UW-PIMS Mathematics Colloquium

Sponsored by the UW Department of Mathematics and the Pacific Institute for the Mathematical Sciences.

April 29, 2016 at 2:30pm
MEB 248
Anthony Várilly-Alvarado
Rice University
K3 surfaces and cubic fourfolds

The group E(Q) of points on an elliptic curve over the rational numbers is known to be finitely generated, by work of Mordell. In an astonishing 1977 paper, Barry Mazur proved there are only 15 possibilities for the torsion subgroup of E(Q). I will discuss a conjecture on the arithmetic of K3 surfaces that I will argue is an analogue to Mazur's theorem, despite the absence of a group law on a K3 surface! I will offer some initial evidence for this conjecture, based on joint work with McKinnie, Sawon, and Tanimoto, as well as separate joint work with Tanimoto on moduli of special cubic fourfolds.

April 22, 2016 at 2:30pm
MEB 248
June Huh
Princeton University
Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids.

Joint work with Karim Adiprasito and Eric Katz. The talk will be accessible to a general audience, see http://matroidunion.org/?p=1664 for some details.

April 1, 2016 at 2:30pm
MEB 248
Katherine E. Stange
University of Colorado, Boulder
A Farey Tour

I will begin with the familiar Farey sequences: the subdivisions of the real line obtained by recursively taking the mediant (a+c)/(b+d) of fractions a/b and c/d. I will take this starting point as an excuse for an ecclectic tour which may include such topics as continued fractions, topographs, hyperbolic geometry, ergodic theory, and even Apollonian circle packings. In particular, I'd like to generalize these classical connections to the case of Gaussian integers and beyond.

March 4, 2016 at 2:30pm
MEB 248
Dusa McDuff
Barnard College
Embedding Questions in Symplectic Geometry

A symplectic structure is a kind of geometric structure that can be put on an even dimensional space. It generalizes the notion of an area form in two dimensions in such a way that many of the special dynamical features of area preserving geometry (such as the existence of extra fixed points) persist. However the global properties of this geometry are still not well understood. This will be a talk for nonspecialists that explains some known facts and some open problems, concentrating on recent progress in understanding symplectic embeddings.

February 12, 2016 at 2:30pm
MEB 248
Brian Conrad
Stanford University
The ABC Conjecture

The ABC Conjecture, formulated in the mid-1980's by Oesterlé and Masser, is one of the most important conjectures in number theory. It has many deep consequences, but its basic formulation can be given in entirely elementary terms. In September 2012, a 500-page solution was announced by Shinichi Mochizuki (building on several thousand pages of work that he has carried out over the last 20 years). I will explain what the conjecture asserts, some evidence that supports it, a few of its consequences, and some of the circumstances surrounding the evaluation of the proof by the number theory community.

November 20, 2015 at 2:30pm
SIG 225
Jennifer McLoud-Mann
University of Washington, Bothell
i-Block Transitive Tilings by Convex Pentagons

This talk will concern tilings of the plane by polygons. A tiling (or tessellation) of the plane is a covering of the plane by shapes (called the tiles of the tiling) in which the interiors of the tiles are disjoint. In particular, in this talk we will discuss monohedral tilings by convex polygons; that is, tilings in which all tiles are congruent to a single convex polygon. Convex n-gons admitting tilings of the plane have been classified for all n ≠ 5. We will discuss the history of the search for all convex pentagons that tile the plane and our recent contribution to this long-standing unsolved problem.

All known types of convex pentagons that admit tilings of the plane also admit tilings with a special kind of symmetry called i-block transitivity. We will present combinatorial results on pentagons that admit i-block transitive tilings. These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings. We will present the methods of this algorithm and the results of our computer search so far, which includes a complete classification of all convex pentagons admitting 1-, 2-, and 3-block transitive tilings, among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.

November 13, 2015 at 2:30pm
SIG 225
Sham M. Kakade
University of Washington
Tensor Decomposition Approaches for Learning Mixture Models

In many applications, we face the challenge of modeling the interactions between multiple observations and hidden causes; such problems range from clustering points in space to document retrieval, where we seek to model the underlying topics, to community detection in social networks. The (unsupervised) learning problem is to accurately estimate the model (e.g. the the underlying clusters, hidden topics, or the hidden communities in a social network) with only samples of the observed variables. In practice, many of these models are fit with local search heuristics. This talk will show how (provably accurate) tensor based approaches provide closed form estimation methods for a wide class of these models.

October 16, 2015 at 2:30pm
SIG 225
Ivan Corwin
Columbia University and
the Clay Mathematics Institute
A Drunk Walk in a Drunk World

In a simple symmetric random walk on Z a particle jumps left or right with 50% chance independently at each time and space location. What if the jump probabilities are taken to be random themselves (e.g. uniformly distributed between 0% and 100%)? In this talk we will describe the effect of this random environment on a random walk, in particular focusing on a new connection to the Kardar-Parisi-Zhang universality class and to the theory of quantum integrable systems. No prior knowledge or background will be expected.

October 2, 2015 at 2:30pm
SIG 225
Amnon Yekutieli
Ben Gurion University
Nonabelian Multiplicative Integration on Surfaces

Nonabelian multiplicative integration on curves is a classical theory, going back to Volterra in the 19th century. In differential geometry this operation can be interpreted as the holonomy of a connection along a curve. In probability theory this is a continuous-time Markov process.

This talk is about the 2-dimensional case. A rudimentary nonabelian multiplicative integration on surfaces was introduced in the 1920's by Schlesinger, but it is not widely known.

I will present a more sophisticated construction, in which there is a Lie group H, together with an action on it by another Lie group G. The multiplicative integral is an element of H, and it is the limit of Riemann products. Each Riemann product involves a fractal decomposition of the surface into kites (triangles with strings). There a twisting of the integrand, that comes from a 1-dimensional multiplicative integral along the strings, with values in the group G.

My main result is a 3-dimensional nonabelian Stokes Theorem. This result is new; only a special case of it was predicted (without proof) in papers in mathematical physics.

The motivation for my work was a problem in twisted deformation quantization. It is related to algebraic geometry (the structure of gerbes), algebraic topology (nerves of 2-groupoids), and mathematical physics (nonabelian gauge theory). I will say a few words about these relations at the end of the talk.

The talk itself is a computer presentation with numerous color pictures. I recommend printing a copy of the notes before the talk, from the link below. The talk should be accessible to a wide mathematical audience.

Lecture notes

Colloquium Video Archive

Archives from Previous Years
    Archive of 2014-2015 colloquia
Archive of 2013-2014 colloquia   Archive of 2012-2013 colloquia
Archive of 2011-2012 colloquia   Archive of 2010-2011 colloquia
Archive of 2009-2010 colloquia   Archive of 2008-2009 colloquia
Archive of 2007-2008 colloquia   Archive of 2006-2007 colloquia
Archive of 2005-2006 colloquia   Archive of 2004-2005 colloquia
Archive of 2003-2004 colloquia   Archive of 2002-2003 colloquia
Archive of 2001-2002 colloquia   Archive of 2000-2001 colloquia
Archive of 1999-2000 colloquia   Archive of 1998-1999 colloquia

The University of Washington is committed to providing access, equal opportunity and reasonable accommodation in its services, programs, activities, education and employment for individuals with disabilities. To request disability accommodation contact the Disability Services Office at least ten days in advance at: 206.543.6450/V, 206.543.6452/TTY, 206.685.7264 (FAX), or e-mail at dso@u.washington.edu.


U of W Website Terms & Conditions    |    PRINTER FRIENDLY FORMAT   |   U of W Online Privacy Statement
Please send comments, corrections, and suggestions to: webmaster[at]math.washington.edu
Last modified: May 6, 2016, 08:23

Bookmark and Share