Upcoming Seminars & Events

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April 2, 2015
12:30pm - 1:30pm
Tropical Curves and Brill-Noether Theory

In the past few years, tropical geometry has established itself as an important new field bridging algebraic geometry and combinatorics whose techniques have been used successfully to attack problems in both fields. I'll talk about the basics of tropical geometry, with an emphasis on tropical curves. If time permits, I'll show how tropical geometry can be applied to prove results in Brill-Noether thoery of line bundles on algebraic curves. 

Speaker: Akash Sengupta , Princeton University
Location:
Fine Hall 314
April 2, 2015
2:00pm - 3:30pm
Minimal Self-Joinings of Substitutions Arising from IETs

In this talk we will discuss substitution systems that have the property of minimal self-joinings. Then we will focus our attention on self-similar interval exchange transformations and their associated substitutions. We will show that 3-IETs have MSJ. This is joint with Giovanni Forni. 

Speaker: Kelly Yancey, University of Maryland
Location:
Fine Hall 601
April 2, 2015
3:00pm - 4:00pm
Motivic Adams spectral sequence

To any field $k$, there is a motivic stable homotopy category of schemes over $k$. In this setting, one can construct a motivic Adams spectral sequence (MASS) which converges to something related to the homotopy groups of spheres. This talk will introduce the motivic stable homotopy category and show how the MASS over the complex numbers relates to the classical Adams spectral sequence. I will then discuss joint work with Paul Ostvaer on the MASS over finite fields.

Speaker: Glen Wilson , Rutgers University
Location:
Fine Hall 314
April 2, 2015
3:00pm - 4:00pm
Arithmetic Tutte polynomials and quasi-polynomials

We will introduce two invariants called the arithmetic Tutte polynomial and the Tutte quasi-polynomial, and we will survey their several applications, including: - generalization of graph colorings and flows to cellular complexes of higher dimension; - combinatorial and topological invariants of toric arrangements; - Ehrhart polynomials of zonotopes; - Hilberts series of some modules related to the vector partition function. The talk is partially based on joint work of the speaker with Petter Brändén, Michele D'Adderio and Alex Fink.

Speaker: Luca Moci, Jussieu, U Paris 7
Location:
Fine Hall 224
April 2, 2015
4:30pm - 5:30pm
Botany of transverse knots

We study transverse representatives of an oriented topological knot type K in the tight contact 3-sphere, where the classical invariant of transverse isotopy introduced by Bennequin is the self-linking number. The botany problem for transverse knots, namely deciphering which transverse isotopy classes exist at a fixed self-linking value, has remained open in general, although for certain knot types a number of interesting botanical features have been shown to exist. In this talk we present partial solutions of the botany problem that hold for arbitrary knot types K.

Speaker: Doug LaFountain, Western Illinois University
Location:
Fine Hall 314
April 2, 2015
4:30pm - 5:30pm
Complex Multiplication and K3 Surfaces over Finite Fields

In this talk I will review CM theory of complex projective K3 surfaces, and show how it can be used to construct K3 surfaces over finite fields. I will discuss work-in-progress where this is applied to describing: (1) the collection of zeta functions of K3 surfaces over a finite field, and (2) the category of ordinary K3 surfaces over a finite field. These are similar to theorems of Honda and Tate resp. Deligne on abelian varieties over finite fields.

Speaker: Lenny Taelman, University of Amsterdam
Location:
Fine Hall 214
April 3, 2015
3:00pm - 4:00pm
On the Bombieri-De Giorgi-Giusti minimal graph and it applications

I will discuss refinements to the Bombieri-De Giorgi-Giusti minimal graph and three applications: 1. De Giorgi's Conjecture for Allen-Cahn equation; 2. Caffarelli-Berestycki-Nirenberg Conjecture for overdetermined problem on epigraphs; 3. Translating graphs of mean curvature flows.

Speaker: Juncheng Wei, University of British Columbia
Location:
Fine Hall 314
April 3, 2015
3:00pm - 4:00pm
On symplectic homology of the complement of a normal crossing divisor

In this talk, we discuss our work in progress about how degeneration of the divisor at infinity into a normal crossing divisor affects the symplectic homology of an affine variety. From an anti-surgery picture, by developing an anti-surgery formula for symplectic homology similar to work by Bourgeois-Ekholm-Eliashberg, we show that essentially, the change in symplectic homology is reflected by the Hochschild invariants of the Fukaya category of a collection of Lagrangian spheres on the smooth divisor.

Speaker: Khoa Nguyen, Stanford University
Location:
IAS Room S-101
April 6, 2015
3:15pm - 4:30pm
TBA - Tim Austin
Speaker: Tim Austin, NYU
Location:
Fine Hall 314
April 6, 2015
4:30pm - 5:30pm
Dictionary Learning and Matrix Recovery with Optimal Rate

Let A be an n×n matrix, X be an n×p matrix and Y = AX.  A challenging and important problem in data analysis, motived by dictionary learning, is to recover both A and X, given Y. Under normal circumstances, it is clear that the problem is underdetermined. However, as showed by Spielman et. al., one can succeed when X is sufficiently sparse and random.  In this talk, we discuss a solution to a conjecture  raised by Spielman et. al. concerning the optimal condition which guarantees efficient recovery. The main  technical ingredient of our analysis is a novel way to use the ε-net argument in high dimensions for proving  matrix concentration, beating the standard union bound. This part is of independent interest. Joint work with K. Luh (Yale). 

Speaker: Van Vu, Yale University
Location:
Fine Hall 214
April 7, 2015
4:30pm - 5:30pm
A simple renormalization flow setup for FK-percolation models

We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is partly based on joint work with Stéphane Benoist and Laure Dumaz.

Speaker: Wendelin Werner, ETH, Zurich
Location:
Jadwin Hall 343
April 8, 2015
3:00pm - 4:00pm
Roots of random polynomials: Universality and Number of Real roots

Estimating the number of real roots of a polynomial is among the oldest and most basic questions in mathematics.The answer to this question depends very strongly on the structure of the coefficients, of course. What happens if we choose the coefficients randomly? In this case, the number of real roots become a random variable, whose value is between 0 and the degree of the polynomial. Can one understand this random variable? What is its mean, variance, and limiting distribution? Random polynomials were first studied by Waring in the 18th century. In the 1930s, Littlewood and Offord started their famous studied which led to the surprising fact that in general a random polynomials with iid coefficients have (with high probability), order log n real roots. Their investigation opened a whole new area of random polynomials and random functions, with deep contributions from several leading mathematicians, including Erdos, Turan, Kac, Kahane, Ibragimov etc. An exciting turn occurred in the 1990s, when physicists Bogomolny, Bohigas and Leboeuf established a link between roots of random polynomials and quantum chaotic dynamics. Since then, random polynomials and random series are also studied intensively by researchers from analysis, probability, and mathematical physics. In these studies, the complex roots become also important. One views the set of all roots as a random point process, and a fundamental problem, among others, is to understand the correlation between nearby points in a small region. (This correlation is often use to model the interaction between particles in certain physics problems.) In this talk, I will try to survey this fascinating area, presenting some of its main questions, results, and ideas. The talk is self-contained, and requires only basic knowledge in analysis and probability.

Speaker: Van Vu , Yale University
Location:
Fine Hall 214
April 9, 2015
2:00pm - 3:30pm
TBA - Dmitry Zakharov
Speaker: Dmitry Zakharov, Courant Institute of Mathematical Sciences
Location:
Fine Hall 601
April 9, 2015
3:00pm - 4:00pm
TBA - Yury Ustinovsky
Speaker: Yury Ustinovsky, Princeton University
Location:
Fine Hall 314
April 9, 2015
3:00pm - 4:00pm
The list decoding radius of Reed-Muller codes over small fields

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well-studied codes, like Reed-Solomon or Reed-Muller codes. The Reed-Muller code F(n,d) for a finite field F is defined by n-variate degree-d polynomials over F. As far as we know, the list decoding radius of Reed-Muller codes can be as large as the minimal distance of the code. This is known in a number of cases: --The Goldreich-Levin theorem and its extensions established it for d=1 (Hadamard codes); --Gopalan, Klivans and Zuckerman established it for the binary field; --and Gopalan established it for d=2 and all fields. In this work (joint with Lovett), we prove the conjecture for all constant prime fields and all constant degrees.  In this talk, I will present the proof for our first main theorem which states that the list decoding radius is exactly equal to the minimum distance of the Reed-Muller code for all constant prime fields and constant degrees. For ease of presentation, we will only see the case when d < char(F). The proof relies on several new ingredients, including higher-order Fourier analysis which I will introduce along the way.

Speaker: Abhishek Bhowmick, UT Austin
Location:
Fine Hall 224
April 9, 2015
4:30pm - 5:30pm
The Andre-Oort conjecture follows from the Colmez conjecture

The Andre-Oort conjecture says that any subvariety of a Shimura Variety with a Zariski dense set of CM points must itself be a shimura subvariety. In recent years, this has been the subject of much work. We explain how this conjecture for the moduli space of Pricnipally polarized abelian varieties of some dimension g follows from current knowledge and a conjecture of Colmez regarding the Faltings heights of CM abelian varieties- in fact, its enough to assume an averaged version of the colmez conjecture.

Speaker: Jacob Tsimerman, University of Toronto
Location:
IAS Room S-101
April 9, 2015
4:30pm - 5:30pm
Instantons and odd Khovanov homology

I will describe a spectral sequence that starts at reduced odd Khovanov homology and converges to a version of instanton homology for double branched covers.

Speaker: Christopher Scaduto, UCLA
Location:
Fine Hall 314
April 9, 2015
4:30pm - 6:00pm
Finite time singularity of a vortex patch model in the half plane

The question of global regularity vs. finite time blow-up remains open for many fluid equations. In this talk, I will discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. This is a joint work with A. Kiselev, L. Ryzhik and A. Zlatos. 

Speaker: Yao Yao , University of Wisconsin
Location:
Fine Hall 322
April 10, 2015
1:30pm - 2:30pm
Smith normal form and combinatorics

Please note special day (Friday) and location (Fine 224).  If $A$ is an $m\times n$ matrix over a PID $R$ (and sometimes more general rings), then there exists an $m\times m$ matrix $P$ and an $n\times n$ matrix $Q$, both invertible over $R$, such that $PAQ$ is a matrix that vanishes off the main diagonal, and whose main diagonal elements $e_1,e_2,\dots,e_m$ satisfy $e_i|e_{i+1}$ in $R$. The matrix $PAQ$ is called a \emph{Smith normal form} (SNF) of $A$. The SNF is unique up to multiplication of the $e_i$'s by units in $R$. We will discuss some aspects of SNF related to combinatorics. In particular, we will give examples of SNF for some combinatorially interesting matrices. We also discuss a theory of SNF for random matrices over the integers recently developed by Yinghui Wang.

Speaker: Richard Stanley, MIT
Location:
Fine Hall 224
April 10, 2015
1:30pm - 2:30pm
Equivalent notions of high-dimensional overtwistedness

In recent joint work with Borman and Eliashberg, a new definition of overtwisted contact structures was given for high-dimensional contact manifolds, which were then classified up to isotopy. However, the definition is fairly cumbersome, so much so that it is essentially impossible to check in any explicit example. This talk will focus on various criteria which are shown to be equivalent to overtwistedness, giving numerous explicit examples of overtwisted contact manifolds, and relating overtwisteness to a number of older works. In particular we show that negatively stabilized open books, looseness of the Legendrian unknot, and existence of a "nice" plastikstufe are all equivalent to overtwistedness. This project is joint work with R. Casals and F. Presas.

Speaker: Emmy Murphy, MIT
Location:
Fine Hall 322

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