Noah Snyder

Email address:

nsnyder math berkeley edu

Curriculum Vitae:

LaTeX PDF

Group Blog:

Secret Blogging Seminar

Office:

1061 Evans Hall.

Advisors:

Mathematical interests:

In many words: Quantum algebra, quantum topology, and representation theory.
In one word: Braidings.
In cryptic symbols: 17B37, 57M27, 18D10.

Papers and Preprints:

Current Research Projects:

The Kac-De Concini quantum groups at roots of unity, with applications to invariants of knots together with a flat connection in the complement:

In the last 30 years the two largest developments in low dimensional topology are the geometrization program of Thurston, and quantum topology in the style of Jones, Witten, Reshetikhin, Turaev, etc. One of the biggest current goals in quantum topology is to find a connection between these two programs. In particular, since a typical prime knot has a unique finite volume hyperbolic structure, one would like to find a connection between certain quantum invariants and this hyperbolic structure.

The biggest breakthrough in this direction is the Hyperbolic Volume Conjecture of Kashaev, and Murakami-Murakami. This says that the behavior as m goes to infinity of the mth colored Jones polynomial (corresponding to the m-dimensional representation of U_q(sl_2)) at an mth root of unity is determined by the hyperbolic volume of the knot.

A different direction where a connection has been found between hyperbolic structures and quantum invariants is by building quantum-style invariants which use the hyperbolic structure as a direct input. A hyperbolic structure is determined by a map Hom(pi_1, SL_2)/SL_2. Such a map is closely related to a choice of coloring of a knot with representations in the Kac-De Concini form of the quantum group at a root of unity. In analogy with the Reshetikhin-Turaev construction of knot invariants from ordinary quantum groups, Kashaev and Reshetikhin constructed invariants of knots together with a choice of hyperbolic structure on the complement. There is a related construction due to Baseilhac and Benedetti which is analogous to the the Turaev-Viro invariants.

Kashaev-Reshetikhin knot invariants (in preparation, ETA: Oct. '08)

This paper is coauthored with Scott Morrison. In the first half we clarify several details from the papers of Kashaev and Reshetikhin to give a fully rigorous construction of their new knot invariants. This invariant is a function on the space Hom(G_m(K), SL_2)/SL_2 where G_m(K) is the generalized knot group, and the action of SL_2 is by conjugation. We prove several basic results, for example, that the value of the function on the trivial point is |J_m(\zeta_m)|^2.

In the second part we give the first computations of this invariant for nontrivial knots. For several small 2-bridge knots we compute explicitly the entire Kashaev-Reshetikhin knot invariant at a third root of unity and at a fifth root of unity. For a larger collection of 2-bridge knots we compute the value of the knot invariant at the finite volume hyperbolic point (again at a third and fifth root of unity).

Braiding for quantum groups at roots of unity (in preparation, ETA: Spring '09)

This paper is coauthored with Nicolai Reshetikhin. In this paper we will extend some of the results of Kashaev and Reshetikhin from sl_2 to a general simple quantum group. In previous papers they show that the R-matrix induces an outer automorphism in a formal neighborhood of the trivial central character. For sl_2 they showed that this outer automorphism is given by rational functions and thus extends to a Zariski open set. For more general quantum groups we show that this outer automorphism is not given by rational functions, but instead by compositions of rational functions and square roots. In particular, we show that the outer automorphism extends to an explicit analytic (rather than just formal) neighborhood of the trivial central character. Thus we get invariants of knots together with certain maps in Hom(pi_1, G)/G for arbitrary Lie group G.

Planar algebras given by generators and relations

A planar algebra is a combinatorial model for a pivotal fusion cateogry (or relatedly, a subfactor) introduced by Jones. Ordinary algebra takes place on a line, you can multiply on the left or on the right, but you can't multiply above or beneath. In a planar algebra there are many more "multiplications." For example, you can multiply three elements by gluing them together in the following way:

The relationship between planar algebras and fusion categories is that the planar algebra describes the Hom spaces between tensor powers of a chosen fundamental object in the fusion category. However, the theory of planar algebras can be built up on its own and gives an independent elementary perspective on quantum groups (I literally taught a class to high school students on U_q(sl_2) this way, which would certainly be impossible from the traditional quantum groups perspective).

This program has two main goals: to describe known fusion categories or subfactors by explicit generators and relations, and to find new fusion categories by generator and relation constructions. The first of these goals has already been accomplished in many instances, starting with Kuperberg's paper on Spiders for rank 2 quantum groups (spiders are essentially another name for planar algebras), and continuing with the work of Westbury (on SO(7)), and my collaborators Scott Morrison (U_q(sl_n)) and Emily Peters (Haagerup's exceptional subfactor). The second of these goals is still in it's infancy, but recent work of Jones and of Peters suggests optimism.

Skein theory for the D_2n planar algebras (preprint)

This paper is joint work with Scott Morrison and Emily Peters. Just as the McKay correspondence classifies all faithful 2-dimensional representations of finite groups, the Quantum McKay correspondence (of Jones, Ocneanu, and Kirillov-Ostrik) classifies all faithful representations of dimension strictly less than 2 of unitary spherical fusion categories. The simplest family, A_n, is related to the quantum group U_q(sl_2) at a root of unity. This category can also be described entirely using pictures as a quotient of the Temperley-Lieb planar algebra given by setting a certain Jones-Wenzl idempotent equal to zero. The next infinite family in the quantum McKay correspondence is D_2n. In this paper we give an explicit construction of the D_2n planar algebra by generators and relations, and prove all of its basic properties directly from these relations. In particular, we show that although the category is not braided in the traditional sense, it is nonetheless braided up to a sign. In the appendix we describe the other infinite family in the quantum McKay correspondence, the T_n planar algebras, using a similar generators and relations construction.

Link invariants from the D_2n planar algebras (in preparation, ETA: Oct. 08)

This paper is joint work with Scott Morrison and Emily Peters. In this paper we use our construction of the D_2n planar algebra to construct knot invariants and to prove several facts about known quantum invariants. We use the D_2n and T_n planar algebras to give a new picture proof of Kirby-Melvin symmetry. Furthermore, we use these new knot invariants to prove new identities between known knot polynomials. The simplest identity is that if K is a knot (not a link!) then the third colored Jones polynomial at a 24th root of unity is always 2.

The atlas of unitary fusion categories and subfactors (preliminary mathematica code)

This project is joint work with Scott Morrison and Emily Peters (and hopefully others, contact us if you want to get involved). The goal is to completely automate the search for new fusion categories and new subfactors. A typical paper we would hope to produce from this project would be ``The classification of simple unitary spherical fusion categories of global dimension smaller than 30." In the end we would hope to have an atlas of all small unitary spherical fusion categories and all small subfactors much like the Knot Atlas.

This program will have four steps.

  1. Produce possible fusion graphs satisfying certain conditions (say, global dimension smaller than 30)
  2. Apply tests which eliminate impossible graphs (for example, if the dimensions of objects fail to be algebraic integers)
  3. For graphs that have not been eliminated, see if they can be constructed with known techniques (e.g. is it a quantum group at some root of unity?)
  4. Find a construction via generators and relations by looking inside the path planar algebra

Commutors coming from half-twists

The ribbon half-twist (in progress, ETA: Sep 09)

This paper is joint work with Peter Tingley. In short, the goal of this paper is to rigorously interpret the following version of a crossing in the category of representations of a quantum group:

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t in H such that (H, R,C) is a ribbon Hopf algebra, where R= (t^{-1} \otimes t^{-1})\Delta(t) and C= t^{-2}. The element t is closely related to the topological `half-twist', which twists a ribbon by 180 degrees. We show that U_q(g) is a (topological) half-ribbon Hopf algebra, but that t^{-2} is not the standard ribbon element. We then discuss some consequences of using this modified ribbon element, mainly in the case of U_q(sl_2).

Older research papers:

Mednykh's formula via lattice topological quantum field theories (Arxiv preprint)

In this note we give a quick application of the theory of lattice TQFTs to prove Mednykh's formula for characters of finite groups. This formula states that for any oriented surface S and any finite group G, if d(V) is the dimension of an irrep V and \chi(S) is the Euler characteristic of S, then:
The proof presented in this paper is especially easy to remember: first compute the invariant of the surface attached to a group algebra in the basis of grouplike elements, then compute the same invariant on the same surface in the basis of matrix elements of irreducible representations.

We modify this theory slightly to prove a related result for unoriented surfaces due to Frobenius and Schur. In the unoriented case we fix a gap in the literature by computing the unoriented lattice TQFT invariant attached to a matrix algebra with an anti-symmetric *-structure.

This paper has been cited multiple times in two of Turaev's recent preprints: Dijkgraaf-Witten invariants of surfaces and projective representations of groups and On certain enumeration problems in two-dimensional topology.

Groups with a character of large degree (Proc. Amer. Math. Soc. 136 (2008), no. 6, 1893--1903. Arxiv preprint)

The research for this paper was done under the advice of Hendrik Lenstra Jr. Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and attempt to classify such groups. For e=1, 2, or 3 we give a complete classification. For any other fixed e we show that there are only finitely many examples.

Cartan subalgebras of root reductive Lie algebras (J. Algebra 308 (2007), no. 2, 583--611. Arxiv preprint)

This paper is coauthored with Ivan Penkov and Elizabeth Dan-Cohen. In this paper we extend the results of Karl-Hermann Neeb and Ivan Penkov in "Cartan subalgebras of gl_\infty" (Canad. Math. Bull. 46 (2003), no. 4, 597--616) to an arbitrary root reductive Lie algebra. Root reductive Lie algebras are limits of finite-dimensional reductive Lie algebras where the inclusions preserve weight spaces. Typical examples of root reductive Lie algebras include gl_\infty and sp_\infty, where all but finitely many of the matrix elements are non-zero. We also give a description of the conjugacy classes of Cartan subalgebras of simple root reductive Lie algebras.

An alternate proof of Mason's theorem (Elem. Math. 55 (2000), no. 3, 93--94)

My senior year of high school Serge Lang mentioned Mason-Stother's theorem (the polynomial version of the famous ABC conjecture) to me. This brief paper gives a simple short proof of this result. It has also appeared in Lang's Math Talks for Undergraduates.

Talks and Seminars

Seminars Organized

In Fall 2005 I organized a weekly seminar on Topological Invariants and Quantum Algebra and gave half of the talks. Here are the notes.

In Spring 2005 Jennifer Berg and I organized a weekly seminar on representation theory under the supervision of Vera Serganova.

Invited Talks

Berkeley Seminars

Expository papers and teaching notes:

In the fall of 2003 I wrote a final paper called "Automorphism Groups of Curves" for Robin Hartshorne's class on algebraic curves. Here it is in pdf.

In the summer of 2002 I taught a tutorial at Harvard on L-functions and zeta functions.
Here are the notes from lectures (in pdf): 1, 2, 3, 4, 5 and 6, 7, 8 and 9, 10
Here are the homeworks (in pdf): 1, 2
Here are the suggested paper topics (in pdf).

In the spring of 2002 I wrote my senior thesis at Harvard under the supervision of Benedict Gross. It was titled "Artin L-Functions: A Historical Approach." It received highest honors from the math department. Here it is in pdf.

In the fall of 2001 I wrote a paper on the distribution of primes in F_p[x] for a seminar taught by Tom Brennan. I received advice on both the content and presentation of this paper from both Tom Brennan and Keith Conrad. Here it is in pdf.

Other interests