(Wednesdays at 2:00 in H705)

The Algebra Seminar meets on Wednesdays, at 2:00-3:00PM in room 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

Fall 2016 Seminars (Wednesdays at 2:00 in H423)
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

Spring 2016 Seminars (Wednesdays at 2:00 in H705)
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

Fall 2015 Seminars (Wednesdays at 2:00 in H425)
 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

Spring 2015 Seminars (Wednesdays at 2:00 in H124)
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

Fall 2014 Seminars (Wednesdays, 3:15-4:15PM in H525)
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

Abstracts of seminar talks

Spring 2016

Noncommutative Poisson Geometry (Semeon Artamonov, April 27, 2016):
One of the major ideas of the noncommutative geometry program consists of replacing the algebra of smooth functions on a manifold with some general associative (not necessary commutative) algebra. It appears that a lot of tools of conventional differential and algebraic geometry can be translated to the noncommutative world. In my talk I will focus on an implication of the noncommutative geometry program to the Poisson manifolds.

I will start by reviewing an $H_0$-Poisson structure --- a noncommutative analog of the Poisson bracket and related notion of double Poisson brackets. We will see how an $H_0$-Poisson structure descends to a usual Poisson bracket on the moduli space of representations of the underlying associative algebra. I will then show how one can substantially modify definition of double Poisson bracket by M. Van den Bergh to provide a number of new nontrivial examples.

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.

The talk is aimed at graduate students. In particular, while some familiarity with Chern classes would be useful, I will introduce the necessary notions during the talk.

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.

Fall 2015

Representations of finite groups and applications (Pham Huu Tiep, Dec. 7, 2015):
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.

The talk will survey some aspects of this ongoing search. The methods for studying this question involve explicit and constructive applications of well known classical theorems in algebra and group theory, for instance Conway's and Pless' application of Burnside's orbit counting theorem and quadratic reciprocity dating back to the 1980's. More recent and partly computational methods are based on representation theory of finite groups.

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.

Spring 2015

The Index Map and Reciprocity Laws for Contou-Carrère Symbols
(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.

Fall 2014

Semiclassical approximation to noncommutative Riemannian geometry
(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 @ / January 1, 2016