's Problem 12 asks which properties of Sylow subgroups can be detected in the character table of a finite group. We will talk about recent progress on this problem.

Effective divisors in the Hodge bundle (Iulia Gheorghita, April 3, 2019):
Computing effective divisor classes can reveal important information about the geometry of the underlying space. For example, in 1982 Harris and Mumford computed the Brill-Noether divisor class and used it to determine the Kodaira dimension of the moduli space of curves.
In this talk I will explain how to compute the divisor class of the locus of canonical divisors in the projectivized Hodge bundle over the moduli space of curves which have a zero at a Weierstrass point. I will also explain the extremality of the divisor class arising from the stratum of canonical divisors with a double zero.

The algebraic theory of systems (Louis Rowen, March 27, 2019):
The notion of ``system'' is introduced to unify classical algebra with tropical mathematics, hyperfields, and other related areas for which we can embed a partial algebraic structure into a fuller structure from which we can extract more information. The main ideas are a generalized negation map since our structures lack classical negatives, and a ``surpassing relation'' to replace equality.
We discuss this theory with emphasis on the main applications, which will be described from the beginning:
1. Classical algebra
2. Supertropical mathematics (used for valuations and tropicalization)
3. Symmetrized systems (used for embedding additively idempotent semi structures into systems)
4. Hyperfields

A Degree Formula for Equivariant Cohomology (Jeanne Duflot, March 13, 2019):
I will talk about a generalization of a result of Lynn on the "degree" of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincaré series, expanded as a Laurent series about t=1. The main theorem, which is joint with Mark Blumstein, is an additivity formula for degree: $$\deg(H^*_G(X)) = \sum_{[A,c] \in \mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} \deg(H^*_{C_G(A,c)}(c)).$$

Volumes and intersection theory on moduli spaces of abelian differentials (Dawei Chen, February 20, 2019):
Computing volumes of moduli spaces has significance in many fields. For instance, the celebrated Witten's conjecture regarding intersection numbers on the Deligne-Mumford moduli space of stable curves has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten's conjecture by Kontsevich and by Okounkov-Pandharipande also used various ideas in ribbon graphs, Gromov-Witten theory, and Hurwitz theory.
In this talk I will introduce an analogous formula of intersection numbers on the moduli spaces of abelian differentials that computes the Masur-Veech volumes. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).

The Real graded Brauer group (Chuck Weibel, February 6, 2019):
We introduce a version of the Brauer--Wall group for Real vector bundles of algebras (in the sense of Atiyah), and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer--Wall and Witt groups.

Paschke Categories, K-homology and the Riemann-Roch Transformation (Khashayar Sartipi, January 30, 2019):
For a separable C*-algebra A, we introduce an exact C*-category called the Paschke Category of A, which is completely functorial in A, and show that its K-theory groups are isomorphic to the topological K-homology groups of the C*-algebra A. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold X, to its topological K-homology spectrum.

Palindromicity and the local invariant cycle theorem (Patrick Brosnan, January 23, 2019):
In its most basic form, the local invariant cycle theorem of Beilinson, Bernstein and Deligne (BBD) gives a surjection from the cohomology of the special fiber of a proper morphism of smooth varieties to the monodromy invariants of the general fiber. This result, which is one of the last theorems stated in the book by BBD, is a relatively easy consequence of their famous decomposition theorem.
In joint work with Tim Chow on a combinatorial problem, we needed a simple condition ensuring that the above surjection is actually an isomorphism. Our theorem is that this happens if and only if the special fiber has palindromic cohomology. I will explain the proof of this theorem and a generalization proved using the (now known) Kashiwara conjecture. I will also say a little bit about the combinatorial problem (the Shareshian-Wachs conjecture on Hessenberg varieties) which motivated our work.

Fall 2018

2-Segal spaces and algebraic K-theory (Julie Bergner, November 28, 2018):
The notion of a 2-Segal space was defined by Dyckerhoff and Kapranov and independently by Galvez-Carrillo, Kock, and Tonks under the name of decomposition space. Although these two sets of authors had different motivations for their work, they both saw that a key example is obtained by applying Waldhausen's S-construction to an exact category, showing that 2-Segal spaces are deeply connected to algebraic K-theory.
In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we show that any 2-Segal space arises from a suitable generalization of this construction. Furthermore, our generalized input has a close relationship to the CGW categories of Campbell and Zakharevich. In this talk, I'll introduce 2-Segal structures and discuss what we know and would like to know about the role they play in algebraic K-theory.

Low Degree Cohomology (Bob Guralnick November 14, 2018):
Let G be a finite group with V an absolutely irreducible kG-module with k a field of positive characteristic. We are interested in bounds on the dimension of the first and second degree cohomology groups of G with coefficients in V. We will discuss some old and new bounds, conjectures and applications.

Low degree points on curves (Isabel Vogt, November 7, 2018):
We will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer $e$ such that the points of residue degree bounded by $e$ are infinite. By work of Faltings, Harris-Silverman and Abramovich-Harris, it is understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria do not generalize to $e$ at least 4. We will focus on scenarios under which we can guarantee that this invariant is actually equal to the gonality using the auxiliary geometry of a surface containing the curve. This is joint work with Geoffrey Smith.

Algebraic groups with good reduction and unramified cohomology (Igor Rapinchuk, October 31, 2018):
Let $G$ be an absolutely almost simple algebraic group over a field K, which we assume to be equipped with a natural set V of discrete valuations. In this talk, our focus will be on the K-forms of $G$ that have good reduction at all v in V . When K is the fraction field of a Dedekind domain, a similar question was considered by G. Harder; the case where $K=\mathbb{Q}$ and V is the set of all p-adic places was analyzed in detail by B.H. Gross and B. Conrad. I will discuss several emerging results in the higher-dimensional situation, where K is the function field $k(C)$ of a smooth geometrically irreducible curve $C$ over a number field k, or even an arbitrary finitely generated field.

These problems turn out to be closely related to finiteness properties of unramified cohomology, and I will present available results over various classes of fields. I will also highlight some connections with other questions involving the genus of $G$ (i.e., the set of isomorphism classes of K-forms of $G$ having the same isomorphism classes of maximal K-tori as $G$), Hasse principles, etc. The talk will be based in part on joint work with V. Chernousov and A. Rapinchuk

Generalized Polar Geometry (Sandra Di Rocco, October 24, 2018):
Polar classes are very classical objects in Algebraic Geometry. A brief introduction to the subject will be presented and ideas and preliminarily results towards generalizations will be explained. These ideas can be applied towards variety sampling and relevant applications in Kinematics and Biochemistry.

Brauer class over the Picard scheme of curves (Qixiao Ma, October 24, 2018):
We study the Brauer class rising from the obstruction to the existence of a tautological line bundle on the Picard scheme of curves. If we consider the universal totally degenerate curve with a fixed dual graph, then, using symmetries of the graph, we give bounds on the period and index of the Brauer classes. As a result, we provide some division algebra of prime degree, serving as candidates for the cyclicity problem.

On singularity properties of convolutions of algebraic morphisms (Yotam Hendel, October 10, 2018):
In analysis, the convolution of two functions results in a smoother, better behaved function. It is interesting to ask whether an analogue of this phenomenon exists in the setting of algebraic geometry. Let $f$ and $g$ be two morphisms from algebraic varieties X and Y to an algebraic group $G$. We define their convolution to be a morphism $f*g$ from $X\times Y$ to $G$ by first applying each morphism and then multiplying using the group structure of $G$.

In this talk, we present some properties of this convolution operation, as well as a recent result which states that after finitely many self convolutions every dominant morphism $f:X\to G$ from a smooth, absolutely irreducible variety X to an algebraic group G becomes flat with reduced fibers of rational singularities (this property is abbreviated FRS). The FRS property is of particular interest since by works of Aizenbud and Avni, FRS morphisms are characterized by having fibers whose point count over the finite rings $Z/p^kZ$ is well-behaved. This leads to applications in probability, group theory, representation growth and more. We will discuss some of these applications, and if time permits, the main ideas of the proof which utilize model-theoretic methods. Joint work with Itay Glazer.

Irrationality of Motivic Zeta Functions (Michael Larsen, October 5, 2018):
It is a remarkable fact that the Riemann zeta function extends to a meromorphic function on the whole complex plane. A conjecture of Weil, proved by Dwork, asserts that the zeta function of any variety over a finite field is likewise meromorphic, from which it follows that it can be expressed as a rational function. In the case of curves, Kapranov observed that this is true in a very strong sense, which continues to hold even in characteristic zero. He asked whether this remains true for higher dimensional varieties. Valery Lunts and I disproved his conjecture fifteen years ago, and recently disproved a weaker conjecture due to Denef and Loeser. This explains, in some sense, why Weil's conjecture was so much easier in dimension 1 than in higher dimension.

Spring 2018

Singularities in geometry, topology and strings, Antonella Grassi, May 9, 2018):
I will discuss a "Brieskorn-Grothendieck" program involving certain singularities, Lie algebras and representations. These singularities arise in many different areas of mathematics and physics. I will focus on the case of complex 3 dimensional spaces relating to algebraic geometry, topology and physics. I will disucss local, global and local-to-global properties of threefolds with certain singularities and crieteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincar\'e duality.
We state a conjecture on the extension of Kodaira's classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal genus one fibered varieties and we give results in this direction.

Hodge theory and o-minimal geometry (Benjamin Bakker, May 2, 2018):
Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients $D/\Gamma$ of periods domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping $D/\Gamma$ with an o-minimal structure, and show that period maps are "definable" with respect to this structure.
As a consequence, we obtain an easy proof of a result of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci, a strong piece of evidence for the Hodge conjecture. The proof of the main theorem relies heavily on work of Schmid, Kashiwara, and Cattani--Kaplan--Schmid on the asymptotics of degenerations of Hodge structures. This is joint work with B. Klingler and J. Tsimerman.

Singularities of general fibers in positive characteristic (Joe Waldron, April 18, 2018):
Generic smoothness fails to hold for some fibrations in positive characteristic. We study consequences of this failure, in particularly by obtaining a canonical bundle formula relating a fiber with the normalization of its maximal reduced subscheme. This has geometric consequences, including that generic smoothness holds on terminal Mori fiber spaces of relative dimension two in characteristic $p\geq 11$. This is joint work with Zsolt Patakfalvi.

Conformal blocks attached to twisted groups (Chiara Damiolini, April 11, 2018):
Let $G$ be a simple and simply connected algebraic group over $\mathbb{C}$. We can attach to $G$ the sheaf of conformal blocks: a vector bundle on $M_{g}$ whose fibres are identified with global sections of a certain line bundle on the stack of $G$-torsors. We generalize the construction of conformal blocks to the case in which $\mathcal{G}$ is a twisted group over a curve which can be defined in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz space and have properties analogous to the classical case.

Derived categories of moduli spaces of stable rational curves (Ana-Maria Castravet, March 28, 2018):
A question of Manin is whether the derived category of the Grothendieck-Knudsen moduli space $M_{0,n}$ of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group $S_n$. I will present an approach towards answering this question. In particular, I will explain a construction of an invariant full exceptional collection on the Losev-Manin space. This is joint work with Jenia Tevelev.

The indeterminacy of universal Abel-Jacobi sections (Nicola Pagani, March 14, 2018):
The (universal) Abel-Jacobi maps are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space $M_{g,n}$ of smooth pointed curves. When the source and target moduli spaces are compactified, these sections are only rational maps, and it is natural to ask for the largest locus where each of them is a well-defined morphism. We explicitly characterize this locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification $\bar{M}_{g,n}$ by means of stable curves). In particular, we deduce that for every Abel-Jacobi map there exists a compactification of the universal Jacobian such that the map extends to a well-defined morphism on $\bar{M}_{g,n}$. We apply this to the problem of defining and computing several different extensions to $\bar{M}_{g,n}$ of the double ramification cycle (= the locus of smooth pointed curves that admit a meromorphic function with prescribed zeroes and poles at the points).
This is a joint work with Jesse Kass.

Algebraic dynamics from topological and holomorphic dynamics (Rohini Ramadas, Feb. 28, 2018):
Let $f:S^2 \to S^2$ be an orientation-preserving branched covering from the 2-sphere to itself whose postcritical set $P := \{ f^n(x) | x\ \mathrm{is\ a\ critical\ point\ of\ f\ and}\ n>0 \}$ is finite. Thurston studied the dynamics of $f$ using an induced holomorphic self-map $T(f)$ of the Teichmuller space of complex structures on $(S^2, P)$. Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems:
1. $T(f)$ always descends to a multivalued self map $H(f)$ of the moduli space $M_{0,P}$ of markings of the Riemann sphere by the finite set $P$
2. When $P$ contains a point $x$ at which $f$ is fully ramified, under certain combinatorial conditions on $f$, the inverse of $T(f)$ descends to a rational self-map $M(f)$ of projective space $P^n$. When, in addition, $x$ is a fixed point of $f$, i.e. $f$ is a `topological polynomial', the induced self-map $M(f)$ is regular.
The dynamics of $H(f)$ and $M(f)$ may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties.
I will introduce the dynamical systems $T(f)$, $H(f)$ and $M(f)$, and dynamical degrees. I will then discuss why it is useful to study $H(f)$ (resp. $M(f)$) simultaneously on several compactifications of $M_{0,P}$. We find that the dynamical degrees of $H(f)$ (resp. $M(f)$) are algebraic integers whose properties are constrained by the dynamics of $f$ on the finite set $P$. In particular, when $M(f)$ exists, then the more $f$ resembles a topological polynomial, the more $M(f): P^n \to P^n$ behaves like a regular map.

Curves, maps, and singularities in genus one (Dhruv Ranganathan, February 21, 2018):
I will outline a new framework based on tropical and logarithmic methods to study genus one curve singularities and discuss its relationship with the geometry of moduli spaces. I will focus on two applications of these ideas.
First, they allow one to explicitly factorize the rational maps among log canonical models of the moduli space of n-pointed elliptic curves. Second, they reveal a modular interpretation for Vakil and Zinger's famous desingularization of the space of elliptic curves in projective space, a short conceptual proof of that result, and several new generalizations.
Time permitting, though it rarely does, I will mention some applications to both classical and virtual enumerative geometry. This is based on work with Len and with Santos-Parker and Wise, as well as ongoing work with Battistella and Nabijou.

Equations of Cartwright-Steger surface (Lev Borisov, February 7, 2018):
Cartwright-Steger surface is an algebraic surface of general type which appeared in the study of fake projective planes. I will describe the technique that allowed us to find equations of it, in its bicanonical embedding. This is a joint work with Sai Kee Yeung.

An arithmetic site of Connes-Consani type for Gaussian integers (Aurélien Sagnier, Jan. 24, 2018):
Connes and Consani proposed to study the action of the multiplicative monoid of positive integers $\mathbb{N}^\times$ on the tropical semiring $(\mathbb{Z},max,+)$, as an approach to the Riemann zeta function. This construction depends upon the ordering on the reals. I will first explain their approach, then give an extension of this construction to the Gaussian integers.

Fall 2017

Gonality of complete intersection curves (Brooke Ullery, Dec. 13, 2017:
The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding.
In my talk, I will discuss recent work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of $\mathbb P^1$ arises in this way.

Equivariant K-theory associated to Kac-Moody groups (Seth Baldwin, Dec. 6, 2017):
The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

Local-global principles for rational points and zero-cycles (Julia Hartmann, Nov. 15, 2017):
Given a variety over a field $F$ and a collection of overfields of $F$, one may ask whether the existence of rational points over each of the overfields (local) implies the existence of a rational point over $F$ (global). Such local-global pinciples are a main tool for understanding the existence of rational points on varieties.
     In this talk, we study varieties that are defined over semi-global fields, i.e., function fields of curves over a complete discretely valued field. A semi-global field admits several natural collections of overfields which are geometrically motivated, and one may ask for local-global principles with respect to each such collection. We exhibit certain cases in which local-global principles for rational points hold. We also show that local-global principles for zero-cycles of degree one hold provided that local-global principles hold for the existence of rational points over extensions of the function field. This last assertion is analogous to a known result for varieties over number fields.
(Joint work with J.-L. Colliot-Thélène, D. Harbater, D. Krashen, R. Parimala, and V. Suresh)

Explicit equations of a fake projective plane (Lev Borisov, Nov. 8, 2017):
Fake projective planes are complex algebraic surfaces of general type whose Betti numbers are the same as that of a usual projective plane. The first example was constructed by Mumford about 40 years ago by 2-adic uniformization. There are 50 complex conjugate pairs of such surfaces, given explicitly as ball quotients (Cartwright+Steger). However, a ball quotient description does not on its own lead to an explicit projective embedding. In a joint work with JongHae Keum, we find equations of one pair of fake projective planes in bicanonical embedding, which is so far the only result of this kind.

Birational geometry of moduli spaces of parabolic bundles (Han-Bom Moon, October 18, 2017):
I will describe a project on birational geometry of the moduli space of parabolic bundles on the projective line in the framework of Mori's program, and its connection with classical invariant theory and conformal blocks. This is joint work with Sang-Bum Yoo.

Linear Systems on General Curves of Fixed Gonality (David Jensen, Oct. 4, 2017:
The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods.
This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Character levels and character bounds (Pham Huu Tiep, September 27, 2017):
We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give various characterizations of the level of a character in terms of its Lusztig's label, of its degree, and of certain dual pairs. This concept is then used to prove exponential bounds for character values, provided that either the level of the character or the centralizer of the element is not too large. This is joint work with R. M. Guralnick and M. Larsen.

K-classes of Brill-Noether loci and a determinantal formula (Nicola Tarasca, September 20, 2017):
I will present a formula for the Euler characteristic of the structure sheaf of Brill-Noether loci of linear series on curves with prescribed vanishing at two marked points.
The formula recovers the classical Castelnuovo number in the zero-dimensional case, and previous work of Eisenbud-Harris, Pirola, Chan-López-Pflueger-Teixidor in the one-dimensional case. The result follows from a new determinantal formula for the K-theory class of certain degeneracy loci of maps of flag bundles.
This is joint work with Dave Anderson and Linda Chen.

A general algebraic structure theory for tropical mathematics (Louis Rowen, September 13, 2017):
We study triples (A,T,-) of a set A with algebraic structure (such as a semiring), a subset T and a negation operator '-' on T. A key example is the max-plus algebra T. This viewpoint enables one to view the tropicalization functor as a morphism, suggesting tropical analogs of classical structures such as Grassmann algebras, Lie algebras, Lie superalgebras, Poisson algebras, and Hopf algebras.

Charles Weibel / weibel @ / January 1, 2020