RESEARCH INTERESTS AND CONFERENCE PHOTOS

Tibor Bisztriczky

Faculty Professor and Professor Emeritus, University of Calgary, tbisztri@ucalgary.ca

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

RESEARCH INTERESTS

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

– development of classes of convex d-polytopes :self-duals , and those with facial structure determined by a total ordering of vertices.

– a study of the geometric and combinatorial properties of convex 4-polytopes.The specific properties of interest are the following :

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

CONFERENCE PHOTOS 1988-2019

AUBURN, AMS SPECIAL SESSION, DISCRETE AND CONVEX GEOMETRY, MARCH 2019

BERLIN, CONVEX AND DISCRETE GEOMETRY, JOERGSHOP, JUNE 2017

ATLANTA, AMS SPECIAL SESSION CONVEX & DISCRETE GEOMETRY, MARCH 2017

OAXACA, CMO-BIRS, GEOMETRY, OCTOBER 2016

BUDAPEST, DISCRETE GEOMETRY DAYS, JUNE 2016

BUDAPEST, DISCRETE GEOMETRY DAYS, JUNE 2016

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

BANFF(BIRS),COMBINATORIAL AND CONVEX GEOMETRY FEST,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 POLYTOPES, MAY 2005

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

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

WORKSHOP ON GEOMETRIC TOMOGRAPHY, ALICANTE , SPAIN, OCTOBER 2004

GAETA, COMBINATORICS CONFERENCE, MAY-JUNE 2000

SEATTLE, MEETING IN HONOUR OF VIC KLEE’S 75’TH BIRTHDAY, AUGUST 2000

HALIFAX, AFFFINE GEOMETRY CONFERENCE, MAY 1996

MSRI, CONVEX GEOMETRY& GEOMETRIC FUNCTIONAL ANALYSIS, MAY 1996

NATO-ASI, POLYTOPES: ABSTRACT, CONVEX AND COMPUTATIONAL, TORONTO,SEPT.1993

CATANIA, COMBINATORICS CONFERENCE, SEPTEMBER 1991

VIENNA, MATHEMATISCHE KOLLOQIUM, TECHNISCHE UNIVERSITAET, APRIL 1991

RAVELLO, COMBINATORICS CONFERENCE, MAY 1988