Distinguished Visitors - Current Events - Department of Mathematics

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Distinguished Visitors

Schedule for 2008-2009

Organizer: Maggy Tomova

March 2009

March 2, 2009

Speaker Maxim Kontsevich, Institut des Hautes Etudes Scientifiques, France;Distinguished Professor, University of Miami, Florida; Fields Metal (1998) & Crafoord Prize (2008)

Title Tropical K3 Surface

Host Laurant Jay and Palle Jorgensen

Room & Time Lecture Room I, Van Allen at 3:30 p.m.

Abstract Tropical geometry is a "limit" of the usual algebraic geometry in which algebraic varieties are replaced by real manifolds with integral affine structures. In a sense, coordinates in these real manifolds are logarithms of norms of coordinates in algebraic varieties defined over non-archimedean fields. The famous mirror symmetry from string theory is most naturally defined in the tropical context, as a duality between tropical Kaehler-Einstein manifolds.In my talk I will focus on the case of K3 surface studied several years ago by M.Gross and B.Siebert, Y.Soibelman and myself, and also discuss some modern development.One of corollaries of our considerations is a construction of a faithful action of an arithmetic subgroup of SO(1,18) by piecewise affine homeomorphisms of 2-dimensional sphere. The central role is played by certain formal group law for symplectic transformations in two dimensions, with remarkable conjectural integrality properties.This group recently reappears in the theory of wall-crossing phenomena in string theory and gauge theory, and in the mathematical counterpart for stabilities in derived non-commutative algebraic geometry.For example, the hyperkahler geometry of Seiberg -Witten integrable system is related to the counting of stable representations of the Kronecker quiver, i.e. pairs of operators form one vector space to another.The annual Crafoord Prize is a science prize established in 1980 and is at the level of the Nobel Prizes. It was awarded in 2008 jointly to Maxim Kontsevich and Edward Witten, Institute for Advanced Study, Princeton, NJ. The two were cited for their important contributions to mathematics inspired by modern theoretical physics. The 1989 recipient of the Crafoord Prize was the Iowa native, world renowned Professor James Van Allen.Maxim Kontsevich's work concentrates on geometric aspects of mathematical physics, most notably on knot theory, quantization, and mirror symmetry.

March 4-6, 2009

Speaker Prof. Marty Golubitsky (Ohio State University)

Titles Symmetry Breaking and Synchrony Breaking; Bifurcations and Dynamics on Networks; Geometric Visual Hallucinations

Host

Room & Time 3/4 at 3:30 in 113 MacLean Hall; 3/5 at 3:30 in 217 MacLean Hall; 3/6 at 1:30 in 61 Schaeffer Hall

Abstracts Symmetry Breaking and Synchrony Breaking: A coupled cell system is a network of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: Which part of the qualitative dynamics observed in coupled cells is the product of network architecture and which part depends on the specific equations? In our theory, local network symmetries replace symmetry as a way of organizing network dynamics, and synchrony breaking replaces symmetry breaking as a basic way in which transitions to complicated dynamics occur. Background on symmetry breaking and some of the more interesting examples will be presented.
Bifurcations and Dynamics on Networks: Identically coupled identical systems of differential equations lead to a very complex class of (synchrony-breaking) bifurcations. We describe some of the theory and some of the interesting resulting
examples. For example, synchrony-breaking Hopf bifurcation in feed-forward chains lead to periodic solutions whose amplitudes grow much more quickly than the amplitudes in standard Hopf bifurcation. This observation leads to a tangential discussion of models of the auditory system.
Geometric Visual Hallucinations: Kluver (circa 1940) proposed that drug-induced geometric visual hallucinations could be divided into four classes or "form constants." Ermentrout and Cowan (circa 1980) proposed that these form constants could result from symmetry-breaking pattern formation on the visual cortex; however only two of the four form constants were found. Bressloff and Cowan (circa 2000) proposed including more of the structure of the visual cortex (namely, orientation tuning and Hubel-Wiesel hypercolumns) in the analysis. This led to the need for new bifurcation theory results and ultimately to the four form constants.

	March 24-26, 2009 Ida Beam Distinguished Visiting Professor
Speaker	Prof. Fanghua Lin (Courant Institute, New York University)
Titles	Public Lecture: Roles of Differential Equations in Mathematics and Sciences: Reflections on Current Developments and Challenges; Nonlocal heat Flows and Partition Problems for Eigenvalues; Topological Vorticity and Conserved Geometric Motions
Host
Room & Time	3/24 at 7:00 p.m. - Lecture Room 1, Van Allen with reception from 6-7 p.m. outside of Room 1; 3/25 at 3:30 p.m. - TBA; 3/26 at 3:30 p.m. - TBA
Abstracts

Archives:

AY 2007-2008
AY 2006-2007
AY 2005-2006

	March 2, 2009
Speaker	Maxim Kontsevich, Institut des Hautes Etudes Scientifiques, France;Distinguished Professor, University of Miami, Florida; Fields Metal (1998) & Crafoord Prize (2008)
Title	Tropical K3 Surface
Host	Laurant Jay and Palle Jorgensen
Room & Time	Lecture Room I, Van Allen at 3:30 p.m.
Abstract	Tropical geometry is a "limit" of the usual algebraic geometry in which algebraic varieties are replaced by real manifolds with integral affine structures. In a sense, coordinates in these real manifolds are logarithms of norms of coordinates in algebraic varieties defined over non-archimedean fields. The famous mirror symmetry from string theory is most naturally defined in the tropical context, as a duality between tropical Kaehler-Einstein manifolds.In my talk I will focus on the case of K3 surface studied several years ago by M.Gross and B.Siebert, Y.Soibelman and myself, and also discuss some modern development.One of corollaries of our considerations is a construction of a faithful action of an arithmetic subgroup of SO(1,18) by piecewise affine homeomorphisms of 2-dimensional sphere. The central role is played by certain formal group law for symplectic transformations in two dimensions, with remarkable conjectural integrality properties.This group recently reappears in the theory of wall-crossing phenomena in string theory and gauge theory, and in the mathematical counterpart for stabilities in derived non-commutative algebraic geometry.For example, the hyperkahler geometry of Seiberg -Witten integrable system is related to the counting of stable representations of the Kronecker quiver, i.e. pairs of operators form one vector space to another.The annual Crafoord Prize is a science prize established in 1980 and is at the level of the Nobel Prizes. It was awarded in 2008 jointly to Maxim Kontsevich and Edward Witten, Institute for Advanced Study, Princeton, NJ. The two were cited for their important contributions to mathematics inspired by modern theoretical physics. The 1989 recipient of the Crafoord Prize was the Iowa native, world renowned Professor James Van Allen.Maxim Kontsevich's work concentrates on geometric aspects of mathematical physics, most notably on knot theory, quantization, and mirror symmetry.

	March 4-6, 2009
Speaker	Prof. Marty Golubitsky (Ohio State University)
Titles	Symmetry Breaking and Synchrony Breaking; Bifurcations and Dynamics on Networks; Geometric Visual Hallucinations
Host
Room & Time	3/4 at 3:30 in 113 MacLean Hall; 3/5 at 3:30 in 217 MacLean Hall; 3/6 at 1:30 in 61 Schaeffer Hall
Abstracts	Symmetry Breaking and Synchrony Breaking: A coupled cell system is a network of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: Which part of the qualitative dynamics observed in coupled cells is the product of network architecture and which part depends on the specific equations? In our theory, local network symmetries replace symmetry as a way of organizing network dynamics, and synchrony breaking replaces symmetry breaking as a basic way in which transitions to complicated dynamics occur. Background on symmetry breaking and some of the more interesting examples will be presented. Bifurcations and Dynamics on Networks: Identically coupled identical systems of differential equations lead to a very complex class of (synchrony-breaking) bifurcations. We describe some of the theory and some of the interesting resulting examples. For example, synchrony-breaking Hopf bifurcation in feed-forward chains lead to periodic solutions whose amplitudes grow much more quickly than the amplitudes in standard Hopf bifurcation. This observation leads to a tangential discussion of models of the auditory system. Geometric Visual Hallucinations: Kluver (circa 1940) proposed that drug-induced geometric visual hallucinations could be divided into four classes or "form constants." Ermentrout and Cowan (circa 1980) proposed that these form constants could result from symmetry-breaking pattern formation on the visual cortex; however only two of the four form constants were found. Bressloff and Cowan (circa 2000) proposed including more of the structure of the visual cortex (namely, orientation tuning and Hubel-Wiesel hypercolumns) in the analysis. This led to the need for new bifurcation theory results and ultimately to the four form constants.