# Colloquium

The Colloquium is held on Mondays at 4pm in Deady 208.

### Spring Quarter, 2017

- April 3,
**Adam Jacob** (UC Davis)

Stable classes and special Lagrangian graphs
**Abstract**: 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.

- April 17,
**Roger Heath-Brown** (Oxford University)

Iteration of quadratic polynomials over finite fields
**Abstract**: Let f(X) be a polynomial over a finite field, and define the iterates f_0(X)=x and f_{n+1}(X)=f(f_n(X)) for n\ge 0. Fix a strating point c in the finite field. What can one say about the trajectory of successive values f_0(c),f_1(c),f_2(c),…? The sequence must eventually recur, but how soon? How should one model this process?

- April 24,
**Elissa Ross** (MESH consultants)

A geometric challenge in digital 3D design: offsetting polygon meshes
**Abstract**: Architectural designs are frequently represented digitally as plane-faced polygon meshes, yet these can be challenging to translate into built structures. Offsetting operations may be used to give thickness to meshes, and are produced by offsetting the faces, edges or vertices of the mesh in an appropriately defined normal direction. Although mesh offsetting operations are a staple of CAD software, current methods are often inappropriate in an architectural setting where the mesh faces and edges may represent architectural elements such as panels, beams or windows.

In this talk I will describe a new algorithm for precisely resolving the details of an offset mesh obtained by face-offsetting, using tools from discrete geometry and algebraic graph theory. I will also give an overview of the activities of MESH Consultants Inc., and describe some of the other geometric challenges encountered in digital design.

- May 1, No colloquium this week – Niven lecture
- May 8,
**Zhiwei Yun** (Yale)

Spectral decomposition in Betti geometric Langlands
**Abstract**: This is joint work with David Nadler. We consider a variant of the geometric Langlands conjecture, which is expected to be of topological nature. It relates constructible sheaves on the moduli space of G-bundles on an algebraic curve (with an important condition on the singular support of the constructible sheaves) to quasi-coherent sheaves on certain character varieties of the dual group. We show that the latter category acts on the former, hence establishing a spectral decomposition of automorphic sheaves in this setting. There is a formal analogy of our result with Vincent Lafforgue’s work on the classical Langlands correspondence over a function field.

- May 15, No colloquium this week – Moursund lecture
- May 22,
**Paul Pollack** (Georgia)

Summing divisors: A status report on the first 2000 years
**Abstract**: For each natural number n, let s(n) be the sum of the proper divisors of n. For example, s(10)=1+2+5=8. Despite having such a simple definition, the function s is the centerpiece of many of the oldest unsolved problems in mathematics. I will survey these problems and discuss what we’ve learned in the past 2000 years.

### Winter Quarter, 2017

- February 17,
**Ivan Losev** (Northeastern University)

Characters
**Abstract**:What representation theorists usually do, they try to study characters (so Representation theory is related to Psychology). Unlike in Psychology, we care about characters of irreducible representations of algebraic objects which usually originate in Lie theory. I’ll review various developments in the subject starting 1900 or so and concentrate on algebraic objects that have to do with the general linear group GL_n: this group itself, its Lie algebra, and the corresponding Rational Cherednik algebra. We will discuss representation theory in characteristic 0, which is well established, and, time permitting, the representation theory in characteristic p, which is of great current interest.

- February 27,
**Tony Pantev** (UPenn)

Foliations, symplectic structures, and potentials
**Abstract**:I will explain how Lagrangian foliations in shifted symplectic geometry give rise to global potentials. I will give natural constructions of isotropic foliations on moduli spaces and will discuss the associated potentials. I will also give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.

### Fall Quarter, 2016

- September 26,
**Yashar Ahmadian** (UO)

Structure and stability in canonical cortical computations
**Abstract**: The cerebral cortex, or the gray matter, is evolutionarily the newest part of the brain, and underlies most of our intelligent behavior. After an introduction to biological neural networks and theoretical approaches to studying their dynamics, I will present my work on the dynamics of local neural circuits in the cortex. The vast majority of cortical neurons are of the excitatory type and they are highly interconnected: a typical neuron receives thousands of excitatory inputs and in turn excites other neurons. How does such a network prevent runaway activity despite this strong positive feedback? Single-neuronal biology provides one stabilizing mechanism: neurons cannot activate at indefinitely high rates for biophysical reasons. But this still leaves open the following question: can cortical networks self-organize into a stable state with moderate activity, without relying on single-neuronal saturation? I will show that fast feedback from the minority of inhibitory neurons is generically sufficient to dynamically stabilize cortical networks even when single-neuronal nonlinearities are of the expansive, non-saturating type. I will then explore the computational consequences of this collective inhibitory stabilization, and show that it accounts for a wide range of “contextual modulation” effects (modulations of responses of neurons to their preferred stimuli by the sensory context). Contextual modulation is a ubiquitous and canonical brain computation, and is an early manifestation of the global integration of sensory information that underlies higher level perception and object recognition.

If time allows I may briefly cover some of the following as well:

Networks studied in neuroscience, and more generally in mathematical biology, have connectivity that includes random disorder about some underlying average regular structure. I will summarize results of my recent work characterizing large NxN random matrices of the form A = M + LJR, where M, L and R are arbitrary deterministic matrices and J is a random matrix of zero-mean i.i.d. elements. The results include general formulae for the spectral distribution of such matrices, as well as for the linear response of networks coupled by A. I will discuss one application of these results in mapping out the phase diagram of a neural network which transitions between a chaotic phase and a “glassy” phase with a large number of random attractors.

- October 3,
**Stefan Tohaneanu** (University of Idaho)

From linear codes to generalized star configurations and beyond
**Abstract**: A code is a mathematical gadget that helps detect and correct errors occurring during message transmissions through insecure channels. A linear code is the image of a linear map, and the invariant that helps with the error-correction goal is the minimum distance. An exercise of De Boer and Pellikaan shows an interesting recursive method to determine the minimum distance of any linear code: if one considers the linear forms dual to the columns of the generating matrix of the code, then the minimum distance can be determined by analyzing whether or not the projective varieties that are the zero locus of fold products of these linear forms are empty or not. Generalized star configurations are projective schemes defined by ideals generated by fold products of such linear forms. My interest is in studying homological properties of these schemes, ultimate goal being to prove the (difficult?) conjecture that these ideals have linear graded free resolutions. One advantage for having this conjecture true is that error-correction can be done via an elegant commutative algebraic simple operation: the colon (quotient) of an ideal by a variable.

- October 10,
**Tyler Helmuth** (UC Berkeley)

Dimensional Reduction for Generalized Continuum Polymers
**Abstract**: A striking example of dimensional reduction established by Brydges and Imbrie is an exact relation between the hard sphere gas in d dimensions and branched polymers in d+2 dimensions. After introducing these probability models I will discuss a new proof of this result, which also establishes dimensional reduction formulas for generalized models of polymers associated to central hyperplane arrangements. The new proof is essentially combinatorial, in contrast to the original proof which uses supersymmetry.

- October 24,
**Scott Baldridge** (LSU)

The Mathematical Foundations of Mathematics Education: A Maturing Field of Study
**Abstract**: In this talk we discuss the field of the mathematical foundations of mathematics education, the fruitful relationships between this field and other fields in mathematics education (Curriculum, Instruction, Assessments, Standards), and how it has contributed to curricula like Eureka Math/EngageNY.

- November 7,
**Peter Samuelson** (U Iowa)

The Homfly skein and elliptic Hall algebras
**Abstract**: The Homfly skein algebra was introduced by Turaev in the early 90’s to quantize the Goldman Lie algebra associated to a surface. It is defined in terms of curves on the surface modulo “skein relations” which come from knot theory. Morton and I recently gave an explicit description of this algebra for the torus. We also showed it is isomorphic to the elliptic Hall algebra of Burban and Schiffmann, which is defined in terms of coherent sheaves over elliptic curves in finite characteristic. (This talk covers joint work with Morton and won’t assume prior knowledge of the objects involved.)

- November 14,
**Jayadev Athreya** (U Washington)

**Special Time:** 3pm in 123 Pacific

How random are the rationals?
**Abstract**: We tell a story involving some basic number theory, Euclidean and hyperbolic geometry, and a smattering of probability theory, to try and understand the very ill-posed question in the title. Expect lots of pictures!

- November 21,
**Nathan Dunfield** (U Illinois)

Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic
**Abstract**:From the revolutionary work of Thurston and Perelman, we know the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions.

After a passing wave at the recent breakthrough results of Agol, I will focus on exciting new questions about the geometric and arithmetic meaning of torsion in the homology of finite covers of hyperbolic 3-manifolds, motivated by the recent work of Bergeron, Venkatesh, Le, and others. I will include some of my own results in this area that are joint work with F. Calegari and J. Brock.

- November 28,
**Ailsa Keating** (Columbia University)

On two-variable singularities, planar graphs and mapping class groups
**Abstract**: Start with a two-variable complex polynomial f with an isolated critical point at the origin. One can associate to it a smooth Riemann surface with boundary: the Milnor fiber of f, M_f, given by a smoothing of f near the origin. This comes equipped with a distinguished collection of S^1s on M_f, called vanishing cycles.

We explain how an algorithm of A’Campo allows one to encode all of this information in a planar graph. Time allowing, we will discuss consequences for the mapping class group of M_f. No prior knowledge of singularity theory will be assumed.

**Previous years:** 2015 2013 2012 2011 2010