Current seminars
December 30, 2012 – January3rd, 2013
Colloquium
Time and Place: Thursday January 3rd, 14:30-15:30, Mathematics Building,
Lecture Hall 2
Speaker: Dror Bar Natan (Toronto)
Title: Meta-Groups, Meta-Bicrossed-Products, and the Alexander
Polynomial
Abstract:
The a priori expectation of first year elementary school students who were just
introduced to the natural numbers, if they would be ready to verbalize it, must
be that soon, perhaps by second grade, they'd master the theory and know all
there is to know about those numbers. But they would be wrong, for number
theory remains a thriving subject, well-connected to practically anything there
is out there in mathematics.
I was a bit more sophisticated when I first heard of knot theory. My first
thought was that it was either trivial or intractable, and most definitely, I wasn't
going to learn it is interesting. But it is, and I was wrong, for the reader of knot
theory is often lead to the most interesting and beautiful structures in topology,
geometry, quantum field theory, and algebra.
Today I will talk about just one minor example, mostly having to do with the
link to algebra: A straightforward proposal for a group-theoretic invariant of
knots fails if one really means groups, but works once generalized to metagroups (to be defined). We will construct one complicated but elementary
meta-group as a meta-bicrossed-product (to be defined), and explain how the
resulting invariant is a not-yet-understood yet potentially significant
generalization of the Alexander polynomial, while at the same time being a
specialization of a somewhat-understood "universal finite type invariant of wknots" and of an elusive "universal finite type invariant of v-knots".
http://www.math.toronto.edu/~drorbn/Talks/Israel-1301/;
Light refreshments will be served after the colloquium in the faculty lounge
at 15:30.
YOU ARE CORDIALLY INVITED
GAME THEORY AND MATHEMATICAL ECONOMICS
RESEARCH SEMINAR
Speaker: Yuval Peres, Microsoft Research
Topic: Random turn games and their connections to percolation, optimal
Lipschitz extensions and bargaining in networks
Place: Elath Hall, 2nd floor, Feldman Building, Givat-Ram Campus
Time: Sunday, December 30, 2012 at 4:00 p.m.
Refreshments available at 3:45 p.m.
YOU ARE CORDIALLY INVITED
Abstract:
In a random-turn game, the right to move in each step is decided by a coin
toss. We will describe some random turn games and their applications:
1. "Hex" has two players who take turns placing stones hexagons of a
hexagonal grid. Black wins by completing a horizontal crossing, while
White wins by completing a vertical crossing. Although ordinary Hex is
famously difficult to analyze, Random-Turn Hex--in which players toss a
coin before each turn to decide who gets to place the next stone--has a
simple optimal strategy, closely related to percolation. Determining the
expected duration of these games is an open problem; the best lower bound
is related to influence of Boolean functions.
2. "Tug of war" is a zero-sum random turn game played in a graph, where in
each turn the player who won the coin toss gets to move the game token
from its current node to a neighboring node of his/her choice. A payoff
function F is given on a set of terminal nodes. The value of the game yields
the unique optimal Lipschitz extension of F to the rest of the graph, and the
scaling limit of this process yields solutions of the Infinity-Laplacian PDE.
3. Bargaining games on exchange networks have been studied extensively.
A balanced outcome for such a game is an equilibrium concept that
combines notions of stability and fairness. We obtain a tight polynomial
bound on the rate of convergence of local balancing dynamics on certain
graphs by relating these dynamics to a tug-of-war game via time reversal.
This lecture is based on the following joint papers:
[1] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Random-turn hex
and other selection games. Amer. Math. Monthly 114 (2007), no. 5, 373387.
http://
arxiv.org/abs/math/0508580
[2] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and
the infinity Laplacian. J. Amer. Math. Soc. 22 (2009), no. 1, 167-210.
[3] E. Celis, N. Devanur and Y. Peres, Local Dynamics in Bargaining
Networks via Random-Turn Games,
In Proc. Workshop on Internet Economics (WINE)
2010, http://research.microsoft.com/enus/um/people/nikdev/pubs/bargainingrtg.pdf
David Kazhdan’s Sunday seminars:
A Topic Course on Algebraic Geometry
Time and Place: Sunday, June 2nd, 9:00-12:00, Math 110
Speaker: Michael Temkin (HUJI)
Abstract:
This course assumes a basic knowledge of algebraic geometry on the level
of a one year introductory course (e.g. chapters II-IV in Hartshorne's book).
The main theme we will study are moduli spaces of smooth curves and their
compactifications. This involves studying various important topics and
techniques, such as stable curves, Hilbert schemes, Grothendieck topologies,
stacks, moduli spaces of stable n-pointed curves, and stable reduction
theorem. As the main application of the developed theory we will prove in
the end of the course de Jong?s theorems on semi-stable families and
desingularization of varieties by alterations.
Seminar: Introduction to Infinity Categories (co-organizer
Emmanuel Farjoun)
General Abstract:
Infinity categories naturally appear in many different fields of
mathematics. The goal of the seminar will be to give an introduction to
infinity categories, using language of complete Segal spaces, introduced by
Rezk.
During the first part of the seminar we will follow a beautiful of paper of
Rezk "A model for the homotopy theory of homotopy
theory" http://arxiv.org/abs/math/9811037.
We will start almost from the scratch, assuming only very basic knowledge
of simplicial sets and model categories.
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Lecture 11
Time and Place: Sunday, June 2nd, 12:00-13:30, Math 110,
Speaker: Yakov Varshavsky (HUJI)
Title: Introduction to (infinity,2)-categories -Continuation
Abstract:
We will follow another beautiful paper of Rezk "A cartesian presentation of
weak n-categories", http://arxiv.org/abs/0901.3602
Seminar on "Mirror symmetry for K3 surfaces”
Time and Place: Sunday, June 2nd, 14:00-16:00, Math 110
Speaker: Michael Temkin (HUJI)
Abstract: continuation
In this talk we will start to discuss the non-archimedean part of the story. I
will review some basic facts about Berkovich analytic spaces. In particular, I
will discuss the structure of analytic curves and, if time permits, abelian
varieties.
Combinatorics seminar
Time and Place: Monday December 31, 11:00-13:00, Math 209
Speaker: Yuval Peres, Microsoft Research
Title: Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
Abstract:
A planar set that contains a unit segment in every direction is called a
Kakeya set. These sets have been studied intensively in geometric measure
theory and harmonic analysis since the work of Besicovich (1928); we find a
new connection to game theory and probability. A hunter and a rabbit move
on a connected n-vertex graph without seeing each other. At each step, the
hunter moves to a neighboring vertex or stays in place, while the rabbit is
free to jump to any node. Thus they are engaged in a zero sum game, where
the payoff is the capture time. On any n-vertex graph, the expected capture
time is between order n and order n log(n), see [1] and [2] . We show that on
the cycle, every rabbit strategy yields a Kakeya set; the optimal rabbit
strategy is based on a discretized Cauchy random walk, and it yields a
Kakeya set K consisting of 4n triangles, that has minimal area among such
sets (the area of K is of order 1/log(n)).
(Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).
In the remaining time, I will discuss some open problems and surprising
facts about search games and random walks on finite graphs:
1) If the rabbit and hunter must both respect the edges of the graph, is the
mean capture time always linear?
2) On which sequences of graphs can lazy simple random walks have cutoff
(a sharp transition in mixing)?
3) Specifically, is cutoff possible on trees? On transitive expanders of
bounded degree?
4) How much can the mixing time of lazy walk on a graph vary when the
edge conductances are changed by a bounded factor?
[1] M. Adler, H. Racke, N. Sivadasan, C. Sohler, and B. Vocking.
Randomized pursuit-evasion in graphs. Combinatorics, probability and
computing 12(3):225--244, 2003.
[2] Y. Babichenko, Y. Peres, R. Peretz, P. Sousi, and P. Winkler. Hunter,
Cauchy Rabbit,
and Optimal Kakeya Sets. Available at arXiv:1207.6389.
Jerusalem Number Theory Seminar
Time and Place: Monday, 31 December 2012, 16:00, Room 209.
Speaker: Dmitry Trushin (HUJI)
Title: Algebraization of a Cartier divisor.
Abstract: I will talk about my recent results extending to pairs classical
theorems of R. Elkik on lifting of homomorphisms and algebraization. This
solves affirmatively a problem raised by M. Temkin and has applications to
desingularization theory.
Dynamics Seminar -no seminar this week
Time and Place: Tuesday, , 14:00, Mathematics Building, Room 209
Speaker:
Title:
Abstract:
The Topology and Geometry Seminar
Time and Place: Wednesday, January 2nd, 11:00, Room TBA
Speaker: Dror Bar Natan (Toronto)
Title: Balloons and hoops and their universal finite type invariant
Abstract:
Balloons are two-dimensional spheres. Hoops are one dimensional loops.
Knotted Balloons and Hoops (KBH) in 4-space behave much like the first
and second fundamental groups of a topological space - hoops can be
composed like in \pi_1, balloons like in \pi_2, and hoops "act" on balloons
as \pi_1 acts on \pi_2. We will observe that ordinary knots and tangles in 3space map into KBH in 4-space and become amalgams of both balloons and
hoops. We give an ansatz for a tree and wheel (that is, free-Lie and cyclic
word) -valued invariant \zeta of KBHs in terms of the said compositions and
action and we explain its relationship with finite type invariants. We
speculate that \zeta is a complete evaluation of the BF topological quantum
field theory in 4D, though we are not sure what that means. We show that a
certain "reduction and repackaging" of \zeta is an "ultimate Alexander
invariant" that contains the Alexander polynomial (multivariable, if you
wish), has extremely good composition properties, is evaluated in a
topologically meaningful way, and is least-wasteful in a computational
sense. If you believe in categorification, that's a wonderful playground.
Mathematical Logic seminar
Time and Place: Wednesday, January 2nd, 15:00, Mathematics Building,
Room 209
Speaker: Salma Kuhlmann
Title: The valuation difference rank of a difference field
Abstract:
The theory of real places and convex valuations is a special chapter in
valuation theory. An important invariant is the rank of a valuation, which
has several equivalent characterizations: via the chain of ideals of the
valuation ring, the chain of convex subgroups of the value group, or the
chain of final segments of the value set of the value group. In this talk, we
explain how to extend these notions to valued fields with extra structure,
giving a characterization completely analogous to the above, but taking into
account the corresponding induced structure on the ideals, convex subgroups
and final segments. For the case of an (ordered) difference field, we develop
the notion of difference compatible valuations, introduce the difference rank,
naturally relating it to the growth rate of the automorphism. We analyze the
difference rank in particular for isometries and $\omega$-increasing
automorphisms.
Joint work with M. Matusinski and F. Point.
Group Dynamics seminar
Time and Place: Thursday, January 3rd, 10:00, Math. 209
Speaker: Amir Yehudayoff (Technion)
Title: Rounding group actions: an example
Abstract:
We shall discuss (numerical) rounding of group actions by following an
example: Consider the group G(F) of one-dimensional affine
transformations over a field F. When F is say the rationals, G(F) is solvable
and amenable. When F = Z/pZ is finite, analogously, G(F) does not yield
Cayley graphs that are expanders [Lubotzky and Weiss]. What happens if
we consider the rounding of the action of G(Q) on G(Z/pZ)? Standard
arguments for solvable groups basically imply that the rounding of the
rational action does not yield expanders (Klawe proved that the rounding of
the action of G(Q) on the line Z/pZ can not yield expanders, by explicitly
constructing a set violating expansion). What happens if we consider the
rounding of the action of G(R) on G(Z/pZ) or Z/pZ? This is unknown but we
shall discuss some connections to Diaphontine approximations.
Special Jerusalem Analysis and PDEs seminar
Speaker: Sylvia Serfaty (Paris 6)
Title: Coulomb gas, Renormalized energy and Abrikosov lattice
Date: Wednesday, January 2nd.
Time: 12:00
Place: TBA
Abstract:
We are interested in the statistical mechanics of (classical) two-dimensional
Coulomb gases and one-dimensional log gases in a confining potential. We connect the
Hamiltonian to the normalized
energy", a way to compute the total Coulomb interaction of an infinite jellium, and whose
minimum is expected to be achieved by the "Abrikosov" triangular lattice in 2D, and is achieved
by the lattice in 1D. We apply this to the study of the finite temperature situation. Results
include computations of the next order term in the partition function, equidistribution of
charges, and concentration to the minimizers of the renormalized energy as the temperature
tends to zero. This is based on joint works, mostly with Etienne Sandier.
Jerusalem Analysis and PDEs seminar
Speaker: Reuven Segev (BGU)
Title: Some Mathematical Aspects of Stress Theory
Time and Place: Thursday, January 3rd, 13:00, Manchester 110.
Abstract:
Stress tensors are used in strength analysis of structures, fluid dynamics,
electromagnetism, and general relativity. Yet, from the theoretical point of
view, the stress tensor object is not a primitive one. It is derived on the basis
of some physically motivated mathematical assumptions. Following a short
introduction to the fundamentals of the classical theory, we will consider the
following applications of mathematical notions---primarily, those related to
duality---to the theoretical aspects of the mechanics of continuous media.
1. A formulation of stress theory on differentiable manifolds, devoid
of any particular Riemannian metric, in which properties of stresses emerge
naturally from the structure of the configuration space, a manifold of
mappings, of a material body in the physical space.
2. Optimal stress distributions that equilibrate given external loadings
and the related notion of load capacity ratio--- a purely geometric property
of bodies that indicates their capacity to sustain a-priori unknown external
loading distributions. The load capacity ratio is related to the norm of the
trace mapping of Sobolev spaces.
3. A formulation of flux theory for irregular bodies, including some
fractal sets, based on Whitney’s geometric integration theory.
Basic Notions Seminar
Time and Place:
Thursday, January 3rd, 16:00, Math., Lecture Hall 2
Speaker: Guy Kindler (CS- Hebrew University)
Title: Continuation: Probabilistically Checkable Proofs.
Abstract: The PCP theorem is one of the most important and profound
achievements of theoretical computer science in the past two decades. It
roughly states that the proof of any mathematical statement, once properly
formatted, can be verified to a high degree of certainty by reading only a
constant number of letters from it.
The proof of the theorem relies on many preceding results methods and
ideas, however by now the proof has been greatly simplified. In the talk I
will explain the formal statement of the theorem and demonstrate some of
the ideas used in the proof, many of which are interesting in their own right.