RUTGERS ALGEBRA SEMINAR - Fall 2018
(Wednesdays at 2:00 in H525)

The Algebra Seminar meets on Wednesdays, at 2:00-3:00PM in room H525
         (in the Hill Center, on Busch Campus of Rutgers University).
A more comprehensive listing of all Math Department seminars is available.
Here is a link to the algebra seminars in previous semesters

Spring 2019 Seminars (Wednesdays at 2:00 in H005)
23 Jan Patrick Brosnan  U.Maryland "Palindromicity and the local invariant cycle theorem"
30 Jan 
 6 Feb
13 Feb
20 Feb
27 Feb
 6 Mar
13 Mar
20 Mar  no seminar      ------------------- Spring Break ----------
27 Mar
 3 Apr
10 Apr
17 Apr
24 Apr
 1 May
Classes end Wednesday 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"
 7 Mar CANCELLED DUE TO SNOW 
15 Mar Nicola Pagani U.Liverpool "The indeterminacy of universal Abel-Jacobi sections" 
21 Mar CANCELLED DUE TO SNOW 
28 Mar Ana-Maria Castravet Northeastern U. "Derived categories of moduli spaces of stable rational curves"
 4 Apr CANCELLED DUE TO SNOW
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

Spring 2017 Seminars (Wednesdays at 2:00 in H705)
22 Feb Ryan Shifler  Virginia Tech "Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian"
 1 Mar Chuck Weibel    Rutgers   "The Witt group of surfaces and 3-folds"
 8 Mar Oliver Pechenik Rutgers   "Decompositions of Grothendieck polynomials"
15 Mar no seminar      ------------------- Spring Break ----------
22 Mar Ilya Kapovich UIUC/Hunter College "Dynamics and polynomial invariants for free-by-cyclic groups"
29 Mar Rachel Levanger Rutgers "Interleaved persistence modules and applications of persistent homology to problems in fluid dynamics"
 5 Apr Cristian Lenart Albany-SUNY "Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications"
19 Apr Anders Buch     Rutgers     "Puzzles in quantum Schubert calculus"
26 Apr Sjuvon Chung    Rutgers     "Equivariant quantum K-theory of projective space"
Classes end May 1; Final Exams are May 4-10, 2017
Here is a link to the algebra seminars in previous semesters

Abstracts of seminar talks


Spring 2019


Palindromicity and the local invariant cycle theorem (Patrick Brosnan, December 12, 2018):
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.

Spring 2017


Equivariant quantum K-theory of projective space (Sjuvon Chung, April 26, 2017):
Recent developments of Buch-Chaput-Mihalcea-Perrin have allowed for a closer look at the quantum K-theory of cominuscule flag varieties. For example, their Chevalley formula allows one to compute quantum K-theoretic products involving Schubert divisor classes. In the special case of projective space, one can extend this Chevalley formula to describe products of arbitrary Schubert classes. We shall discuss this extension along with some of its potential combinatorial and representation-theoretic consequences.


Schubert calculus (Anders Buch, April 19, 2017):
The cohomology ring of a flag variety has a natural basis of Schubert classes. The multiplicative structure constants with respect to this basis count solutions to enumerative geometric problems; in particular they are non-negative. For example, the structure constants of a Grassmannian are the classical Littlewood-Richardson coefficients, which show up in numerous branches of mathematics.
I will speak about a new puzzle-counting formula for the structure constants of 3-step partial flag varieties that describes products of classes that are pulled back from 2-step flag varieties. By using a relation between quantum cohomology of Grassmannians and classical cohomology of 2-step flag varieties, this produces a new combinatorial formula for the (3 point, genus zero) Gromov-Witten invariants of Grassmannians, which is in some sense more economical than earlier formulas.


Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications (Christian Lenart, April 5, 2017):
This talk is largely self-contained.
In a series of papers with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we developed a uniform combinatorial model for (tensor products of one-column) Kirillov-Reshetikhin (KR) modules of affine Lie algebras. We also showed that their graded characters coincide with the specialization of symmetric Macdonald polynomials at $t=0$, and extended this result to non-symmetric Macdonald polynomials. I will present a survey of this work and of the recent applications, which include: computations related to KR crystals, crystal bases of level 0 extremal weight modules, Weyl modules (local, global, and generalized), $q$-Whittaker functions, and the quantum $K$-theory of flag varieties.


Dynamics and polynomial invariants for free-by-cyclic groups (Ilya Kapovich, March 22, 2017):
We develop a counterpart of the Thurston-Fried-McMullen "fibered face" theory in the setting of free-by-cyclic groups, that is, mapping tori groups of automorphisms of finite rank free groups. We obtain information about the BNS invariant of such groups, and construct a version of McMullen's "Teichmuller polynomial" in the free-by-cyclic context. The talk is based on joint work with Chris Leininger and Spencer Dowdall.


Decompositions of Grothendieck polynomials (Oliver Pechenik, March 8, 2017):
Finding a combinatorial rule for the Schubert structure constants in the K-theory of flag varieties is a long-standing problem. The Grothendieck polynomials of Lascoux and Sch├╝tzenberger (1982) serve as polynomial representatives for K-theoretic Schubert classes, but no positive rule for their multiplication is known outside of the Grassmannian case.
We contribute a new basis for polynomials, give a positive combinatorial formula for the expansion of Grothendieck polynomials in these "glide polynomials", and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. A specialization of the glide basis recovers the fundamental slide polynomials of Assaf and Searles (2016), which play an analogous role with respect to the Chow ring of flag varieties. Additionally, the stable limits of another specialization of glide polynomials are Lam and Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, K-theoretic analogues of I. Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials. (Joint work with D. Searles).


The Witt group of surfaces and 3-folds (Chuck Weibel, March 1, 2017):
If V is an algebraic variety, the Witt group is formed from vector bundles equipped with a nondegenerate symmetric bilinear form. When it has dimension <4, it embeds into the more classical Witt group of the function field (Witt 1934). When V is defined over the reals, versions of the discriminant and Hasse invariant enable us to determine W(V).


Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian (Ryan Shifler, February 23, 2017):
The odd symplectic Grassmannian IG:=IG(k, 2n+1) parametrizes k dimensional subspaces of $\mathbb{C}^{2n+1}$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n+2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k=2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring. The current work is joint with L. Mihalcea.


Charles Weibel / weibel @ math.rutgers.edu / January 1, 2019