Seminars and Colloquia by Series

Monday, January 5, 2015 - 11:00 , Location: Skiles 005 , Michael Damron , Indiana University , Organizer: Christian Houdre
In first-passage percolation (FPP), one places random non-negative weights on the edges of a graph and considers the induced weighted graph metric. Of particular interest is the case where the graph is Z^d, the standard d-dimensional cubic lattice, and many of the questions involve a comparison between the asymptotics of the random metric and the standard Euclidean one. In this talk, I will survey some of my recent work on the order of fluctuations of the metric, focusing on (a) lower bounds for the expected distance and (b) our recent sublinear bound for the variance for edge-weight distributions that have 2+log moments, with corresponding concentration results. This second work addresses a question posed by Benjamini-Kalai-Schramm in their celebrated 2003 paper, where such a bound was proved for only Bernoulli weights using hypercontractivity. Our techniques draw heavily on entropy methods from concentration of measure.
Thursday, December 11, 2014 - 11:00 , Location: Skiles 006 , Gang Zhou , California Institute of Technology , Organizer: Chongchun Zeng
It is known that certain medium, for example electromagnetic field and Bose Einstein condensate, has positive speed of sound. It is observed that if the medium is in its equilibrium state, then an invading subsonic particle will slow down due to friction; and the speed of a supersonic particle will slow down to the speed of sound and the medium will radiate. This is called Cherenkov radiation. It has been widely discussed in physical literature, and applied in experiments. In this talk I will present some rigorous mathematical results. Joint works with Juerg Froehlich, Israel Michael Sigal, Avy Soffer, Daniel Egli, Arick Shao.  
Tuesday, December 9, 2014 - 14:05 , Location: Skiles 006 , Wei-Kuo Chen , University of Chicago , Organizer: Christian Houdre
Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. This talk will be focused on one of the famous mean-field spin glasses, the Sherrington-Kirkpatrick model. We will present results on the conjectured properties of the Parisi measure including its uniqueness and quantitative behaviors. This is based on joint works with A. Auffinger.
Thursday, December 4, 2014 - 14:00 , Location: Skiles 005 , Choongbum Lee , MIT , Organizer: William T. Trotter
The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated result of Shelah from 1988 gives a significantly improved bound for this theorem. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. Hoping to further improve Shelah's result, more than twenty years ago, Graham, Rothschild and Spencer asked whether there exists a polynoimal bound for this lemma. In this talk, we present the answer to their question and discuss numerous connections of the cube lemma with other problems in Ramsey theory. Joint work with David Conlon (Oxford), Jacob Fox (MIT), and Benny Sudakov (ETH Zurich).
Tuesday, December 2, 2014 - 11:00 , Location: Skiles 005 , Eviatar Procaccia , University of California, Los Angeles , Organizer: Heinrich Matzinger
Experimentalists observed that microscopically disordered systems exhibit homogeneous geometry on a macroscopic scale. In the last decades elegant tools were created to mathematically assert such phenomenon. The classical geometric results, such as asymptotic graph distance and isoperimetry of large sets, are restricted to i.i.d. Bernoulli percolation. There are many interesting models in statistical physics and probability theory, that exhibit long range correlation. In this talk I will survey the theory, and discuss a new result proving, for a general class of correlated percolation models, that a random walk on almost every configuration, scales diffusively to Brownian motion with non-degenerate diffusion matrix. As a corollary we obtain new results for the Gaussian free field, Random Interlacements and the vacant set of Random Interlacements. In the heart of the proof is a new isoperimetry result for correlated models.
Tuesday, November 25, 2014 - 11:00 , Location: Skiles 006 , Dr Anna Vershynina , Institute for quantum information, RWTH University Aachen, Germany , , Organizer: Jean Bellissard

Anna Vershynina is a job candidate. She is a Mathematical Physicist working on the rigorous mathematical theory of N-body problem and its relation with quantum information.

Entanglement is one of the crucial phenomena in quantum theory. The existence of entanglement between two parties allows for notorious protocols, like quantum teleportation and super dense coding. Finding a running time for many quantum algorithms depends on how fast a system can generate entanglement. This raises the following question: given some Hamiltonian and dissipative interactions between two or more subsystems, what is the maximal rate at which an ancilla-assisted entanglement can be generated in time. I will review a recent progress on  bounding the entangling rate in a closed bipartite system. Then I will generalize the problem first to open system and then to a higher multipartite system, presenting the most recent results in both cases.
Tuesday, November 18, 2014 - 12:00 , Location: Skiles 006 , Galyna Livshyts , Kent State University , Organizer: Karim Lounici
The perimeter of a convex set in R^n with respect to a given measure is the measure's density averaged against the surface measure of the set. It was proved by Ball in 1993 that the perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of the measure's natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related questions on the geometry and isoperimetric properties of log-concave measures.
Thursday, November 13, 2014 - 13:00 , Location: Skiles 006 , Shahaf Nitzan , Kent State University , Organizer:
Orthonormal bases (ONB) are used throughout mathematics and its applications. However, in many settings such bases are not easy to come by. For example, it is known that even the union of as few as two intervals may not admit an ONB of exponentials. In cases where there is no ONB, the next best option is a Riesz basis (i.e. the image of an ONB under a bounded invertible operator). In this talk I will discuss the following question: Does every finite union of rectangles in R^d, with edges parallel to the axes, admit a Riesz basis of exponentials? In particular, does every finite union of intervals in R admit such a basis? (This is joint work with Gady Kozma).
Thursday, February 13, 2014 - 15:05 , Location: Skiles 005 , Vladimir Itskov , U. of Nebraska , Organizer:
Experimental neuroscience is achieving rapid progress in the ability to collect neural activity and connectivity data. This holds promise to directly test many theoretical ideas, and thus advance our understanding of "how the brain works." How to interpret this data, and what exactly it can tell us about the structure of neural circuits, is still not well-understood.  A major obstacle is that these data often measure quantities that are related to more "fundamental" variables by an unknown nonlinear transformation. We find that combinatorial topology can be used to obtain meaningful answers to questions about the structure of neural activity. In this talk I will first introduce a new method, using tools from computational topology, for detecting structure in correlation matrices that is obscured by an unknown nonlinear transformation.  I will illustrate its use by testing the "coding space" hypothesis on neural data. In the second part of my talk I will attempt to answer a simple question: given a complete set of binary response patterns of a network, can we rule out that the network functions as a collection of disconnected discriminators (perceptrons)?  Mathematically this translates into questions about the combinatorics of hyperplane arrangements and convex sets.
Tuesday, February 4, 2014 - 11:00 , Location: Skiles 006 , Carina Curto , University of Nebraska-Lincoln , Organizer: Christine Heitsch
Synapses in many cortical areas of the brain are dominated by local, recurrent connections.  It has long been suggested, therefore, that cortical networks may serve to restore a noisy or incomplete signal by evolving it towards a stored pattern of activity.  These "preferred" activity patterns are constrained by the excitatory connections, and comprise the neural code of the recurrent network. In this talk I will briefly review the permitted and forbidden sets model for cortical networks, first introduced by Hahnloser et. al. (Nature, 2000), in which preferred activity patterns are modeled as "permitted sets" - that is, as subsets of neurons that co-fire at stable fixed points of the network dynamics. I will then present some recent results that provide a geometric handle on the relationship between permitted sets and network connectivity.  This allows us to precisely characterize the structure of neural codes that arise from a simple learning rule. In particular, we find "natural codes" that can be learned from few examples, and that closely mimic receptive field codes that have been observed in the brain. Finally, we use our geometric description of permitted sets to prove that these networks can perform error correction and pattern completion for a wide range of connectivities.