Mathematics Colloquium Fall 2016
For further information please contact Professor Junecue Suh or call the Mathematics Department at 459-2969
Tuesday, September 27, 2016
"Covering the Langlands Program"
Martin Weissman, University of California, Santa Cruz
The Langlands program is a network of theorems and conjectures linking arithmetic geometry to representation theory. I will give a short introduction to the Langlands program, as a grand organizer of results in number theory. This part should be accessible to graduate students, and will have examples of historical importance. The most cited results in the Langlands program are links between Galois representations and modular forms. Unfortunately, modular forms of half-integer weight do not fit into the Langlands program (as imagined by Langlands), even though there are tantalizing clues that they should fit well. I will finish by describing an extension of the Langlands program to "covering groups" which incorporates modular forms of half-integer weight and beyond.
Tuesday, October 4, 2016
"On the classification of holomorphic vertex operator algebras of central charge 24"
Ching Hung Lam, Institute of Mathematics, Academia Sinica, Taiwan
A simple vertex operator algebra (VOA) V is said to be holomorphic if it has only one irreducible, up to isomorphism and all V-modules are completely reducible. In 1993, Schellekens determined the possible Lie algebra structures for the weight one subspaces of holomorphic vertex operator algebras of central charge 24. There are 71 cases in his list but not all cases were constructed at that time. Moreover, it is believed that the VOA structure of a holomorphic VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace.
In this talk, we will discuss the recent progress on the classification. We will first discuss the constructions of all 71 cases in Schellenkens' list using various types of orbifold construction. We will also discuss the uniqueness conjecture. In particular, we will show that the VOA structures of certain holomorphic VOAs of central charge 24 are uniquely determined by the Lie algebra structures of their weight one subspaces using a technique which we call "reverse orbifold construction".
Tuesday, October 11, 2016
"Linear algebra as a natural language for special relativity and its paradoxes"
John de Pillis, University of California, Riverside
Using basic linear algebra and the single assumption that the speed of light in a vacuum is the same for all observers (regardless of speeds of observer or source), we quickly develop the Lorentz Transformation.
We graphically explore one well-known paradox, the Twin Paradox, and the lesser well-known Bug-Rivet paradox. Using original cartoons and animations, we see how, among other things, Special Relativity is inconsistent with a body's enjoying the property of perfect rigidity ... unless we are willing to accept time reversal (effect precedes its cause).
Click here for a full description of the presentation.
Tuesday, October 18, 2016
"Jacobi's geodesic problem and integrable hamiltonian systems on lie algebras"
Velimir Jurdevic, University of Toronto
This lecture will introduce an affine-quadratic optimal control problem on a Lie group G with a semisimple Lie algebra that admits the usual Cartan decomposition.
We will find the necessary and sufficient conditions that the Hamiltonian (associated with a positive definite operator and a regular representation) admits an isospectral representation of a particular form. Then we will correlate these findings with the seminal works of S.M. Manakov, A.T. Fomenko, V.V. Trofimov and O. Bogoyavlensky on the integrability of the n-dimensional mechanical tops.
Additionally, we will single out an affine-quadratic Hamiltonian whose spectral invariants lead to the integrals of motion for the Jacobi's geodesic problem on the ellipsoid. More explicitly, we will link Jacobi's problem with the elliptic geodesic problem on the sphere and C. Newmann's mechanical problem on the sphere share the same integrals of motion inherited from the affine Hamiltonian on the Lie algebra of matrices of zero trace.
Click here for a full description of the presentation.
Tuesday, October 25, 2016
"L-functions and regulators"
Samit Dasgupta, University of California, Santa Cruz
In this talk, I will introduce the concept of L-functions in number theory, mainly through lots of examples. L-functions can be attached to Galois characters, elliptic curves, algebraic varieties, or in general “motives.” In every case, there is a conjectural formula for the values of these L-functions at certain integer points. Examples of these special value formulae include the famous conjectures of Stark, Birch—Swinnerton-Dyer, Beilinson, and Bloch—Kato. The conjectures typically equate special L-values to a product of three terms—an algebraic number, a regulator, and a period. The regulator is the determinant of a matrix whose entries are logarithmic heights. The conjectures have been proven in very few cases, with the higher rank settings (i.e. when the matrix has dimension greater than 1) appearing to be the most difficult.
For the second half of the talk, I will discuss the analogs of these conjectures for p-adic L-functions. I will introduce the concepts of the p-adic numbers and p-adic L-functions along the way. I will conclude with a brief description of my recent proof of the Gross—Stark conjecture (joint with Mahesh Kakde and Kevin Ventullo). This result gives an exact formula for the leading terms of Deligne—Ribet p-adic L-functions at 0 in terms of p-adic regulators of p-units. Time permitting, I will conclude with the statements of famous and mysterious open conjectures regarding the non-vanishing of certain p-adic regulators.
Tuesday, November 1, 2016
NO COLLOQUIUM
Tuesday, November 8, 2016
"Can quantum mechanics solve optimization problems?"
Peter Young, University of California, Santa Cruz
Optimization problems require one to minimize a function of many variables with constraints. Many important problems are of this type, e.g. voice recognition, image recognition, protein folding in biology, constraint satisfaction problems in computer science, and disordered systems in condensed matter physics called spin glasses (what I work on). I'll denote the cost function to be minimized by the "energy". A "greedy" algorithm goes along the route which lowers the energy as fast as possible. However, the system then gets trapped in local minima of the energy which are not the ground state. (We say that problems where no set of values minimizes separately each term in the energy are "frustrated". These are the ones of interest. Unfrustrated problems are trivial.) One physics-motivated approach to try to overcome the problem of local minima in problems with frustration is to put in a fictitious temperature, with the result that most of the time one still goes downhill in energy, but sometimes one goes uphill and so one has the chance of escaping from a local minimum. The temperature is gradually lowered to zero. This is called "thermal annealing". More recently people have suggested using quantum, rather than thermal, fluctuations to escape from a local minimum, an idea called "quantum annealing". I will discuss that a quantum particle can "tunnel" through a barrier rather than have to be activated over it. A lot of interest is in problems with binary variables (bits). In the quantum case these are called qubits, the ingredients of a quantum computer. A company, D-Wave has produced a quantum "computer" for solving optimization problems with over 1000 qubits. The results of recent computer simulations on using quantum annealing to solve hard optimization problems, and experiments on the D-Wave machine will be discussed.
Tuesday, November 15, 2016
"Affine Springer fibers, compactified Jacobians and their asymptotic property"
Cheng-Chiang Tsai, Massachusetts Institute of Technology
Affine Springer fibers, being the zero loci of vector fields on the affine flag variety given by elements in the Lie algebra, enjoy interpretations in many other areas. They serve as a geometric tool for orbital integrals important for number theory. In the $GL_n$ case they appear as compactified Jacobians of planar curve singularities, which are then related to Hilbert schemes and knot invariants for the knots associated to the singularities. The latter is furthermore related to representations of graded double affine Hecke algebras. We will introduce the objects and these connections, and present an asymptotic property of affine Springer fibers.
Tuesday, November 22, 2016
"Stables classes and special Lagrangian graphs"
Adam Jacob, University of California, Davis
In their famous work on calibrated geometry, Harvey-Lawson introduce an equation called the special Lagrangian graph equation with potential, and outline how it defines an area minimizing surface with high codimension. In this talk, I will describe a complex version of this equation, and demonstrate how it can be derived using mirror symmetry. I will discuss criteria for existence of solutions on complex manifolds, and introduce a conjecture relating existence in general to an algebro-geometric notion of stability, which is inspired by slope stability for holomorphic vector bundles. This is joint work with Tristan C. Collins and S.-T. Yau.
Tuesday, November 29, 2016 *SPECIAL COLLOQUIUM*
"Harry Potter's Cloak Via Transformation Optics"
Gunther Uhlmann, University of Washington
Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last thirteen years transformation optics, a very simple mathematical method, was proposed as a general procedure to achieve invisibility for several types of waves. We will describe this method and applications in this talk.
Tuesday, December 6, 2016
NO COLLOQUIUM
