Tuesday, September 21
MIT - Room 56-114
Map of MIT

3:00 p.m.
Kai-Wen Lan - Princeton/IAS
"Vanishing theorems for torsion automorphic sheaves"
abstract

4:30 p.m.
Michael Rapoport - Bonn
"The Langlands-Kottwitz method for the zeta function, beyond the parahoric case"
abstract

Tuesday, October 19
BC - 9 Lake Street, Room 035 (north of Commonwealth Avenue near the "B" line)
Directions

3:00 p.m.
Eyal Goren - McGill
“Canonical subgroups over Hilbert modular varieties”
abstract

4:30 p.m.
Pierre Colmez - Jussieu
“On the p-adic local Langlands correspondence for GL2”

Tuesday, November 16
MIT - Room 56-114
Map of MIT

3:00 p.m.
Brian Smithling - Toronto
“On some local models for Shimura varieties”
abstract

4:30 p.m.
Matt Baker - Georgia Tech
“Tropical and Berkovich analytic curves”
abstract

Tuesday, February 8
BC - Campion 009
Note room change
Directions

3:00 p.m.
Amanda Folsom - Yale
"ell-adic properties of the partition function"
abstract

4:30 p.m.
Jordan Ellenberg - Wisconsin
"Expander graphs, gonality, and Galois representations"
abstract

Tuesday, March 1
MIT - 4-159
Map of MIT

3:00 p.m.
Michael Harris - Jussieu
"The Taylor-Wiles Method for Coherent Cohomology"
abstract

4:30 p.m.
Laurent Clozel - Orsay
“Presentation of the Iwasawa algebra of Gamma_1 SL_(2, Z_p)”
abstract

Tuesday, April 12
BC - Campion 009
Note room change
Directions

3:00 p.m.
Freydoon Shahidi - Purdue
"Arthur Packets and the Ramanujan Conjecture"
abstract

4:30 p.m.
William Duke - UCLA
“The interpretation and  distribution of cycle integrals of modular functions”
abstract

Professor Martin Bridgeman (Boston College) will speak on “The orthospectra of finite volume hyperbolic manifolds with totally geodesic boundary and associated volume identities.”

Abstract: Given a finite volume hyperbolic n-manifold $M$ with totally geodesic boundary, an orthogeodesic of $M$ is a geodesic arc which is perpendicular to the boundary. For each dimension n, we show there is a real valued function $F_n$ such that the volume of any $M$ is the sum of values of $F_n$ on the orthospectrum (length of orthogeodesics). For $n=2$ the function $F_2$ is the Rogers L-function and the summation identities give dilogarithm identities on the Moduli space of surfaces.

Professor Daniel Mathews (Boston College) will speak on “Sutured topological quantum field theory and contact elements in sutured Floer homology.”

Abstract: We consider a type of topological quantum field theory, a “sutured TQFT”, inspired by the work of Honda-Kazez--Matic on sutured Floer homology: contact elements in the sutured Floer homology of product manifolds forms a sutured TQFT. This theory has curious connections to structures seen in physics and representation theory. As an application, we obtain a “contact geometry free” proof that the contact element in sutured Floer homology of a contact structure with Giroux torsion is zero.

Professor Genevieve Walsh (Tufts) will speak on “Knot commensurability and the Berge Conjecture.”

Abstract: We discuss the problem of understanding commensurability classes of hyperbolic knots in S^3. We show that generically, there are at most three knots in a commensurability class. If there is more than one knot in such a commensurability class, the knots are fibered. We also discuss how this relates to understanding lens space surgeries along knots in lens spaces. This is joint work with M. Boileau, S. Boyer, and R. Cebanu.

Thursday, October 7
Carney 309
2:00 p.m.

Professor Gabriel Katz (MIT) will speak on "Topological Invariants of Gradient Flows on Manifolds with Boundary."

Abstract: Let f: X —> R be a Morse function on a manifold X and v its gradient-like vector field. Classically, the topology of a closed X can be described in terms of the spaces of v-trajectories that link the singular points of f. On manifolds with boundary, the situation is somewhat different: there, a massive set of nonsingular functions is available. For such Morse data (f, v), the interactions of the gradient flow with the boundary dX take central stage.

We will introduce and measure the convexity and concavity of a v-flow relative to dX. “Some manifolds are intrinsically more concave than others with respect to any gradient flow” is the main slogan of the talk. Stated differently, the intrinsic concavity of X is a reflection of its complexity.

We will explain how this approach leads to new topological invariants, both of the flow v and of the manifold X. In 3D, we have a good grasp of these invariants and their connection to the classification of 3-folds.

Professor Refik Baykur (Brandeis) will speak on "Round handles and smooth four-manifolds."

Abstract: In this talk, we will unfold the strong affiliation of round handles with smooth four-manifolds. Several essential topics that appear in the study of smooth four-manifolds, such as logarithmic transforms along tori, exotic smooth structures, cobordisms, handlebodies, broken Lefschetz fibrations, one and all, will come into play as we discuss the relevant interactions between them.

Thursday, October 21
Carney 309
2:00 p.m.

Professor Tao Li (Boston College) will speak on “Rank and genus of amalgamated 3-manifolds."

Abstract: The rank conjecture says that, for a hyperbolic 3-manifold, the rank of its fundamental group equals its Heegaard genus. We will discuss constructions of counterexamples involving hyperbolic JSJ pieces and candidate hyperbolic counterexamples to this conjecture.

Thursday, October 28
Carney 309
2:00 p.m.

Professor Sucharit Sarkar (Columbia) will speak on “Grid diagrams and the Ozsvath-Szabo tau-invariant.”

Abstract: The Ozsvath-Szabo knot invariant $\tau$ satisfies the inequality that $|\tau(K_1)-\tau(K_2)|\leq g$, whenever there is a genus $g$ knot cobordism joining $K_1$ to $K_2$. We will give a new proof of this fact using grid diagrams. This will lead to a new and entirely grid diagram-based proof of Milnor's conjecture that the unknotting number the torus knot $T(p,q)$ is $\frac{(p-1)(q-1)}{2}$.

Thursday, November 4
Carney 309
2:00 p.m.

Professor Adam Levine (Brandeis) will speak on "A Combinatorial Spanning Tree Model for Knot Floer Homology."

Abstract: We provide an explicit description of complex, based on spanning trees of the black graph of a diagram of a knot K in S^3, that computes the knot Floer homology of K. The strategy is to iterate Manolescu's unoriented skein exact sequence for knot Floer homology, using twisted coefficients in a Novikov ring, to form a cube of resolutions in which the only nonzero groups correspond to the connected resolutions. This construction has intriguing similarities with Ozsvath and Szabo's spectral sequence from the reduced Khovanov homology of K to the Heegaard Floer homology of the double branched cover of K. This is joint work with John Baldwin.

Professor Vera Vertesi (MIT) will speak on “Invariants for Legendrian knots in Heegaard Floer Homology.”

Abstract: This talk will concentrate on invariants for contact 3--manifolds in Heegaard Floer homology. They can be defined both for closed 3--manifolds, in this case they live in Heegaard Floer homology and for 3--manifolds with boundary, when the invariant is in sutured Floer homology. There are two natural generalizations of these invariants for a Legendrian knot K in a contact manifold M. One can directly generalize the definition of the contact invariant to obtain an invariant L(K), or one can take the complement of the knot, and compute the invariant for that: EH(M-K). At the end of the talk I would like to describe a map that sends EH(M-K) to L(K). This is a joint work with Andras Stipsicz.

Thursday, November 18
Carney 309
2:00 p.m.

Professor Joshua Greene (Columbia) will speak on “The lens space realization problem.”

Abstract: I will discuss the classification of the lens spaces which arise by integral Dehn surgery along a knot in the three-sphere. A related result is that if surgery along a knot produces a connected sum of lens spaces, then the knot is either a torus knot or a cable thereof, confirming the cabling conjecture in this case. The proofs rely on Floer homology and lattice theory.

Tuesday, February 8
Carney 309
4:00 p.m.

Prof. Andy Cotton-Clay (Harvard) will speak on “Sharp fixed point bounds for surface symplectomorphisms in each mapping class.”

Abstract: Let S be a closed, oriented surface, possibly with boundary, and consider a connected component H of the space of symplectomorphisms of S with no fixed points on the boundary. We give sharp bounds on the number of fixed points for symplectomorphisms f: S to S with f in mapping class H, both with and without nondegeneracy assumptions on the fixed points of f. These bounds often exceed those for non-area-preserving maps coming from Nielsen theory. This generalizes the Poincaré-Birkhoff fixed point theorem, which states that area-preserving twist maps of the cylinder have at least two fixed points, to arbitrary surfaces and mapping classes.

For the nondegenerate case, our techniques involve Floer homology computations with certain twisted coefficients plus a method for obtaining fixed point bounds on entire symplectic mapping classes from such. In the possibly degenerate case, we additionally use quantum-cup-length-type arguments for certain cohomology operations we define on Nielsen summands of the Floer homology.

Tuesday, March 1
Carney 309
4:00 p.m.

Prof. Stephan Wehrli (Syracuse)will speak on "On Quiver Algebras and Floer homology."

Abstract: In this talk, I will discuss a connection between certain Khovanov- and Heegaard Floer-type homology theories for knots, braids, and 3-manifolds. Specifically, I plan to explain how the bordered Floer homology bimodule associated to the branched double cover of a braid is related to a similar bimodule defined by Khovanov and Seidel. This is joint work with D. Auroux and E. Grigsby.

Tuesday, March 15
Carney 309
4:00 p.m.

Professor Tejas Kalelkar (Washington University, St. Louis) will speak on “Normal surfaces and incompressible surfaces in 3-manifolds.”

Abstract: Let S be a surface embedded in a triangulated 3-manifold M. S is said to be normal if it intersects each tetrahedron of this triangulation 'nicely'. S is said to be incompressible if it is \pi_1 injective. Haken showed that if S is incompressible then with respect to each triangulation of M, the minimal PL-area surface isotopic to S is a normal surface. In this talk the converse will be proved, that is, if with respect to each triangulation of M, a minimal PL-area surface isotopic to S is normal then in fact S is incompressible.

Tuesday, April 12
Carney 309
4:00 p.m.

Abstract: Abstract: There exist proteins, such as topoisomerases and recombinases, that change the topology of DNA. These changes can inhibit or aid in biological processes that involve the structure of DNA. Because the mechanism of many proteins involves interaction with double stranded DNA, applications of knot theory to problems involving these proteins have been extensively studied. 

In the 1980's, DeWitt Sumners and Claus Ernst developed the tangle model of protein-DNA complexes, using the mathematics of tangles to model DNA-protein binding. An n-string tangle is a pair (B,t) where B is a 3-dimensional ball and t is a collection of n non-intersecting curves properly embedded in B. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. In this talk, I will give definitions and a description of the tangle model with a biological example.

Tuesday, April 26
Carney 309
4:00 p.m.

Professor Solomon Friedberg (Boston College) will speak on “Eisenstein series and crystal graphs.”

Abstract: The study of the Whittaker coefficients of Eisenstein series on reductive groups led Langlands to formulate his Conjectures. But the study of Whittaker coefficients on covers of such groups has not been carried out. In this talk I present a theorem for the simplest Eisenstein series on such a cover, showing that these series may be computed in a surprising way that involves the theory of crystal graphs.

David Hansen (Boston College) will speak on "Ranks of elliptic curves over (nearly) abelian extensions"

Abstract: Given a modular elliptic curve E over a number field K, the theory of L-functions provides a powerful tool for studying the rank of E over K and over varying families of extensions of K. Most results of this flavor analyze the rank of E over "vertical" towers of abelian extensions of K.

I will review these results, and then explain some recent progress on the corresponding question for some interesting "horizontal" families of abelian extensions.

Professor George McNinch (Tufts/MIT) will speak on “The special fiber of a parahoric group scheme.”

Abstract: Let G be a connected and reductive algebraic group over the field of fractions K of a complete discrete valuation ring A with residue field k. Bruhat and Tits have associated with G certain smooth A-group schemes P -- called parahoric group schemes -- which have generic fiber P/K = G. The special fiber P/k of such a group scheme is a linear algebraic group over k, and in general it is not reductive.

In some recent work, it was proved that P/k has a Levi factor in case G splits over an unramified extension of K. Even more recently, this result was (partially) extended to cover the case where G splits over a tamely ramified extension.

The talk will discuss these results and some applications. In particular, it will mention possible applications to the description of the scheme-theoretic centralizer of suitable nilpotent sections in Lie(P)(A).

Thursday, October 28

Professor Avner Ash (Boston College) will speak on "Reducible Galois representations and Hecke eigenclasses."

Abstract: Serre's conjecture (now a theorem) was stated for irreducible Galois representations, but it could have been stated as well for reducible ones. When Warren Sinnott and I generalized the niveau 1 case to GL(n), we stated a conjecture for irreducible and reducible Galois representations. I have proved this conjecture for direct sums of one-dimensional characters with pairwise relatively prime conductors. This talk will describe the background and proof of this theorem.

Thursday, November 11

Professor Andrew Ledoan (Boston College) will speak on “Zeros of partial sums of the Riemann zeta function.”

Abstract: The Riemann zeta-function zeta(s) of the complex variable is defined in the half plane Re(s) > 1 by an absolutely convergent Dirichlet series 1+1/2^s+1/3^s+...which can be continued analytically to a meromorphic function in the complex plane with solely a simple pole situated at s = 1 with residue 1. The critical strip 0< Re(s) < 1 is the most important and mysterious region for zeta(s), and much attention has been given to the right half of the strip. Although a great deal is known and conjectured about the distribution of zeros of zeta(s), little is known about the zeros of its partial sums F_X(s) = 1+1/2^s+...+1/X^s, where X>1. By the absolute convergence of the Dirichlet series one sees that, even for X not very large, F_X(s) gives (at least away from the pole) a rather good approximation to zeta(s) with a remainder which is o(1) as X goes to infinity. To be more precise, zeta(s) is well-approximated unconditionally by arbitrarily short truncations of its Dirichlet series in the region sigma>1, |s-1| > 1/10. This is also true in the right half of the critical strip, if one assumes the Lindelof Hypothesis. In this talk, I will present recent results obtained in collaboration with S. M. Gonek on the distribution of zeros of F_X(s), in which we estimate the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of their horizontal distribution.

Thursday, November 18

Professor Sawyer Tabony (Boston College) will speak on “Finding Representation Theory in a Statistical Mechanical Model.”

Abstract:  In a recent paper, Gross and Reeder have described an interesting class of smooth representations of reductive p-adic groups, which they call simple supercuspidal representations. Guided by the conjectural framework of the Langlands correspondence, they analyse the structure of the expected Langlands parameters for these representations. These so called simple wild parameters are wildly ramified, but in a minimal way. In this talk we will report on a construction which explicitly associates to each simple wild parameter a finite set of simple supercuspidal representations, and furthermore provides a description of this set in terms of the Langlands dual group.

Prof. Benedict Gross (Harvard) will speak on "Stable orbits and the arithmetic of curves."

Abstract: Manjul Bhargava has recently made a great advance in the arithmetic of elliptic curves, giving the first bounds on the average rank of the group of rational points. He shows that the average order of the 2-Selmer group is equal to 3, by studying the stable orbits of the group PGL(2,Z) acting on the lattice of binary quartic forms.  In this talk, I will begin by reviewing some basic material on elliptic curves, defining the 2-Selmer group, and describing the stable orbits in this representation, whose invariants were determined by Hermite. If time permits, I will discuss a possible generalization of Bhargava's result to hyperelliptic curveswith a rational Weierstrass point.

Prof. Danny Calegari (Caltech) will speak on "Stable commutator length in free groups."

Abstract: Stable commutator length (scl) answers the question: “what is the simplest surface in a given space with prescribed boundary?” where “simplest” is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures).

I will discuss how scl in free groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, phase locking for nonlinear oscillators, and the theory of multi-dimensional continued fractions and Klein polyhedra.

September 22 MIT

3:00 p.m.
Yiannis Sakellaridis - University of Toronto
"A 'relative' Langlands program and periods of automorphic forms"

4:30 p.m.
Matthew Emerton - Northwestern University
"p-adically completed cohomology and the p-adic Langlands program"

October 20
BC

3:00 p.m.
Ze'ev Rudnick - Tel-Aviv University and IAS
"Statistics of the zeros of zeta functions over a function field"

4:30 p.m.
Haruzo Hida - UCLA
"Characterization of abelian components of the 'big' Hecke algebra"

November 17
MIT

3:00 p.m.
Akshay Venkatesh - Stanford University
"Torsion in the homology of arithmetic groups"

4:30 p.m.
Ken Ono - University of Wisconsin
"p-adic coupling of harmonic Maass forms"

February 9
BC

3:00 p.m.
Gautam Chinta - CUNY
"Orthogonal periods of Eisenstein series"

4:30 p.m.
Mihran Papikian - Pennsylvania State University
"On the arithmetic of modular varieties of D-elliptic sheaves"

March 9
MIT

3:00 p.m.
Elena Mantovan - Caltech
"l-adic etale cohomology of PEL Shimura varieties with non-trivial coefficients"

4:30 p.m.
Karl Rubin - UC Irvine
"Selmer ranks of elliptic curves in families of quadratic twists"

April 13
BC

3:00 p.m.
Shou-Wu Zhang - Columbia University
"Calabi-Yau theorem and algebraic dynamics"

4:30 p.m.
Ching-Li Chai - University of Pennsylvania
"CM lifting of abelian varieties"

Professor Benson Farb (University of Chicago) will be speaking this spring as the department's third annual Boston College Distinguished Lecturer in Mathematics. Professor Farb is an internationally renowned mathematician who specializes in the interaction between geometry, topology and group theory. 

March 10
4:00 p.m.

McGuinn 121

"Geometry and the Imagination (with applications)"

Abstract: Geometry and geometric reasoning underlie all of science. In this talk I will explore a few fundamental geometric notions, including symmetry, dimension (including dimensions bigger than 3), and orientation (i.e. left-handed vs. right-handed). I will give some examples illustrating important applications in chemistry, biology and physics, from the weak nuclear force to understanding the Thalidomide tragedy. Some questions to ponder before the talk: How can you turn a left sneaker into a right sneaker without ripping or bending the sneaker at all? Why do mirrors reflect left/right but not up/down? This talk is intended for all who are interested in mathematics.

March 11
4:00 p.m.

Cushing 212

"Topology, dynamics and geometry of surfaces (and their remarkable relationships)"

Abstract: Surfaces can be considered from many different angles: their shape (i.e. topological structure), their geometry (e.g. curvature), and the behavior of fluid flows on them. In this talk I will describe three beautiful theorems, one for each of these aspects of surfaces. I will also try to explain the remarkable fact that these seemingly completely different viewpoints are intimately related. This talk will be geared towards those with some familiarity with calculus.

March 12
3:00 p.m.

Higgins 265

Abstract: Homological stability is a remarkable phenomenon in the study of groups and spaces. For certain sequences G_n of groups, for example G_n=GL(n,Z), it states that the homology group H_i(G_n) does not depend on n for big enough n. There are many natural sequences G_n, from pure braid groups to congruence groups to Torelli groups, for which homological stability fails horribly. In these cases the rank of H_i(G_n) blows up to infinity, and in many (e.g. the latter two) cases almost nothing is known about H_i(G_n); indeed there may be no nice "closed form" for the answers.

While doing some homology computations for the Torelli group, Tom Church and I found what looked to us like the shadow of a broad pattern. In order to explain it and formulate a specific conjecture, we came up with a notion of "stability of a sequence of representations of group G_n". We began to realize that this notion can be used to make other predictions: from group representations to Malcev Lie algebras to the homology of congruence groups. Some of these predicitions are known results, while others are not known. In this talk I will explain our broad conjectural picture via some of its many instances. No knowledge of either representation theory or group homology will be assumed. This talk is intended for a mathematically sophisticated audience.

7:30 p.m.

Dr. Garvey is Chief Scientist, and a Director, for the Center for Acquisition and Systems Analysis - a division at The MITRE Corporation. He is internationally recognized and widely published in cost analysis, cost uncertainty analysis, and in the application of advanced decision analytic methods to problems in engineering systems risk analysis and management. He is an alumnus of the BC Mathematics department.

April 7  

5:00 p.m.
Carney

Abstract: In late February this year, the Large Hadron Collider at the international physics laboratory in Switzerland, CERN, began crashing protons at energy levels never seen before since the Big Bang, and will increase these levels over the next few years. The reason for this unprecedented $10 billion effort is the search for new particles, including the mysterious Higgs boson, the so-called "God particle," believed to give all particles in the universe their mass. If the Higgs is found, along with other possible particles, this will be a major triumph not only for physics, but also for mathematics: Mathematical theories, including Lie groups, underlie much of the foundation that allows physicists to predict the existence of new particles. We will survey this fascinating topic.

September 24  

2:00 p.m.
Carney 309

Abstract: Khovanov and Heegaard Floer homology, two theories inspired by ideas in physics, have transformed the landscape of low-dimensional topology in the past decade. The philosophies underlying the theories' constructions are quite different, yet there are intriguing connections between the two.

In this talk, I will focus on one such connection: a relationship between a reduced version of Khovanov homology and a relative version of Heegaard Floer homology recently developed by Andras Juhasz. This relationship can be used to prove that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot when n>1; furthermore, the relationship, in its most general form, satisfies nice naturality properties with respect to standard TQFT-type operations like cutting and stacking.

This is joint work with Stephan Wehrli.

2:00 p.m.
Carney 309

Abstract: The Giroux Correspondence is a one-to-one correspondence between contact structures up to isotopy and open book decompositions up to positive stabilization. An open book decomposition of a 3-manifold is a link with a fibration of its exterior such that each fiber is a Seifert surface for the link. Cabling a link component produces a new open book decomposition (with few exceptions). We will describe how the contact structure supported by an open book changes under cabling, generalizing Hedden's result for open books in S^3. We'll also define rational open books and discuss their cablings.

This is joint work with John Etnyre and Jeremy Van Horn-Morris.

2:00 p.m.
Carney 309

Abstract: I will describe recent work (joint with Maggy Tomova) which develops a new kind of thin position for graphs in 3-manifolds. I will outline the theory and describe how it can be used to level edges of certain graphs in 3-maniforld. The main theorem is a generalization of an old theorem by Casson and Gordon to bridge surfaces for graphs in 3-manifolds.

2:00 p.m.
Carney 309

Abstract: Cluster algebras were developed by Fomin and Zelevinsky in 2002. Many cluster algebras arise as the coordinate rings of varieties, with the key feature - known as the Laurent Phenomenon - that the transition functions for any pair of charts (clusters) are Laurent polynomials in the coordinates (cluster algebras). The work of Gekhtman, Shapiro and Vainshtein related Teichmuller theory to cluster algebras. There is a non-commutative deformation of the rational functions on the Teichmuller space, called quantum Teichmuller space. In this talk we study the relation between the quantum Teichmuller space and quantum cluster algebra, in accordance with the technique introduced by Berenstein and Zelevinsky. This is joint work with Francis Bonahon.

2:00 p.m.
Carney 309

Abstract: Links obtained using the operations of Whitehead and Bing doubling (and combinations thereof) are of great interest in the study of concordance, since they play a fundamental role in the work of Freedman on topological 4-manifolds. I will discuss recent work on this topic and prove some new results on the smooth sliceness of such links. For example, we can prove that the positive Whitehead double of the Borromean rings is not smoothly slice; whether or not it is topologically slice remains a major unsolved question.

4:00 p.m. 
Carney 309

Abstract: We prove that fundamental group of a closed hyperbolic 3-manifold contains a surface subgroup. The subgroups are quasifuchsian groups 1 + eplilon close to a fuchsian group. We prove this result by showing via mixing of the geodesic flow that randomly determined pairs of pants are sufficiently uniformly distributed to fit together into a closed almost flat surface. This is joint work with Vladimir Markovic.

2:00 p.m.
Carney 309

Abstract: I'll describe new invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction draws on a surprising connection between the topology of link surgeries and the combinatorics of polytopes called graph associahedra. For a classical link L in S^3, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, Z/2Z Khovanov homology of L and a version of the monopole Floer homology of the branched double cover. This perspective also yields a simple, topological proof that odd Khovanov homology is mutation invariant. I'll emphasize low-dimensional topology through lots of pictures, and not the technical details of Floer homology.

Paper reference:  arxiv.org/abs/0903.3746, arxiv.org/abs/0909.0816

3:00 p.m.
Carney 309

Abstract: Let M be a compact orientable 3-manifold which contains a closed incompressible surface F. We denote by N(F) an open regular neighborhood of F in M. If each component of M-N(F) has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting, i.e. the amalgamation or self-amalgamation of the minimal Heegaard splittings of M-N(F).

2:00 p.m.
Carney 309

Abstract: We are interested in some group of hyperbolic knots in S^3 which lie on Heegaard surface of genus 2 of S^3. We define primitive/primitive knots and primitive/Seifert-fibered knots from this group. The former admits lens surgeries and the latter admits small Seifert-fibered space surgeries. The goal of this talk is to provide some idea to give a complete list of all doubly primitive/primitive and all primitive/Seifert knots. The idea is based on the R-R diagrams introduced by Osborne and Stevens.

2:00 p.m.
Carney 103

Abstract: Suppose F is a Heegaard surface of a closed, compact, orientable 3-manifold M, such that F bounds handlebodies H and H'. A choice of complete sets of cutting disks v of H and v' of H' yields a Heegaard diagram carried by F. The complexity of such a diagram is the total number of points of essential intersection of disks in v' with disks in v.

We will show that it is usually possible, and surprisingly easy, to locate and identify all minimal complexity Heegaard diagrams carried by F, when F has genus two.

Some of the consequences of the ability to identify the minimal complexity Heegaard diagrams carried by F are:
1) One can locate all essential tori in M.
2) One can find all alternative Heegaard surfaces in M, up to isotopy.
3) If F' is a genus two Heegaard surface of a closed orientable 3-manifold N, one can determine if there is a homeomorphism h:M->N such that H(F)= F'.
4) If all genus two Heegaard surfaces of M and N have identifiable minimal complexity Heegaard diagrams, which is usually the case, one can determine if M and N are homeomorphic.

4:00 p.m.
Carney 309

Abstract: The log 3 Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least log 3 by one of the non-commuting isometries ξ or η provided that ξ and η generate a torsion-free, discrete group which is not co-compact and contains no parabolic.

In my talk, I'll introduce a technique which determines a lower bound for the maximum of displacements under a given set of isometries. In particular, I'll show that every point in the hyperbolic 3-space is moved a distance at least (1/2)log(5+3 √ 2) by one of the isometries ξ, η or ξη when ξ and η satisfy the conditions given in the log 3 Theorem. 

3:00 p.m.
Carney 309

Abstract: We consider the asymptotic structure induced by a pseudo-Anosov flow in the universal cover of the underlying 3-manifold. In particular we consider untwisted flows: this means that no closed orbit is freely homotopic to the inverse of another orbit. In this case we use the dynamics of the flow to produce a flow ideal boundary to the universal cover of the manifold. We show that the action of the fundamental group G of the manifold on the flow ideal boundary is a uniform convergence group. This implies that G is Gromov hyperbolic and the action of G on the flow ideal sphere is conjugate to the action of G on its Gromov ideal boundary. This implies that untwisted pseudo-Anosov flows are quasigeodesic. This also has consequences for the asymptotic behavior of certain foliations. 

2:00 p.m.
Carney 309

Abstract: We discuss a program to show that a topologically minimal surface (of arbitrary index) in a compact 3-manifold can be isotoped to meet a triangulation so that it meets each tetrahedron in precisely the same way that a geometrically minimal surface (of the same index) can meet a ball. We will then discuss the immediate applications to topology, as well as potential applications to geometry. 

2:00 p.m.
Carney 309

Abstract: TBA

4:00 p.m.
Carney 309

Abstract: I will discuss how the notion of Gelfand pairs from the representation theory leads to various identities for automorphic periods. These include the classical Rankin-Selberg integral, its anisotropic analog, and many other identities. Time permitting, I will discuss some applications towards bounds for L-functions.

3:00 p.m.
Carney 309

Abstract: The classical Shintani lift is the adjoint of the Shimura correspondence. It realizes periods of even weight cusp forms as Fourier coefficients of a half-integral modular form. In this talk we revisit the Shintani lift from a (co)homological perspective. In particular, we extend the lift to Eisenstein series and give a geometric interpretation of this extension.

This is joint work with John Millson.

4:00 p.m.
McGuinn 521 Lounge

Abstract: I will discuss the existence and uniqueness theorems for broken fibrations of arbitrary orientable, smooth 4-manifolds over either S^2, B^2, or S^1 x I. Existence always holds, and there is a nice set of moves relating different broken fibrations for a given 4-manifold.

3:00 p.m.
Carney 309

Abstract: TBA 

4:00 p.m.
Carney 309

Abstract: The unknotting number u(K) of a knot K is the minimal number of times you must allow K to pass through itself in order to unknot it. Although this is one of the oldest and most natural knot invariants, it remains mysterious. We will survey known results on u(K), including relations with 4-dimensional smooth topology, and describe some joint work with John Luecke on algebraic knots with u(K)=1. We will also discuss several open questions.

October 8

McGuinn 334  

October 29

McGuinn 334

December 3

McGuinn 334

February 4

Canceled

February 25

McGuinn 334

April 15

McGuinn 521

April 27 

McGuinn 334

3:00 p.m.
Wee Teck Gan - UC San Diego

4:30 p.m.
Daniel Bump - Stanford

3:00 p.m.
Steve Kudla - Toronto

4:30 p.m.
Chris Skinner - Princeton

3:00 p.m.
Henri Darmon - McGill

4:30 p.m.
Peter Sarnak - Princeton

3:00 p.m.
Brooke Feigon - Toronto

4:30 p.m.
Kartik Prasanna - Maryland

3:00 p.m.
Dorian Goldfeld - Columbia

4:30 p.m.
Brian Conrad - Stanford

Matt Papanikolas
Texas A&M

Dinakar Ramakrishnan
Caltech

March 31

"Hidden polynomials in geometry"

Abstract: A number of recent developments in geometry have hinged on unexpected polynomials appearing in geometric phenomena. Interpreted appropriately, this behavior is in retrospect visible in the doodling I did as a child. I'll use this doodle as a jumping-off point. It will lead inevitably to ideas from topology, geometry, a Hilbert problem, and the work of several Fields Medalists.

Gasson Hall, Room 202 at 3:00 p.m.

This talk is intended for all who are interested in mathematics.

March 31

"Murphy's Law in algebraic geometry: Badly-behaved moduli spaces"

Abstract: We consider the question: "How bad can the deformation space of an object be?" (Alternatively: "What singularities can appear on a moduli space?") The answer seems to be: "Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad." I show this for a number of important moduli spaces, parametrizing objects such as smooth curves in projective space, smooth projective surfaces, and plane curves with nodes and cusps. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. This is good news, not bad: we now have a complete constructive understanding of the singularities of these fundamental spaces.

I will begin by telling you what "moduli spaces" and "deformation spaces" are, and then explain our question and its answer.

Gasson Hall, Room 202 at 4:30 p.m.

This talk is intended for a broad but mathematically sophisticated audience.

April 1

"A geometric Littlewood-Richardson rule"

Abstract: I will describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux and Knutson and Tao's puzzles.

This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. It has a host of geometric consequences, which I may describe, time permitting. The rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory, and in fact a Littlewood-Richardson rule in equivariant K-theory (ongoing work with Knutson). The rule suggests a natural approach to the open question of finding a Littlewood-Richardson rule for the flag variety, leading to a conjecture, shown to be true up to dimension 5. Finally, the rule suggests approaches to similar open problems, such as Littlewood-Richardson rules for the symplectic Grassmannian and two-step flag varieties. 

McElroy Conference Room at 3:00 p.m.

This talk is intended for a mathematically sophisticated audience.

2008-2009

Thomas Banchoff (Brown University)

February 25, 2009

"The Four-Dimensional Geometry and Theology of Salvador Dali"

Co-sponsored by the Department of Mathematics, the Boston College Mathematics Society, the Department of Fine Arts, the Department of Theology, and the Jesuit Institute

Abstract: Throughout his career, Salvador Dali was fascinated by mathematics and science, and he incorporated many geometric ideas and symbols into his paintings, especially his religious paintings. Where did he get his ideas and how did he carry them out? This presentation will feature images and stories from ten years of conversations with Dali, about the Fourth Dimension, impossible perspectives, catastrophe theory, art history and medieval philosophy. The talk will be illustrated by computer-generated images and animations, and is intended for a broad audience.

September 23

Yi Ni (AIM and MIT) will speak in 251 Carney Hall at 2:00 p.m.

"Dehn surgeries that reduce the Thurston norm of a fibered manifold"

Abstract: Suppose K is a knot on the fiber of a surface bundle over the circle. If we do surgery on K with slope specified by the fiber, then the Thurston norm of the homology class of the fiber will decrease in the new manifold. We will show that the converse is also true. Namely, if a Dehn surgery on a winding number 0 knot in a fibered manifold reduces the Thurston norm of the homology class of the fiber, then the knot must lie on the fiber and the slope is the natural one.

September 25  

Scott Taylor (Colby College) will speak in 251 Carney Hall at 2:00 p.m.

"Adding a 2-handle to a sutured manifold"

Abstract: Sutured manifold theory has long been used to study Dehn surgery on knots in 3-manifolds. It has not often been used to study 2-handle addition, a natural generalization of Dehn surgery. If a component F of a simple 3-manifold N has genus two, sutured manifold theory is particularly effective for studying degenerating separating curves on F. (A curve is degenerating if attaching a 2-handle to it creates a non-simple 3-manifold N[a].) For example, suppose that the boundary of N consists of tori and the genus two surface F containing essential separating curves a and b. Then if N[a] is reducible and N[b] is non-simple, a and b are istopic on F.

Similar sutured manifold theory techniques are useful for studying knots and links obtained by "boring" a split link or unknot. Such a perspective allows a theorem to be proved which is a generalization of two seemingly unrelated theorems. The first theorem generalized is the superadditivity of genus under band connect sum (Gabai, Scharlemann) and the second is the fact that a tunnel for a tunnel number one knot or link can be slid and isotoped to be disjoint from a minimal genus Seifert surface (Scharlemann, Thompson). As time permits, I will discuss other applications of sutured manifold theory to questions about bored split links and unknots.

December 8 

Yoav Moriah (Technion and Yale University) will speak in 251 Carney Hall at 3:00 p.m.

"Horizontal Dehn surgery and distance of Heegaard splittings"

Abstract: Given a 3-manifold M with a Heegaard surface S of genus g at least 2 and an essential simple closed curve c in S, we can obtain a new Heegaard splitting by changing the gluing of the two handlebodies/compression bodies by a Dehn twist to some power m along c. If c is "sufficiently complicated", measured a priori by a parameter n, then there is at most a single value so that the obtained Heegaard splitting is of smaller distance than n-1. Furthermore, the curves c with this property are "generic" in the set of essential simple closed curves c in S. (Joint with M. Lustig)

March 17

Bill Menasco (SUNY at Buffalo) will speak in 251 Carney Hall at 1:00 p.m.

"Legedrian and Lorenz knots"

April 15

Elmas Irmak (Bowling Green State University) will speak in 251 Carney Hall at 1:00 p.m.

"Mapping Class Groups and Complexes of Arcs on Surfaces"

Abstract: I will talk about a joint work with J.D. McCarthy: Each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface, and the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center. I will also talk about my similar results on nonorientable surfaces.

September 18   

Mark Reeder (Boston College) will speak in 309 Carney Hall at 3:15 p.m.

October 23

Benjamin Howard (Boston College) will speak in 309 Carney Hall at 3:15 p.m.

November 6  

Avner Ash (Boston College) will speak in 309 Carney Hall at 3:15 p.m.

November 13

Jay Pottharst (Boston College) will speak in 309 Carney Hall at 3:15 p.m.

April 23

Riad Masri (University of Wisconsin) will speak in 309 Carney Hall at 4:00 p.m.

"Equidistribution of Heegner points and integer partitions"

Abstract: A classical problem in number theory concerns the asymptotic growth of the function p(n) which counts the number of partitions of a positive integer n. This problem led Hardy and Ramanujan to invent what is now known as the "circle method". In this talk, I will explain how the equidistribution of Galois orbits of Heegner points on X_0(6) can be used to obtain a new asymptotic formula for p(n). The resulting error term sharpens those obtained by Rademacher and D.H. Lehmer in the 1930s. This is joint with Amanda Folsom.

October 2

Dan Margalit (Tufts University) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Homologies of mapping class groups"

Abstract: The mapping class group is the group of topological symmetries of a surface. By understanding the homology and cohomology of the mapping class group and its subgroups, we gain insight into its finiteness properties (finite generation, finite presentability, etc.) and we can also classify topological invariants of surface bundles. In this talk, we will introduce basic notions about the mapping class group and explain how to compute its low dimensional homology groups. Then, we will explain some recent work with Mladen Bestvina and Kai-Uwe Bux concerning the homology of the Torelli subgroup of the mapping class group, the group of elements acting trivially on the homology of the surface. In particular, we answer a question of Mess by proving that the cohomological dimension of the Torelli group for a genus g surface is 3g-5.

November 4 

Eriko Hironaka (Florida State University) will speak in 309 Carney Hall at 1:00 p.m..

"Families of mapping classes with small dilatation"

Abstract: R. Penner showed that the logarithm of the least dilatation of mapping classes on an oriented genus g surface is asymptotic to 1/g. In joint work with E. Kin, we construct a sequence of mapping classes with small dilatations improving on explicit bounds found previously by Penner and Bauer. Our examples arise as mapping classes associated to labeled graphs. For such mapping classes, we discuss the relation between dilatation and the spectral radius of the graph, and show how dilatation is affected by edge subdivision.

December 4  

Richard Kenyon (Brown University) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Dimers and Harnack curves"

Abstract: A polynomial P(z,w) with real coefficients is said to be Harnack if the real components of P(z,w)=0 satisfy a certain simple geometric property. These polynomials are somewhat analogous to one-variable polynomials with only real, negative roots. We describe a surprising parameterization of the space of all Harnack polynomials, coming from the dimer model of statistical mechanics.

March 25

Bill Goldbloom Bloch (Wheaton College) will speak in 309 Carney Hall.  Talk at 4:30.

"Navigating the Mathematical and Literary Labyrinths in Jorge Luis Borges' story "The Library of Babel" "

Abstract: Jorge Luis Borges, the poet, essayist, librarian, and master crafter of short stories, was arguably the most influential writer in Spanish in the 20th century. An autodidact, he read and reread works by (among others) Bertrand Russell on the foundations and philosophy of mathematics, and these kinds of considerations explicitly directed the arcs of many of his short stories. "The Library of Babel" is perhaps his most famous story, and in its scant seven pages, he deploys simple combinatorial ideas to help create a miasmic atmosphere in the service of raising issues about the meaningfulness of our existence. The story also evokes ideas from three-dimensional manifold theory, real analysis, and graph theory; and, moreover, it is open to an interpretation from the theory of computation.

This talk will touch on a number of these themes and along the way illustrate how a mathematician can become (to everyone's surprise) a literary theorist.

April 7

Teruyoshi Yoshida (Harvard University and Cambridge University) will speak in 309 Carney Hall. Talk at 3:00, followed by refreshments at 4:00.

"Arithmetic Geometry related to Local Langlands Correspondence"

Abstract: To be announced