Tibor Bisztriczky Homepage

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Tibor Bisztriczky

Faculty Professor and Professor Emeritus
Rm# 422 Mathematical Sciences Building 
2500 University Drive N.W. Calgary, AB T2N 1N4
Phone: (403)220-6312

B.Sc., 1970, McMaster University
M.Sc., 1971, McMaster University
Ph.D., 1974, University of Toronto

Associate Member, Renyi Institute, Hungarian Academy of Sciences,2011

Adjunct Professor of Mathematics & Statistics, York University,2012-2018

A list of my publications
My links page

 

 

Research Interests and Conference Photos

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Current research is in the fields of convex polytopes, and discrete geometry . Specific topics of interest include:

 

Edge-antipodality (A set V or, a polytope P with vertex set V, is antipodal if any two elements of V are antipodal. Antipodal d-polytopes have been studied extensively over the past fifty years, and it is known that any such P has at most 2d vertices. In the last decade I. Talata introduced the concept of an edge-antipodal P: any two vertices of P, that lie on an edge of P, are antipodal. It is known that edge-antipodal 3-polytopes are antipodal, and that for each d ≥ 4, there is an edge-antipodal P that is not antipodal. With K. Boroczky, we have began a program for the study (classification, and determining the maximum of the number of vertices) of edge-antipodal d-polytopes, d ≥ 4. Most progress has been achieved in the case that d = 4. There the focus is presently on strongly edge-antipodal P (any two vertices of P, that lie on the edge of P, are contained in distinct parallel facets of P).

 

Separation (One of the most famous conjectures in Discrete Geometry is attributed to H. Hadwiger, I.Z. Gohberg and A.S. Markus. It is that the smallest number h(K), of smaller homothetic copies of a compact convex set K in real d-space, d ≥ 2, with which it is possible to cover K, is at most 2d only if K is d-dimensional parallelotope. The G-M-H conjecture is confirmed for d = 2, open for d ≥ 3, and has a rich history with various equivalent formulations. The formulation of particular interest is due to K. Bezdek: if the origin is in the interior of K then h(K) is the smallest number of hyperplanes required to strictly separate the origin from any face of the polar K* of K. This formulation is particularly attractive in the case of polytopes and leads naturally to the following Separation Problem: Let P be a convex d-polytope. Determine the smallest number s(P) such that any facet of P is strictly separated from an arbitrary fixed interior point of P by one of s(P) hyperplanes. Again, most progress on the separation problem has been achieved in the case that d = 4 and the 4-polytope P is neighborly (any two vertices of P determine an edge of P). In particular, it is known that for certain classes of neighborly 4-polytopes P that s(P) ≤ 16. The big question now is if s(P) ≤ 16 for any neighborly 4-polytope?

 

Topics in Combinatorial Geometry: Erdos-Szekeres type theorems, Transversal properties of families of ovals in the plane, Edge-antipodal 4-polytopes and Triangulations of simple convex polygons.

 

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CONFERENCE PHOTOS 1988-2017

BERLIN, CONVEX & DISCRETE GEOMETRY, JOERGSHOP, JUNE, 2017

 

ATLANTA, AMS, DISCRETE GEOMETRY & CONVEXITY SPECIAL SESSION, JANUARY 2017

 

OAXACA, CMO-BIRS, GEOMETRY,COMBINATORICS&TOPOLOGY,OCTOBER,2016

 

BUDAPEST, DISCRETE GEOMETRY DAYS, JUNE 2016

 

BANFF (BIRS), DISCRETE GEOMETRY & SYMMETRY, FEB. 2015

 

HALIFAX, CMS, DISCRETE & COMBINATORIAL GEOMETRY SPECIAL SESSION,JUNE 2013

 

BANFF (BIRS), TRANSVERSAL THEORY & HELLY TYPE THEOREMS, OCT. 2012

 

SZEGED, DISCRETE & CONVEX GEOMETRY CONFERENCE, MAY 2012

 

RICHMOND, AMS, CONVEXITY & COMBINATORICS SPECIAL SESSION, NOV. 2010

 

KELOWNA, PRAIRIE DISCRETE MATH WORKSHOP, AUGUST 2009

 

SEVILLA, PHENOMINA IN HIGH DIMENSIONS CONFERENCE, JUNE 2008

 

BANFF (BIRS) & CALGARY, INTUITIVE GEOMETRY, SEPTEMBER 2007

 

BUDAPEST, GEOMETRY FEST, JUNE 2007

 

BANFF (BIRS), CONVEX GEOMETRY & APPLICATIONS, MARCH 2006

 

SAN ANTONIO, AMS, CONVEX & DISCRETE GEOMETRY SPECIAL SESSION, JANUARY 2006

 

BANFF (BIRS) & CALGARY, CONVEX AND ABSTRACT POLYOTPES, MAY 2005

 

PITTSBURGH, AMS, CONVEXITY & COMBINATORICS SPECIAL SESSION, NOVEMBER 2004

 

GAETA, COMBINATORICS CONFERENCE, MAY-JUNE 2000

 

HALIFAX, AFFFINE GEOMETRY CONFERENCE, MAY 1996

MSRI, CONVEX GEOMETRY& GEOMETRIC FUNCTIONAL ANALYSIS, MAY 1996

CATANIA, COMBINATORICS CONFERENCE, SEPTEMBER 1991

 

VIENNA, MATHEMATISCHE KOLLOQIUM, TECHNISCHE UNIVERSITAET, APRIL 1991

 

RAVELLO, COMBINATORICS CONFERENCE, MAY 1988