(Wednesdays at 2:00 in H705)

(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

07 Sept 23 Nov --- no seminar --- Thanksgiving is Nov. 24; Friday class schedule 14 Dec Classes start Sept 7 and end December 14; Final Exams are December 16-23, 2016

20 Jan Louis Rowen Bar-Ilan Univ "Symmetrization in tropical algebra" 3 Feb Volodia Retakh Rutgers "Generalized adjoint actions" 10 Feb Omer Bobrowski Duke (@noon!) "Random Topology and its Applications" 17 Feb Lisa Carbone Rutgers "Arithmetic constructions of hyperbolic Kac-Moody groups" 2 Mar Chuck Weibel Rutgers "Relative Cartier divisors" 9 Mar Lev Borisov Rutgers "Elliptic genera of singular varieties and related topics" 16 Mar no seminar ------------------- Spring Break ---------- 23 Mar Rachel Levanger Rutgers "Auslander-Reiten quivers of finite-dimensional algebras" 30 Mar Richard Lyons Rutgers "Aspects of the Classification of simple groups" 6 Apr Richard Lyons Rutgers "Aspects of the Classification, continued" 13 Apr Siddhartha Sahi Rutgers "Eigenvalues of generalized Capelli operators" 20 Apr Ed Karasiewicz Rutgers "Some Aspects of p-adic Representations & the Casselman-Shalika Formula" 27 Apr Semeon Artamonov Rutgers "Noncommutative Poisson Geometry" Classes end May 2; Final Exams are May 4-10

7 Oct Chuck Weibel Rutgers "Monoids, monoid rings and monoid schemes" 14 Oct Lev Borisov Rutgers "Introduction to A-D-E singularities" 21 Oct Dylan Allegretti Yale "Quantization of Fock and Goncharov's canonical basis" 28 Oct Volodia Retakh Rutgers "Noncommutative Cross Ratios" 4 Nov Gabriele Nebe U.Aachen "Automorphisms of extremal codes" 11 Nov Chuck Weibel Rutgers "Relative Cartier divisors and polynomials" 18 Nov Glen Wilson Rutgers "Motivic stable homotopy over finite fields" 25 Nov --- no seminar --- Thanksgiving is Nov. 26; Friday class schedule 2 Dec Anders Buch Rutgers "The Thom Porteous formula" 9 Dec Pham Huu Tiep U. Arizona "Representations of finite groups and applications " Classes end Dec. 10; Final Exams are December 15-22

27 Jan --- 4 Feb Jesse Wolfson Chicago "The Index Map and Reciprocity Laws for Contou-Carrère Symbols" 18 Feb Justin Lynd Rutgers "Fusion systems and centric linking systems" 25 Feb Lev Borisov Rutgers "Zero divisors in the Grothendieck ring of varieties" 4 Mar Volodia Retakh Rutgers "Noncommutative triangulations and the Laurent phenomenon" 6 MarC Burt Totaro UCLA/IAS "Birational geometry and algebraic cycles" (Colloquium) 11 Mar Anders Buch Rutgers "TK" 18 Mar no seminar ------------------- Spring Break ------------------ 22 Apr Howard Neuer Rutgers "On special cubic 4-folds" Classes end May 4; Spring Final Exams are May 7-13

17 Sep Edwin Beggs U.Swansea "Semiclassical approximation to noncommutative Riemannian geometry" 24 Sep Anders Buch Rutgers "Equivariant quantum cohomology and puzzles" 8 Oct Lev Borisov Rutgers "Cut and paste approaches to rationality of cubic fourfolds" 15 Oct Chuck Weibel Rutgers "The Witt group of real varieties" 22 Oct Ed Karasiewicz Rutgers "Jacobians of modular curves" 29 Oct Charlie Siegel (IPMU Japan) "A Modular Operad of Embedded Curves" 5 Nov no seminar 12 Nov Marvin Tretkoff Texas A&M "Some non-compact Riemann surfaces branched over three points" 19 Nov Ashley Rall U. Virginia "Property T for Kac-Moody groups" 26 Nov (Thanksgiving is Nov. 27) no seminar 3 Dec Alex Lubotzky NYU/Hebrew U. (Israel) "Sieve methods in group theory"

Here is a link to the algebra seminars in previous semesters

**Auslander-Reiten quivers of finite-dimensional algebras
(Rachel Levanger, March 9, 2016):**

We summarize the construction of Auslander-Reiten quivers for
finite-dimensional algebras over an algebraically closed field.
We give an example in the category of commutative diagrams of
vector spaces.

**Elliptic genera of singular varieties and related topics
(Lev Borisov, March 9, 2016):**

A two-variable (Krichever-Hohn) elliptic genus is an invariant
of complex compact manifolds. It associates to such manifold $X$ a
function in two variables. I will describe the various properties of
elliptic genus. In particular, I will explain why it is a (weak) Jacobi
modular form if the canonical class of $X$ is numerically trivial. I will
then talk about extensions of the elliptic genus to some singular
varieties.

**Relative Cartier divisors (Chuck Weibel, March 2, 2016):**

If $B/A$ is a commutative ring extension, we consider the group
$I(B/A)$ of invertible $A$-submodules of $B$. If $A$ is a domain and
$B$ is its field of fractions, this is the usual Cartier divisor group.
The group $I(B[x]/A[x])$ has a very interesting structure, one which
is related to $K$-theory.

**Arithmetic constructions of hyperbolic Kac-Moody groups
(Lisa Carbone, Feb. 17, 2016):**

Tits defined Kac-Moody groups over commutative rings, providing
infinite dimensional analogues of the Chevalley-Demazure group
schemes. Tits' presentation can be simplified considerably when the
Dynkin diagram is hyperbolic and simply laced. In joint work with
Daniel Allcock, we have obtained finitely many generators and defining
relations for simply laced hyperbolic Kac-Moody groups over $\mathbb{Z}$. We
compare this presentation with a representation theoretic construction
of Kac-Moody groups over $\mathbb{Z}$. We also present some preliminary results
with Frank Wagner about uniqueness of representation theoretic
hyperbolic Kac-Moody groups.

**Generalized adjoint actions (Volodia Retakh, Feb. 3, 2016:**

We generalize the classical formula for expanding the
conjugation of $y$ by $\exp(x)$ by replacing $\exp(x)$ with any formal power
series. We also obtain combinatorial applications to $q$-exponentials,
$q$-binomials, and Hall-Littlewood polynomials.

(This is joint work with A. Berenstein from U. of Oregon.)

**Symmetrization in tropical algebra (Louis Rowen, Jan. 21, 2015):**

Tropicalization involves an ordered group, usually taken to be
$(\mathbb R, +)$ or $(\mathbb Q, +)$, viewed as a semifield.
Although there is a rich theory arising from this
viewpoint, idempotent semirings possess a restricted algebraic
structure theory, and also do not reflect important
valuation-theoretic properties, thereby forcing researchers to rely
often on combinatoric techniques.

A max-plus algebra not only lacks negation, but it is not even
additively cancellative. We introduce a general way to artificially
insert negation, similar to group completion. This leads
to the possibility of defining many auxiliary tropical structures,
such as Lie algebras and exterior algebras, and also providing a key
ingredient for a module theory that could enable one to use standard
tools such as homology.

In the first part of the talk we will survey some recent results on representations of finite groups. In the second part we will discuss applications of these results to various problems in group theory, number theory, and algebraic geometry.

**Relative Cartier divisors and polynomials
(Charles Weibel, Nov. 11, 2015):**

If A is a subring of a commutative ring B, a *relative Cartier
divisor* is an invertible A-submodule of B. These divisors form a group $I(A,B)$
related to the units and Picard groups of A and B. We decompose the
groups $I(A[t],B[t])$ and $I(A[t,1/t],B[t,1/t])$ and relate this
construction to the global sections of an étale sheaf.
This is joint work with Vivek Sadhu.

**Automorphisms of extremal codes
(Gabriele Nebe, Nov. 4, 2015):**

Extremal codes are self-dual binary codes with largest possible minimum
distance. In 1973 Neil Sloane published a short note asking whether there
is an extremal code of length 72. Since then many mathematicians search
for such a code, developing new tools to narrow down the structure of
its automorphism group. We now know that, if such a code exists,
then its automorphism group has order ≤5.

**Noncommutative Cross Ratios
(Volodia Retakh, Oct. 28, 2015):**

This is an introductory talk aimed at graduate students.
We will introduce cross ratios and use them to define a
noncommutative version of the Shear coordinates used in theoretical physics.

**Quantization of Fock and Goncharov's canonical basis
(Dylan Allegratti, Oct. 21, 2015):**

In a famous paper from 2003, Fock and Goncharov defined a version of the
space of $PGL_2(\mathbb C)$-local systems on a surface and showed that the
algebra of functions on this space has a canonical basis parametrized by
points of a dual moduli space. This algebra of functions can be canonically
quantized, and Fock and Goncharov conjectured that their canonical basis
could be deformed to a canonical set of elements of the quantized algebra.
In this talk, I will describe my recent work with Hyun Kyu Kim proving
Fock and Goncharov's conjecture.

**Introduction to A-D-E singularities
(Lev Borisov, Oct. 14, 2015):**

This is an introductory talk aimed at graduate students. ADE
singularities are remarkable mathematical objects which are studied from
multiple perspectives. They are indexed by the so-called Dynkin diagrams
$A_n$, $D_n$, $E_6$, $E_7$, $E_8$ and can be viewed as quotients of a
two-dimensional complex space $\mathbb C^2$ by a finite subgroup of the special
linear group $SL_2(\mathbb C)$. I will explain this correspondence as well as the
relationship between ADE singularities and the Platonic solids.

**Monoids, monoid rings and monoid schemes
(Chuck Weibel, Oct. 7, 2015):**

This is an introductory talk aimed at graduate students.
If $A$ is a pointed abelian monoid, we can talk about the topological
space of prime ideals in $A$, the monoid ring $k[A]$
and the topological space Spec(k[A]). Many of the theorems about
commutative rings have analogues for monoids, and just as schemes
are locally Spec(R), we can define monoid schemes.
I will explain some of the neaterr aspects of this dictionary.

(Jesse Wolfson, Feb. 4, 2015):

In the 1960s, Atiyah and Janich constructed a natural "index" map from the space of Fredholm operators on Hilbert space to the classifying space of topological K-theory. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. The index map allows us to relate the Contou-Carrère symbol, a local analytic invariant of families of schemes, to algebraic K-theory. Using this, we prove reciprocity laws for Contou-Carrère symbols in all dimensions. This extends previous results, of Anderson and Pablos Romo in dimension 1, and of Osipov and Zhu, in dimension 2.

**Zero divisors in the Grothendieck ring of varieties
(Lev Borisov, Feb. 25, 2015):**

I will explain the motivation and results of my recent preprint
that shows that the class of the affine line is a zero divisor in the
Grothendieck ring of varieties.

(Edwin Beggs, Sept. 17, 2014):

I will consider the first order deformation of a Riemannian manifold, including the vector bundles, differential calculus and metric. One example will be the Schwarzschild solution, which illustrates that not all the properties of the classical case can be simply carried into the quantum case. The other example is quantising the Kahler manifold, complex projective space. This case is much simpler, and here the complex geometry is also preserved. I will end with some comments on the connection between noncommutative complex geometry and noncommutative algebraic geometry.

**Equivariant quantum cohomology and puzzles
(Anders Buch, Sept. 24, 2014):**

The "classical equals quantum" theorem states that any equivariant
Gromov-Witten invariant (3 point, genus zero) of a Grassmann variety
can be expressed as a triple intersection of Schubert classes on a
two-step partial flag variety. An equivariant triple intersection on
a two-step flag variety can in turn be expressed as a sum over puzzles
that generalizes both Knutson and Tao's puzzle rule for Grassmannians
and the cohomological puzzle rule for two-step flag varieties. These
results together give a Littlewood-Richardson rule for the equivariant
quantum cohomology of Grassmannians. I will speak about geometric and
combinatorial aspects of this story, which is based on papers with
Kresch, Purbhoo, Mihalcea, and Tamvakis.

**Cut and paste approaches to rationality of cubic fourfolds
(Lev Borisov, Oct. 8, 2014):**

A random 4-dimensional hypersurface of degree 3 is widely
expected to be nonrational, but no proof of this statement currently
exists. Moreover, there is no clear understanding precisely which such
fourfolds are ratonal. An approach to this problem developed in a recent
preprint of Galkin and Shinder gives an unexpected necessary condition of
rationality modulo a variant of cancellation conjecture. This is a
surprisingly short and clean argument that involves the Grothendieck ring
of varieties. I will aim to make my talk accessible to the audience with
very limited algebraic geometry background.

**The Witt group of real varieties (Chuck Weibel, Oct. 15, 2014):**

We approximate the Witt groups of a variety V over the reals,
using a topological invariant: the Witt groups
of Real vector bundles on the space of complex points of V.
This is a better approximation than one might expect, and
has the advantage of being finitely generated.
This is joint work with Max Karoubi.

**Jacobians of modular curves (Ed Karasiewicz, Oct. 22, 2014):**

We study the Jacobian variety of a modular curve $C$
over an elliptic curve, and its Hecke operators.
The goal is to show that the $L$-function of a weight 2 cusp form
for $C$ is the same as the $L$-function of the elliptic curve.

**A Modular Operad of Embedded Curves (Charles Siegel, Oct. 29, 2014):**

Modular operads were introduced by Getzler and Kapranov to formalize the
structure of gluing maps between moduli of stable marked curves. We present
a construction of analogous gluing maps between moduli of
pluri-log-canonically embedded marked curves, which fit together to give
a modular operad of embedded curves.
This is joint work with Satoshi Kondo and Jesse Wolfson.

**Some non-compact Riemann surfaces branched over three points
(Marvin Tretkoff, Nov. 12, 2014):**

Recall that the Riemann surface of the multi-valued function $\log(z)$ is an infinite sheeted covering of the z-sphere branched over the two points $z=0$ and $z=\infty$ and has an ''infinite spiral ramp'' over each of them. Consequently, its monodromy group is infinite cyclic.

Today, we construct Riemann surfaces as infinite sheeted coverings of the z-sphere that are branched over precisely three points on the z-sphere. Moreover, each of these Riemann surfaces has a single ''infinite spiral ramp'' over each of its branch points. The monodromy groups of such surfaces are infinite two generator groups of permutations of the set of integers. Our construction yields many non-isomorphic groups with varying algebraic properties. In this lecture, we shall discuss one of these in some detail.

**Property T for Kac-Moody groups (Ashley Rall, Nov. 19, 2014):**

I will give a brief introduction to Kac-Moody groups, infinite dimensional
analogues of Chevalley groups, and Kazhdan's property (T) and then discuss
joint work with Mikhail Ershov establishing property (T) for Kac-Moody groups
over rings. We expand upon previous results by Dymara and Januszkiewicz
establishing property (T) for Kac-Moody groups over finite fields and by
Ershov, Jaikin, & Kassabov establishing property (T) for Chevalley groups
over commutative rings to prove that given any indecomposable 2-spherical
generalized Cartan matrix A there is an integer m (depending solely on A)
such that if R is a finitely generated commutative unital ring with no
ideals of index less than m then the Kac-Moody group over R associated
to A property (T).

**Sieve methods in group theory (Alex Lubotsky, Dec. 3, 2014):**

The sieve methods are classical methods in number theory.
Inspired by the 'affine sieve method' developed by Sarnak, Bourgain,
Gamburd and others, as well as by works of Rivin and Kowalsky, we develop
in a systemtic way a 'sieve method' for group theory. This method is
especially useful for groups with 'property tau'. Hence the recent results
of Breuillard-Green-Tao, Pyber-Szabo, Varju and Salehi-Golsefidy are
very useful and enables one to apply them for linear groups.

We will present the method and some of its applications to linear
groups and to the mapping class groups.
[Based on joint with Chen Meiri (JAMS) and with Lior Rosenzweig (to
appear in Amer. J. of Math.) ].

Charles Weibel / weibel @ math.rutgers.edu / January 1, 2016