**Tibor Bisztriczky**

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:

– combinatorial constructions 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 2

^{d}vertices. In the last decade I. Talata introduced the concept of an

**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**

*edge-antipodal***P (any two vertices of P, that lie on the edge of P, are contained in distinct parallel facets of P).**

*strongly edge-antipodal**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 2

^{d}only if K is d-dimensional

*. 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*

**parallelotope****: 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?**

*Separation Problem*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**