Wednesdays at 2:00

(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

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

The Fall 2019 seminar will be in Hill 525

Fall 2019 classes begin Tuesay September 3 and end Wednesday Dec.11.

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 POSTPONEDClasses end Wednesday Dec. 12; Finals begin Dec. 15, 2018

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

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

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

**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.

**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.

**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 $G$.

**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.

**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.

**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.

**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 Schützenberger (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